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from __future__ import annotations |
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from sympy.core.add import Add |
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from sympy.core.cache import cacheit |
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from sympy.core.expr import Expr |
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from sympy.core.function import DefinedFunction, ArgumentIndexError, PoleError, expand_mul |
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from sympy.core.logic import fuzzy_not, fuzzy_or, FuzzyBool, fuzzy_and |
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from sympy.core.mod import Mod |
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from sympy.core.numbers import Rational, pi, Integer, Float, equal_valued |
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from sympy.core.relational import Ne, Eq |
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from sympy.core.singleton import S |
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from sympy.core.symbol import Symbol, Dummy |
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from sympy.core.sympify import sympify |
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from sympy.functions.combinatorial.factorials import factorial, RisingFactorial |
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from sympy.functions.combinatorial.numbers import bernoulli, euler |
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from sympy.functions.elementary.complexes import arg as arg_f, im, re |
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from sympy.functions.elementary.exponential import log, exp |
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from sympy.functions.elementary.integers import floor |
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from sympy.functions.elementary.miscellaneous import sqrt, Min, Max |
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from sympy.functions.elementary.piecewise import Piecewise |
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from sympy.functions.elementary._trigonometric_special import ( |
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cos_table, ipartfrac, fermat_coords) |
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from sympy.logic.boolalg import And |
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from sympy.ntheory import factorint |
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from sympy.polys.specialpolys import symmetric_poly |
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from sympy.utilities.iterables import numbered_symbols |
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def _imaginary_unit_as_coefficient(arg): |
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""" Helper to extract symbolic coefficient for imaginary unit """ |
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if isinstance(arg, Float): |
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return None |
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else: |
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return arg.as_coefficient(S.ImaginaryUnit) |
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class TrigonometricFunction(DefinedFunction): |
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"""Base class for trigonometric functions. """ |
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unbranched = True |
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_singularities = (S.ComplexInfinity,) |
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def _eval_is_rational(self): |
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s = self.func(*self.args) |
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if s.func == self.func: |
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if s.args[0].is_rational and fuzzy_not(s.args[0].is_zero): |
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return False |
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else: |
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return s.is_rational |
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def _eval_is_algebraic(self): |
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s = self.func(*self.args) |
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if s.func == self.func: |
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if fuzzy_not(self.args[0].is_zero) and self.args[0].is_algebraic: |
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return False |
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pi_coeff = _pi_coeff(self.args[0]) |
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if pi_coeff is not None and pi_coeff.is_rational: |
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return True |
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else: |
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return s.is_algebraic |
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def _eval_expand_complex(self, deep=True, **hints): |
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re_part, im_part = self.as_real_imag(deep=deep, **hints) |
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return re_part + im_part*S.ImaginaryUnit |
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def _as_real_imag(self, deep=True, **hints): |
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if self.args[0].is_extended_real: |
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if deep: |
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hints['complex'] = False |
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return (self.args[0].expand(deep, **hints), S.Zero) |
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else: |
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return (self.args[0], S.Zero) |
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if deep: |
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re, im = self.args[0].expand(deep, **hints).as_real_imag() |
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else: |
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re, im = self.args[0].as_real_imag() |
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return (re, im) |
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def _period(self, general_period, symbol=None): |
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f = expand_mul(self.args[0]) |
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if symbol is None: |
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symbol = tuple(f.free_symbols)[0] |
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if not f.has(symbol): |
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return S.Zero |
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if f == symbol: |
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return general_period |
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if symbol in f.free_symbols: |
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if f.is_Mul: |
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g, h = f.as_independent(symbol) |
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if h == symbol: |
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return general_period/abs(g) |
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if f.is_Add: |
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a, h = f.as_independent(symbol) |
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g, h = h.as_independent(symbol, as_Add=False) |
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if h == symbol: |
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return general_period/abs(g) |
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raise NotImplementedError("Use the periodicity function instead.") |
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@cacheit |
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def _table2(): |
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return { |
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12: (3, 4), |
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20: (4, 5), |
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30: (5, 6), |
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15: (6, 10), |
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24: (6, 8), |
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40: (8, 10), |
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60: (20, 30), |
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120: (40, 60) |
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} |
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def _peeloff_pi(arg): |
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r""" |
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Split ARG into two parts, a "rest" and a multiple of $\pi$. |
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This assumes ARG to be an Add. |
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The multiple of $\pi$ returned in the second position is always a Rational. |
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Examples |
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======== |
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>>> from sympy.functions.elementary.trigonometric import _peeloff_pi |
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>>> from sympy import pi |
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>>> from sympy.abc import x, y |
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>>> _peeloff_pi(x + pi/2) |
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(x, 1/2) |
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>>> _peeloff_pi(x + 2*pi/3 + pi*y) |
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(x + pi*y + pi/6, 1/2) |
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""" |
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pi_coeff = S.Zero |
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rest_terms = [] |
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for a in Add.make_args(arg): |
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K = a.coeff(pi) |
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if K and K.is_rational: |
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pi_coeff += K |
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else: |
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rest_terms.append(a) |
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if pi_coeff is S.Zero: |
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return arg, S.Zero |
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m1 = (pi_coeff % S.Half) |
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m2 = pi_coeff - m1 |
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if m2.is_integer or ((2*m2).is_integer and m2.is_even is False): |
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return Add(*(rest_terms + [m1*pi])), m2 |
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return arg, S.Zero |
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def _pi_coeff(arg: Expr, cycles: int = 1) -> Expr | None: |
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r""" |
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When arg is a Number times $\pi$ (e.g. $3\pi/2$) then return the Number |
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normalized to be in the range $[0, 2]$, else `None`. |
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When an even multiple of $\pi$ is encountered, if it is multiplying |
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something with known parity then the multiple is returned as 0 otherwise |
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as 2. |
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Examples |
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======== |
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>>> from sympy.functions.elementary.trigonometric import _pi_coeff |
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>>> from sympy import pi, Dummy |
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>>> from sympy.abc import x |
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>>> _pi_coeff(3*x*pi) |
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3*x |
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>>> _pi_coeff(11*pi/7) |
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11/7 |
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>>> _pi_coeff(-11*pi/7) |
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3/7 |
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>>> _pi_coeff(4*pi) |
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0 |
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>>> _pi_coeff(5*pi) |
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1 |
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>>> _pi_coeff(5.0*pi) |
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1 |
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>>> _pi_coeff(5.5*pi) |
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3/2 |
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>>> _pi_coeff(2 + pi) |
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>>> _pi_coeff(2*Dummy(integer=True)*pi) |
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2 |
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>>> _pi_coeff(2*Dummy(even=True)*pi) |
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0 |
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""" |
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if arg is pi: |
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return S.One |
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elif not arg: |
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return S.Zero |
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elif arg.is_Mul: |
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cx = arg.coeff(pi) |
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if cx: |
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c, x = cx.as_coeff_Mul() |
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if c.is_Float: |
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f = abs(c) % 1 |
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if f != 0: |
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p = -int(round(log(f, 2).evalf())) |
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m = 2**p |
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cm = c*m |
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i = int(cm) |
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if equal_valued(i, cm): |
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c = Rational(i, m) |
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cx = c*x |
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else: |
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c = Rational(int(c)) |
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cx = c*x |
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if x.is_integer: |
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c2 = c % 2 |
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if c2 == 1: |
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return x |
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elif not c2: |
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if x.is_even is not None: |
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return S.Zero |
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return Integer(2) |
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else: |
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return c2*x |
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return cx |
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elif arg.is_zero: |
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return S.Zero |
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return None |
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class sin(TrigonometricFunction): |
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r""" |
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The sine function. |
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Returns the sine of x (measured in radians). |
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Explanation |
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=========== |
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This function will evaluate automatically in the |
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case $x/\pi$ is some rational number [4]_. For example, |
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if $x$ is a multiple of $\pi$, $\pi/2$, $\pi/3$, $\pi/4$, and $\pi/6$. |
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Examples |
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======== |
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>>> from sympy import sin, pi |
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>>> from sympy.abc import x |
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>>> sin(x**2).diff(x) |
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2*x*cos(x**2) |
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>>> sin(1).diff(x) |
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0 |
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>>> sin(pi) |
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0 |
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>>> sin(pi/2) |
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1 |
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>>> sin(pi/6) |
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1/2 |
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>>> sin(pi/12) |
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-sqrt(2)/4 + sqrt(6)/4 |
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See Also |
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======== |
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csc, cos, sec, tan, cot |
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asin, acsc, acos, asec, atan, acot, atan2 |
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References |
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========== |
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.. [1] https://en.wikipedia.org/wiki/Trigonometric_functions |
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.. [2] https://dlmf.nist.gov/4.14 |
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.. [3] https://functions.wolfram.com/ElementaryFunctions/Sin |
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.. [4] https://mathworld.wolfram.com/TrigonometryAngles.html |
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""" |
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def period(self, symbol=None): |
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return self._period(2*pi, symbol) |
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def fdiff(self, argindex=1): |
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if argindex == 1: |
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return cos(self.args[0]) |
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else: |
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raise ArgumentIndexError(self, argindex) |
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@classmethod |
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def eval(cls, arg): |
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from sympy.calculus.accumulationbounds import AccumBounds |
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from sympy.sets.setexpr import SetExpr |
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if arg.is_Number: |
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if arg is S.NaN: |
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return S.NaN |
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elif arg.is_zero: |
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return S.Zero |
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elif arg in (S.Infinity, S.NegativeInfinity): |
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return AccumBounds(-1, 1) |
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if arg is S.ComplexInfinity: |
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return S.NaN |
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if isinstance(arg, AccumBounds): |
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from sympy.sets.sets import FiniteSet |
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min, max = arg.min, arg.max |
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d = floor(min/(2*pi)) |
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if min is not S.NegativeInfinity: |
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min = min - d*2*pi |
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if max is not S.Infinity: |
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max = max - d*2*pi |
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if AccumBounds(min, max).intersection(FiniteSet(pi/2, pi*Rational(5, 2))) \ |
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is not S.EmptySet and \ |
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AccumBounds(min, max).intersection(FiniteSet(pi*Rational(3, 2), |
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pi*Rational(7, 2))) is not S.EmptySet: |
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return AccumBounds(-1, 1) |
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elif AccumBounds(min, max).intersection(FiniteSet(pi/2, pi*Rational(5, 2))) \ |
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is not S.EmptySet: |
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return AccumBounds(Min(sin(min), sin(max)), 1) |
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elif AccumBounds(min, max).intersection(FiniteSet(pi*Rational(3, 2), pi*Rational(8, 2))) \ |
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is not S.EmptySet: |
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return AccumBounds(-1, Max(sin(min), sin(max))) |
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else: |
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return AccumBounds(Min(sin(min), sin(max)), |
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Max(sin(min), sin(max))) |
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elif isinstance(arg, SetExpr): |
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return arg._eval_func(cls) |
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if arg.could_extract_minus_sign(): |
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return -cls(-arg) |
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i_coeff = _imaginary_unit_as_coefficient(arg) |
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if i_coeff is not None: |
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from sympy.functions.elementary.hyperbolic import sinh |
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return S.ImaginaryUnit*sinh(i_coeff) |
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pi_coeff = _pi_coeff(arg) |
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if pi_coeff is not None: |
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if pi_coeff.is_integer: |
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return S.Zero |
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if (2*pi_coeff).is_integer: |
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if pi_coeff.is_even is False: |
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return S.NegativeOne**(pi_coeff - S.Half) |
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if not pi_coeff.is_Rational: |
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narg = pi_coeff*pi |
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if narg != arg: |
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return cls(narg) |
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return None |
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if pi_coeff.is_Rational: |
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x = pi_coeff % 2 |
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if x > 1: |
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return -cls((x % 1)*pi) |
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if 2*x > 1: |
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return cls((1 - x)*pi) |
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narg = ((pi_coeff + Rational(3, 2)) % 2)*pi |
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result = cos(narg) |
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if not isinstance(result, cos): |
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return result |
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if pi_coeff*pi != arg: |
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return cls(pi_coeff*pi) |
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return None |
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if arg.is_Add: |
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x, m = _peeloff_pi(arg) |
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if m: |
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m = m*pi |
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return sin(m)*cos(x) + cos(m)*sin(x) |
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if arg.is_zero: |
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return S.Zero |
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if isinstance(arg, asin): |
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return arg.args[0] |
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if isinstance(arg, atan): |
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x = arg.args[0] |
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return x/sqrt(1 + x**2) |
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if isinstance(arg, atan2): |
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y, x = arg.args |
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return y/sqrt(x**2 + y**2) |
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if isinstance(arg, acos): |
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x = arg.args[0] |
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return sqrt(1 - x**2) |
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if isinstance(arg, acot): |
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x = arg.args[0] |
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return 1/(sqrt(1 + 1/x**2)*x) |
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if isinstance(arg, acsc): |
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x = arg.args[0] |
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return 1/x |
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if isinstance(arg, asec): |
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x = arg.args[0] |
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return sqrt(1 - 1/x**2) |
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@staticmethod |
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@cacheit |
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def taylor_term(n, x, *previous_terms): |
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if n < 0 or n % 2 == 0: |
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return S.Zero |
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else: |
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x = sympify(x) |
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if len(previous_terms) > 2: |
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p = previous_terms[-2] |
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return -p*x**2/(n*(n - 1)) |
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else: |
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return S.NegativeOne**(n//2)*x**n/factorial(n) |
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def _eval_nseries(self, x, n, logx, cdir=0): |
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arg = self.args[0] |
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if logx is not None: |
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arg = arg.subs(log(x), logx) |
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if arg.subs(x, 0).has(S.NaN, S.ComplexInfinity): |
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raise PoleError("Cannot expand %s around 0" % (self)) |
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return super()._eval_nseries(x, n=n, logx=logx, cdir=cdir) |
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def _eval_rewrite_as_exp(self, arg, **kwargs): |
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from sympy.functions.elementary.hyperbolic import HyperbolicFunction |
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I = S.ImaginaryUnit |
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if isinstance(arg, (TrigonometricFunction, HyperbolicFunction)): |
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arg = arg.func(arg.args[0]).rewrite(exp) |
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return (exp(arg*I) - exp(-arg*I))/(2*I) |
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def _eval_rewrite_as_Pow(self, arg, **kwargs): |
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if isinstance(arg, log): |
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I = S.ImaginaryUnit |
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x = arg.args[0] |
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return I*x**-I/2 - I*x**I /2 |
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def _eval_rewrite_as_cos(self, arg, **kwargs): |
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return cos(arg - pi/2, evaluate=False) |
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def _eval_rewrite_as_tan(self, arg, **kwargs): |
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tan_half = tan(S.Half*arg) |
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return 2*tan_half/(1 + tan_half**2) |
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def _eval_rewrite_as_sincos(self, arg, **kwargs): |
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return sin(arg)*cos(arg)/cos(arg) |
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def _eval_rewrite_as_cot(self, arg, **kwargs): |
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cot_half = cot(S.Half*arg) |
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return Piecewise((0, And(Eq(im(arg), 0), Eq(Mod(arg, pi), 0))), |
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(2*cot_half/(1 + cot_half**2), True)) |
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def _eval_rewrite_as_pow(self, arg, **kwargs): |
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return self.rewrite(cos, **kwargs).rewrite(pow, **kwargs) |
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def _eval_rewrite_as_sqrt(self, arg, **kwargs): |
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return self.rewrite(cos, **kwargs).rewrite(sqrt, **kwargs) |
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def _eval_rewrite_as_csc(self, arg, **kwargs): |
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return 1/csc(arg) |
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def _eval_rewrite_as_sec(self, arg, **kwargs): |
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return 1/sec(arg - pi/2, evaluate=False) |
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def _eval_rewrite_as_sinc(self, arg, **kwargs): |
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return arg*sinc(arg) |
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def _eval_rewrite_as_besselj(self, arg, **kwargs): |
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from sympy.functions.special.bessel import besselj |
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return sqrt(pi*arg/2)*besselj(S.Half, arg) |
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def _eval_conjugate(self): |
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return self.func(self.args[0].conjugate()) |
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def as_real_imag(self, deep=True, **hints): |
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from sympy.functions.elementary.hyperbolic import cosh, sinh |
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re, im = self._as_real_imag(deep=deep, **hints) |
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return (sin(re)*cosh(im), cos(re)*sinh(im)) |
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def _eval_expand_trig(self, **hints): |
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from sympy.functions.special.polynomials import chebyshevt, chebyshevu |
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arg = self.args[0] |
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x = None |
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if arg.is_Add: |
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|
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x, y = arg.as_two_terms() |
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sx = sin(x, evaluate=False)._eval_expand_trig() |
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sy = sin(y, evaluate=False)._eval_expand_trig() |
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cx = cos(x, evaluate=False)._eval_expand_trig() |
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cy = cos(y, evaluate=False)._eval_expand_trig() |
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return sx*cy + sy*cx |
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elif arg.is_Mul: |
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n, x = arg.as_coeff_Mul(rational=True) |
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if n.is_Integer: |
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if n.is_odd: |
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return S.NegativeOne**((n - 1)/2)*chebyshevt(n, sin(x)) |
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else: |
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return expand_mul(S.NegativeOne**(n/2 - 1)*cos(x)* |
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chebyshevu(n - 1, sin(x)), deep=False) |
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return sin(arg) |
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|
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def _eval_as_leading_term(self, x, logx, cdir): |
|
from sympy.calculus.accumulationbounds import AccumBounds |
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arg = self.args[0] |
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x0 = arg.subs(x, 0).cancel() |
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n = x0/pi |
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if n.is_integer: |
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lt = (arg - n*pi).as_leading_term(x) |
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return (S.NegativeOne**n)*lt |
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if x0 is S.ComplexInfinity: |
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x0 = arg.limit(x, 0, dir='-' if re(cdir).is_negative else '+') |
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if x0 in [S.Infinity, S.NegativeInfinity]: |
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return AccumBounds(-1, 1) |
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return self.func(x0) if x0.is_finite else self |
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|
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def _eval_is_extended_real(self): |
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if self.args[0].is_extended_real: |
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return True |
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|
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def _eval_is_finite(self): |
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arg = self.args[0] |
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if arg.is_extended_real: |
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return True |
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|
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def _eval_is_zero(self): |
|
rest, pi_mult = _peeloff_pi(self.args[0]) |
|
if rest.is_zero: |
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return pi_mult.is_integer |
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|
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def _eval_is_complex(self): |
|
if self.args[0].is_extended_real \ |
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or self.args[0].is_complex: |
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return True |
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|
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|
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class cos(TrigonometricFunction): |
|
""" |
|
The cosine function. |
|
|
|
Returns the cosine of x (measured in radians). |
|
|
|
Explanation |
|
=========== |
|
|
|
See :func:`sin` for notes about automatic evaluation. |
|
|
|
Examples |
|
======== |
|
|
|
>>> from sympy import cos, pi |
|
>>> from sympy.abc import x |
|
>>> cos(x**2).diff(x) |
|
-2*x*sin(x**2) |
|
>>> cos(1).diff(x) |
|
0 |
|
>>> cos(pi) |
|
-1 |
|
>>> cos(pi/2) |
|
0 |
|
>>> cos(2*pi/3) |
|
-1/2 |
|
>>> cos(pi/12) |
|
sqrt(2)/4 + sqrt(6)/4 |
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|
|
See Also |
|
======== |
|
|
|
sin, csc, sec, tan, cot |
|
asin, acsc, acos, asec, atan, acot, atan2 |
|
|
|
References |
|
========== |
|
|
|
.. [1] https://en.wikipedia.org/wiki/Trigonometric_functions |
|
.. [2] https://dlmf.nist.gov/4.14 |
|
.. [3] https://functions.wolfram.com/ElementaryFunctions/Cos |
|
|
|
""" |
|
|
|
def period(self, symbol=None): |
|
return self._period(2*pi, symbol) |
|
|
|
def fdiff(self, argindex=1): |
|
if argindex == 1: |
|
return -sin(self.args[0]) |
|
else: |
|
raise ArgumentIndexError(self, argindex) |
|
|
|
@classmethod |
|
def eval(cls, arg): |
|
from sympy.functions.special.polynomials import chebyshevt |
|
from sympy.calculus.accumulationbounds import AccumBounds |
|
from sympy.sets.setexpr import SetExpr |
|
if arg.is_Number: |
|
if arg is S.NaN: |
|
return S.NaN |
|
elif arg.is_zero: |
|
return S.One |
|
elif arg in (S.Infinity, S.NegativeInfinity): |
|
|
|
|
|
|
|
|
|
return AccumBounds(-1, 1) |
|
|
|
if arg is S.ComplexInfinity: |
|
return S.NaN |
|
|
|
if isinstance(arg, AccumBounds): |
|
return sin(arg + pi/2) |
|
elif isinstance(arg, SetExpr): |
|
return arg._eval_func(cls) |
|
|
|
if arg.is_extended_real and arg.is_finite is False: |
|
return AccumBounds(-1, 1) |
|
|
|
if arg.could_extract_minus_sign(): |
|
return cls(-arg) |
|
|
|
i_coeff = _imaginary_unit_as_coefficient(arg) |
|
if i_coeff is not None: |
|
from sympy.functions.elementary.hyperbolic import cosh |
|
return cosh(i_coeff) |
|
|
|
pi_coeff = _pi_coeff(arg) |
|
if pi_coeff is not None: |
|
if pi_coeff.is_integer: |
|
return (S.NegativeOne)**pi_coeff |
|
|
|
if (2*pi_coeff).is_integer: |
|
|
|
|
|
if pi_coeff.is_even is False: |
|
return S.Zero |
|
|
|
if not pi_coeff.is_Rational: |
|
narg = pi_coeff*pi |
|
if narg != arg: |
|
return cls(narg) |
|
return None |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
if pi_coeff.is_Rational: |
|
q = pi_coeff.q |
|
p = pi_coeff.p % (2*q) |
|
if p > q: |
|
narg = (pi_coeff - 1)*pi |
|
return -cls(narg) |
|
if 2*p > q: |
|
narg = (1 - pi_coeff)*pi |
|
return -cls(narg) |
|
|
|
|
|
|
|
|
|
|
|
table2 = _table2() |
|
if q in table2: |
|
a, b = table2[q] |
|
a, b = p*pi/a, p*pi/b |
|
nvala, nvalb = cls(a), cls(b) |
|
if None in (nvala, nvalb): |
|
return None |
|
return nvala*nvalb + cls(pi/2 - a)*cls(pi/2 - b) |
|
|
|
if q > 12: |
|
return None |
|
|
|
cst_table_some = { |
|
3: S.Half, |
|
5: (sqrt(5) + 1) / 4, |
|
} |
|
if q in cst_table_some: |
|
cts = cst_table_some[pi_coeff.q] |
|
return chebyshevt(pi_coeff.p, cts).expand() |
|
|
|
if 0 == q % 2: |
|
narg = (pi_coeff*2)*pi |
|
nval = cls(narg) |
|
if None == nval: |
|
return None |
|
x = (2*pi_coeff + 1)/2 |
|
sign_cos = (-1)**((-1 if x < 0 else 1)*int(abs(x))) |
|
return sign_cos*sqrt( (1 + nval)/2 ) |
|
return None |
|
|
|
if arg.is_Add: |
|
x, m = _peeloff_pi(arg) |
|
if m: |
|
m = m*pi |
|
return cos(m)*cos(x) - sin(m)*sin(x) |
|
|
|
if arg.is_zero: |
|
return S.One |
|
|
|
if isinstance(arg, acos): |
|
return arg.args[0] |
|
|
|
if isinstance(arg, atan): |
|
x = arg.args[0] |
|
return 1/sqrt(1 + x**2) |
|
|
|
if isinstance(arg, atan2): |
|
y, x = arg.args |
|
return x/sqrt(x**2 + y**2) |
|
|
|
if isinstance(arg, asin): |
|
x = arg.args[0] |
|
return sqrt(1 - x ** 2) |
|
|
|
if isinstance(arg, acot): |
|
x = arg.args[0] |
|
return 1/sqrt(1 + 1/x**2) |
|
|
|
if isinstance(arg, acsc): |
|
x = arg.args[0] |
|
return sqrt(1 - 1/x**2) |
|
|
|
if isinstance(arg, asec): |
|
x = arg.args[0] |
|
return 1/x |
|
|
|
@staticmethod |
|
@cacheit |
|
def taylor_term(n, x, *previous_terms): |
|
if n < 0 or n % 2 == 1: |
|
return S.Zero |
|
else: |
|
x = sympify(x) |
|
|
|
if len(previous_terms) > 2: |
|
p = previous_terms[-2] |
|
return -p*x**2/(n*(n - 1)) |
|
else: |
|
return S.NegativeOne**(n//2)*x**n/factorial(n) |
|
|
|
def _eval_nseries(self, x, n, logx, cdir=0): |
|
arg = self.args[0] |
|
if logx is not None: |
|
arg = arg.subs(log(x), logx) |
|
if arg.subs(x, 0).has(S.NaN, S.ComplexInfinity): |
|
raise PoleError("Cannot expand %s around 0" % (self)) |
|
return super()._eval_nseries(x, n=n, logx=logx, cdir=cdir) |
|
|
|
def _eval_rewrite_as_exp(self, arg, **kwargs): |
|
I = S.ImaginaryUnit |
|
from sympy.functions.elementary.hyperbolic import HyperbolicFunction |
|
if isinstance(arg, (TrigonometricFunction, HyperbolicFunction)): |
|
arg = arg.func(arg.args[0]).rewrite(exp, **kwargs) |
|
return (exp(arg*I) + exp(-arg*I))/2 |
|
|
|
def _eval_rewrite_as_Pow(self, arg, **kwargs): |
|
if isinstance(arg, log): |
|
I = S.ImaginaryUnit |
|
x = arg.args[0] |
|
return x**I/2 + x**-I/2 |
|
|
|
def _eval_rewrite_as_sin(self, arg, **kwargs): |
|
return sin(arg + pi/2, evaluate=False) |
|
|
|
def _eval_rewrite_as_tan(self, arg, **kwargs): |
|
tan_half = tan(S.Half*arg)**2 |
|
return (1 - tan_half)/(1 + tan_half) |
|
|
|
def _eval_rewrite_as_sincos(self, arg, **kwargs): |
|
return sin(arg)*cos(arg)/sin(arg) |
|
|
|
def _eval_rewrite_as_cot(self, arg, **kwargs): |
|
cot_half = cot(S.Half*arg)**2 |
|
return Piecewise((1, And(Eq(im(arg), 0), Eq(Mod(arg, 2*pi), 0))), |
|
((cot_half - 1)/(cot_half + 1), True)) |
|
|
|
def _eval_rewrite_as_pow(self, arg, **kwargs): |
|
return self._eval_rewrite_as_sqrt(arg, **kwargs) |
|
|
|
def _eval_rewrite_as_sqrt(self, arg: Expr, **kwargs): |
|
from sympy.functions.special.polynomials import chebyshevt |
|
|
|
pi_coeff = _pi_coeff(arg) |
|
if pi_coeff is None: |
|
return None |
|
|
|
if isinstance(pi_coeff, Integer): |
|
return None |
|
|
|
if not isinstance(pi_coeff, Rational): |
|
return None |
|
|
|
cst_table_some = cos_table() |
|
|
|
if pi_coeff.q in cst_table_some: |
|
rv = chebyshevt(pi_coeff.p, cst_table_some[pi_coeff.q]()) |
|
if pi_coeff.q < 257: |
|
rv = rv.expand() |
|
return rv |
|
|
|
if not pi_coeff.q % 2: |
|
pico2 = pi_coeff * 2 |
|
nval = cos(pico2 * pi).rewrite(sqrt, **kwargs) |
|
x = (pico2 + 1) / 2 |
|
sign_cos = -1 if int(x) % 2 else 1 |
|
return sign_cos * sqrt((1 + nval) / 2) |
|
|
|
FC = fermat_coords(pi_coeff.q) |
|
if FC: |
|
denoms = FC |
|
else: |
|
denoms = [b**e for b, e in factorint(pi_coeff.q).items()] |
|
|
|
apart = ipartfrac(*denoms) |
|
decomp = (pi_coeff.p * Rational(n, d) for n, d in zip(apart, denoms)) |
|
X = [(x[1], x[0]*pi) for x in zip(decomp, numbered_symbols('z'))] |
|
pcls = cos(sum(x[0] for x in X))._eval_expand_trig().subs(X) |
|
|
|
if not FC or len(FC) == 1: |
|
return pcls |
|
return pcls.rewrite(sqrt, **kwargs) |
|
|
|
def _eval_rewrite_as_sec(self, arg, **kwargs): |
|
return 1/sec(arg) |
|
|
|
def _eval_rewrite_as_csc(self, arg, **kwargs): |
|
return 1/sec(arg).rewrite(csc, **kwargs) |
|
|
|
def _eval_rewrite_as_besselj(self, arg, **kwargs): |
|
from sympy.functions.special.bessel import besselj |
|
return Piecewise( |
|
(sqrt(pi*arg/2)*besselj(-S.Half, arg), Ne(arg, 0)), |
|
(1, True) |
|
) |
|
|
|
def _eval_conjugate(self): |
|
return self.func(self.args[0].conjugate()) |
|
|
|
def as_real_imag(self, deep=True, **hints): |
|
from sympy.functions.elementary.hyperbolic import cosh, sinh |
|
re, im = self._as_real_imag(deep=deep, **hints) |
|
return (cos(re)*cosh(im), -sin(re)*sinh(im)) |
|
|
|
def _eval_expand_trig(self, **hints): |
|
from sympy.functions.special.polynomials import chebyshevt |
|
arg = self.args[0] |
|
x = None |
|
if arg.is_Add: |
|
x, y = arg.as_two_terms() |
|
sx = sin(x, evaluate=False)._eval_expand_trig() |
|
sy = sin(y, evaluate=False)._eval_expand_trig() |
|
cx = cos(x, evaluate=False)._eval_expand_trig() |
|
cy = cos(y, evaluate=False)._eval_expand_trig() |
|
return cx*cy - sx*sy |
|
elif arg.is_Mul: |
|
coeff, terms = arg.as_coeff_Mul(rational=True) |
|
if coeff.is_Integer: |
|
return chebyshevt(coeff, cos(terms)) |
|
return cos(arg) |
|
|
|
def _eval_as_leading_term(self, x, logx, cdir): |
|
from sympy.calculus.accumulationbounds import AccumBounds |
|
arg = self.args[0] |
|
x0 = arg.subs(x, 0).cancel() |
|
n = (x0 + pi/2)/pi |
|
if n.is_integer: |
|
lt = (arg - n*pi + pi/2).as_leading_term(x) |
|
return (S.NegativeOne**n)*lt |
|
if x0 is S.ComplexInfinity: |
|
x0 = arg.limit(x, 0, dir='-' if re(cdir).is_negative else '+') |
|
if x0 in [S.Infinity, S.NegativeInfinity]: |
|
return AccumBounds(-1, 1) |
|
return self.func(x0) if x0.is_finite else self |
|
|
|
def _eval_is_extended_real(self): |
|
if self.args[0].is_extended_real: |
|
return True |
|
|
|
def _eval_is_finite(self): |
|
arg = self.args[0] |
|
|
|
if arg.is_extended_real: |
|
return True |
|
|
|
def _eval_is_complex(self): |
|
if self.args[0].is_extended_real \ |
|
or self.args[0].is_complex: |
|
return True |
|
|
|
def _eval_is_zero(self): |
|
rest, pi_mult = _peeloff_pi(self.args[0]) |
|
if rest.is_zero and pi_mult: |
|
return (pi_mult - S.Half).is_integer |
|
|
|
|
|
class tan(TrigonometricFunction): |
|
""" |
|
The tangent function. |
|
|
|
Returns the tangent of x (measured in radians). |
|
|
|
Explanation |
|
=========== |
|
|
|
See :class:`sin` for notes about automatic evaluation. |
|
|
|
Examples |
|
======== |
|
|
|
>>> from sympy import tan, pi |
|
>>> from sympy.abc import x |
|
>>> tan(x**2).diff(x) |
|
2*x*(tan(x**2)**2 + 1) |
|
>>> tan(1).diff(x) |
|
0 |
|
>>> tan(pi/8).expand() |
|
-1 + sqrt(2) |
|
|
|
See Also |
|
======== |
|
|
|
sin, csc, cos, sec, cot |
|
asin, acsc, acos, asec, atan, acot, atan2 |
|
|
|
References |
|
========== |
|
|
|
.. [1] https://en.wikipedia.org/wiki/Trigonometric_functions |
|
.. [2] https://dlmf.nist.gov/4.14 |
|
.. [3] https://functions.wolfram.com/ElementaryFunctions/Tan |
|
|
|
""" |
|
|
|
def period(self, symbol=None): |
|
return self._period(pi, symbol) |
|
|
|
def fdiff(self, argindex=1): |
|
if argindex == 1: |
|
return S.One + self**2 |
|
else: |
|
raise ArgumentIndexError(self, argindex) |
|
|
|
def inverse(self, argindex=1): |
|
""" |
|
Returns the inverse of this function. |
|
""" |
|
return atan |
|
|
|
@classmethod |
|
def eval(cls, arg): |
|
from sympy.calculus.accumulationbounds import AccumBounds |
|
if arg.is_Number: |
|
if arg is S.NaN: |
|
return S.NaN |
|
elif arg.is_zero: |
|
return S.Zero |
|
elif arg in (S.Infinity, S.NegativeInfinity): |
|
return AccumBounds(S.NegativeInfinity, S.Infinity) |
|
|
|
if arg is S.ComplexInfinity: |
|
return S.NaN |
|
|
|
if isinstance(arg, AccumBounds): |
|
min, max = arg.min, arg.max |
|
d = floor(min/pi) |
|
if min is not S.NegativeInfinity: |
|
min = min - d*pi |
|
if max is not S.Infinity: |
|
max = max - d*pi |
|
from sympy.sets.sets import FiniteSet |
|
if AccumBounds(min, max).intersection(FiniteSet(pi/2, pi*Rational(3, 2))): |
|
return AccumBounds(S.NegativeInfinity, S.Infinity) |
|
else: |
|
return AccumBounds(tan(min), tan(max)) |
|
|
|
if arg.could_extract_minus_sign(): |
|
return -cls(-arg) |
|
|
|
i_coeff = _imaginary_unit_as_coefficient(arg) |
|
if i_coeff is not None: |
|
from sympy.functions.elementary.hyperbolic import tanh |
|
return S.ImaginaryUnit*tanh(i_coeff) |
|
|
|
pi_coeff = _pi_coeff(arg, 2) |
|
if pi_coeff is not None: |
|
if pi_coeff.is_integer: |
|
return S.Zero |
|
|
|
if not pi_coeff.is_Rational: |
|
narg = pi_coeff*pi |
|
if narg != arg: |
|
return cls(narg) |
|
return None |
|
|
|
if pi_coeff.is_Rational: |
|
q = pi_coeff.q |
|
p = pi_coeff.p % q |
|
|
|
table10 = { |
|
1: sqrt(1 - 2*sqrt(5)/5), |
|
2: sqrt(5 - 2*sqrt(5)), |
|
3: sqrt(1 + 2*sqrt(5)/5), |
|
4: sqrt(5 + 2*sqrt(5)) |
|
} |
|
if q in (5, 10): |
|
n = 10*p/q |
|
if n > 5: |
|
n = 10 - n |
|
return -table10[n] |
|
else: |
|
return table10[n] |
|
if not pi_coeff.q % 2: |
|
narg = pi_coeff*pi*2 |
|
cresult, sresult = cos(narg), cos(narg - pi/2) |
|
if not isinstance(cresult, cos) \ |
|
and not isinstance(sresult, cos): |
|
if sresult == 0: |
|
return S.ComplexInfinity |
|
return 1/sresult - cresult/sresult |
|
|
|
table2 = _table2() |
|
if q in table2: |
|
a, b = table2[q] |
|
nvala, nvalb = cls(p*pi/a), cls(p*pi/b) |
|
if None in (nvala, nvalb): |
|
return None |
|
return (nvala - nvalb)/(1 + nvala*nvalb) |
|
narg = ((pi_coeff + S.Half) % 1 - S.Half)*pi |
|
|
|
|
|
cresult, sresult = cos(narg), cos(narg - pi/2) |
|
if not isinstance(cresult, cos) \ |
|
and not isinstance(sresult, cos): |
|
if cresult == 0: |
|
return S.ComplexInfinity |
|
return (sresult/cresult) |
|
if narg != arg: |
|
return cls(narg) |
|
|
|
if arg.is_Add: |
|
x, m = _peeloff_pi(arg) |
|
if m: |
|
tanm = tan(m*pi) |
|
if tanm is S.ComplexInfinity: |
|
return -cot(x) |
|
else: |
|
return tan(x) |
|
|
|
if arg.is_zero: |
|
return S.Zero |
|
|
|
if isinstance(arg, atan): |
|
return arg.args[0] |
|
|
|
if isinstance(arg, atan2): |
|
y, x = arg.args |
|
return y/x |
|
|
|
if isinstance(arg, asin): |
|
x = arg.args[0] |
|
return x/sqrt(1 - x**2) |
|
|
|
if isinstance(arg, acos): |
|
x = arg.args[0] |
|
return sqrt(1 - x**2)/x |
|
|
|
if isinstance(arg, acot): |
|
x = arg.args[0] |
|
return 1/x |
|
|
|
if isinstance(arg, acsc): |
|
x = arg.args[0] |
|
return 1/(sqrt(1 - 1/x**2)*x) |
|
|
|
if isinstance(arg, asec): |
|
x = arg.args[0] |
|
return sqrt(1 - 1/x**2)*x |
|
|
|
@staticmethod |
|
@cacheit |
|
def taylor_term(n, x, *previous_terms): |
|
if n < 0 or n % 2 == 0: |
|
return S.Zero |
|
else: |
|
x = sympify(x) |
|
|
|
a, b = ((n - 1)//2), 2**(n + 1) |
|
|
|
B = bernoulli(n + 1) |
|
F = factorial(n + 1) |
|
|
|
return S.NegativeOne**a*b*(b - 1)*B/F*x**n |
|
|
|
def _eval_nseries(self, x, n, logx, cdir=0): |
|
i = self.args[0].limit(x, 0)*2/pi |
|
if i and i.is_Integer: |
|
return self.rewrite(cos)._eval_nseries(x, n=n, logx=logx) |
|
return super()._eval_nseries(x, n=n, logx=logx) |
|
|
|
def _eval_rewrite_as_Pow(self, arg, **kwargs): |
|
if isinstance(arg, log): |
|
I = S.ImaginaryUnit |
|
x = arg.args[0] |
|
return I*(x**-I - x**I)/(x**-I + x**I) |
|
|
|
def _eval_conjugate(self): |
|
return self.func(self.args[0].conjugate()) |
|
|
|
def as_real_imag(self, deep=True, **hints): |
|
re, im = self._as_real_imag(deep=deep, **hints) |
|
if im: |
|
from sympy.functions.elementary.hyperbolic import cosh, sinh |
|
denom = cos(2*re) + cosh(2*im) |
|
return (sin(2*re)/denom, sinh(2*im)/denom) |
|
else: |
|
return (self.func(re), S.Zero) |
|
|
|
def _eval_expand_trig(self, **hints): |
|
arg = self.args[0] |
|
x = None |
|
if arg.is_Add: |
|
n = len(arg.args) |
|
TX = [] |
|
for x in arg.args: |
|
tx = tan(x, evaluate=False)._eval_expand_trig() |
|
TX.append(tx) |
|
|
|
Yg = numbered_symbols('Y') |
|
Y = [ next(Yg) for i in range(n) ] |
|
|
|
p = [0, 0] |
|
for i in range(n + 1): |
|
p[1 - i % 2] += symmetric_poly(i, Y)*(-1)**((i % 4)//2) |
|
return (p[0]/p[1]).subs(list(zip(Y, TX))) |
|
|
|
elif arg.is_Mul: |
|
coeff, terms = arg.as_coeff_Mul(rational=True) |
|
if coeff.is_Integer and coeff > 1: |
|
I = S.ImaginaryUnit |
|
z = Symbol('dummy', real=True) |
|
P = ((1 + I*z)**coeff).expand() |
|
return (im(P)/re(P)).subs([(z, tan(terms))]) |
|
return tan(arg) |
|
|
|
def _eval_rewrite_as_exp(self, arg, **kwargs): |
|
I = S.ImaginaryUnit |
|
from sympy.functions.elementary.hyperbolic import HyperbolicFunction |
|
if isinstance(arg, (TrigonometricFunction, HyperbolicFunction)): |
|
arg = arg.func(arg.args[0]).rewrite(exp) |
|
neg_exp, pos_exp = exp(-arg*I), exp(arg*I) |
|
return I*(neg_exp - pos_exp)/(neg_exp + pos_exp) |
|
|
|
def _eval_rewrite_as_sin(self, x, **kwargs): |
|
return 2*sin(x)**2/sin(2*x) |
|
|
|
def _eval_rewrite_as_cos(self, x, **kwargs): |
|
return cos(x - pi/2, evaluate=False)/cos(x) |
|
|
|
def _eval_rewrite_as_sincos(self, arg, **kwargs): |
|
return sin(arg)/cos(arg) |
|
|
|
def _eval_rewrite_as_cot(self, arg, **kwargs): |
|
return 1/cot(arg) |
|
|
|
def _eval_rewrite_as_sec(self, arg, **kwargs): |
|
sin_in_sec_form = sin(arg).rewrite(sec, **kwargs) |
|
cos_in_sec_form = cos(arg).rewrite(sec, **kwargs) |
|
return sin_in_sec_form/cos_in_sec_form |
|
|
|
def _eval_rewrite_as_csc(self, arg, **kwargs): |
|
sin_in_csc_form = sin(arg).rewrite(csc, **kwargs) |
|
cos_in_csc_form = cos(arg).rewrite(csc, **kwargs) |
|
return sin_in_csc_form/cos_in_csc_form |
|
|
|
def _eval_rewrite_as_pow(self, arg, **kwargs): |
|
y = self.rewrite(cos, **kwargs).rewrite(pow, **kwargs) |
|
if y.has(cos): |
|
return None |
|
return y |
|
|
|
def _eval_rewrite_as_sqrt(self, arg, **kwargs): |
|
y = self.rewrite(cos, **kwargs).rewrite(sqrt, **kwargs) |
|
if y.has(cos): |
|
return None |
|
return y |
|
|
|
def _eval_rewrite_as_besselj(self, arg, **kwargs): |
|
from sympy.functions.special.bessel import besselj |
|
return besselj(S.Half, arg)/besselj(-S.Half, arg) |
|
|
|
def _eval_as_leading_term(self, x, logx, cdir): |
|
from sympy.calculus.accumulationbounds import AccumBounds |
|
from sympy.functions.elementary.complexes import re |
|
arg = self.args[0] |
|
x0 = arg.subs(x, 0).cancel() |
|
n = 2*x0/pi |
|
if n.is_integer: |
|
lt = (arg - n*pi/2).as_leading_term(x) |
|
return lt if n.is_even else -1/lt |
|
if x0 is S.ComplexInfinity: |
|
x0 = arg.limit(x, 0, dir='-' if re(cdir).is_negative else '+') |
|
if x0 in (S.Infinity, S.NegativeInfinity): |
|
return AccumBounds(S.NegativeInfinity, S.Infinity) |
|
return self.func(x0) if x0.is_finite else self |
|
|
|
def _eval_is_extended_real(self): |
|
|
|
return self.args[0].is_extended_real |
|
|
|
def _eval_is_real(self): |
|
arg = self.args[0] |
|
if arg.is_real and (arg/pi - S.Half).is_integer is False: |
|
return True |
|
|
|
def _eval_is_finite(self): |
|
arg = self.args[0] |
|
|
|
if arg.is_real and (arg/pi - S.Half).is_integer is False: |
|
return True |
|
|
|
if arg.is_imaginary: |
|
return True |
|
|
|
def _eval_is_zero(self): |
|
rest, pi_mult = _peeloff_pi(self.args[0]) |
|
if rest.is_zero: |
|
return pi_mult.is_integer |
|
|
|
def _eval_is_complex(self): |
|
arg = self.args[0] |
|
|
|
if arg.is_real and (arg/pi - S.Half).is_integer is False: |
|
return True |
|
|
|
|
|
class cot(TrigonometricFunction): |
|
""" |
|
The cotangent function. |
|
|
|
Returns the cotangent of x (measured in radians). |
|
|
|
Explanation |
|
=========== |
|
|
|
See :class:`sin` for notes about automatic evaluation. |
|
|
|
Examples |
|
======== |
|
|
|
>>> from sympy import cot, pi |
|
>>> from sympy.abc import x |
|
>>> cot(x**2).diff(x) |
|
2*x*(-cot(x**2)**2 - 1) |
|
>>> cot(1).diff(x) |
|
0 |
|
>>> cot(pi/12) |
|
sqrt(3) + 2 |
|
|
|
See Also |
|
======== |
|
|
|
sin, csc, cos, sec, tan |
|
asin, acsc, acos, asec, atan, acot, atan2 |
|
|
|
References |
|
========== |
|
|
|
.. [1] https://en.wikipedia.org/wiki/Trigonometric_functions |
|
.. [2] https://dlmf.nist.gov/4.14 |
|
.. [3] https://functions.wolfram.com/ElementaryFunctions/Cot |
|
|
|
""" |
|
|
|
def period(self, symbol=None): |
|
return self._period(pi, symbol) |
|
|
|
def fdiff(self, argindex=1): |
|
if argindex == 1: |
|
return S.NegativeOne - self**2 |
|
else: |
|
raise ArgumentIndexError(self, argindex) |
|
|
|
def inverse(self, argindex=1): |
|
""" |
|
Returns the inverse of this function. |
|
""" |
|
return acot |
|
|
|
@classmethod |
|
def eval(cls, arg): |
|
from sympy.calculus.accumulationbounds import AccumBounds |
|
if arg.is_Number: |
|
if arg is S.NaN: |
|
return S.NaN |
|
if arg.is_zero: |
|
return S.ComplexInfinity |
|
elif arg in (S.Infinity, S.NegativeInfinity): |
|
return AccumBounds(S.NegativeInfinity, S.Infinity) |
|
|
|
if arg is S.ComplexInfinity: |
|
return S.NaN |
|
|
|
if isinstance(arg, AccumBounds): |
|
return -tan(arg + pi/2) |
|
|
|
if arg.could_extract_minus_sign(): |
|
return -cls(-arg) |
|
|
|
i_coeff = _imaginary_unit_as_coefficient(arg) |
|
if i_coeff is not None: |
|
from sympy.functions.elementary.hyperbolic import coth |
|
return -S.ImaginaryUnit*coth(i_coeff) |
|
|
|
pi_coeff = _pi_coeff(arg, 2) |
|
if pi_coeff is not None: |
|
if pi_coeff.is_integer: |
|
return S.ComplexInfinity |
|
|
|
if not pi_coeff.is_Rational: |
|
narg = pi_coeff*pi |
|
if narg != arg: |
|
return cls(narg) |
|
return None |
|
|
|
if pi_coeff.is_Rational: |
|
if pi_coeff.q in (5, 10): |
|
return tan(pi/2 - arg) |
|
if pi_coeff.q > 2 and not pi_coeff.q % 2: |
|
narg = pi_coeff*pi*2 |
|
cresult, sresult = cos(narg), cos(narg - pi/2) |
|
if not isinstance(cresult, cos) \ |
|
and not isinstance(sresult, cos): |
|
return 1/sresult + cresult/sresult |
|
q = pi_coeff.q |
|
p = pi_coeff.p % q |
|
table2 = _table2() |
|
if q in table2: |
|
a, b = table2[q] |
|
nvala, nvalb = cls(p*pi/a), cls(p*pi/b) |
|
if None in (nvala, nvalb): |
|
return None |
|
return (1 + nvala*nvalb)/(nvalb - nvala) |
|
narg = (((pi_coeff + S.Half) % 1) - S.Half)*pi |
|
|
|
|
|
cresult, sresult = cos(narg), cos(narg - pi/2) |
|
if not isinstance(cresult, cos) \ |
|
and not isinstance(sresult, cos): |
|
if sresult == 0: |
|
return S.ComplexInfinity |
|
return cresult/sresult |
|
if narg != arg: |
|
return cls(narg) |
|
|
|
if arg.is_Add: |
|
x, m = _peeloff_pi(arg) |
|
if m: |
|
cotm = cot(m*pi) |
|
if cotm is S.ComplexInfinity: |
|
return cot(x) |
|
else: |
|
return -tan(x) |
|
|
|
if arg.is_zero: |
|
return S.ComplexInfinity |
|
|
|
if isinstance(arg, acot): |
|
return arg.args[0] |
|
|
|
if isinstance(arg, atan): |
|
x = arg.args[0] |
|
return 1/x |
|
|
|
if isinstance(arg, atan2): |
|
y, x = arg.args |
|
return x/y |
|
|
|
if isinstance(arg, asin): |
|
x = arg.args[0] |
|
return sqrt(1 - x**2)/x |
|
|
|
if isinstance(arg, acos): |
|
x = arg.args[0] |
|
return x/sqrt(1 - x**2) |
|
|
|
if isinstance(arg, acsc): |
|
x = arg.args[0] |
|
return sqrt(1 - 1/x**2)*x |
|
|
|
if isinstance(arg, asec): |
|
x = arg.args[0] |
|
return 1/(sqrt(1 - 1/x**2)*x) |
|
|
|
@staticmethod |
|
@cacheit |
|
def taylor_term(n, x, *previous_terms): |
|
if n == 0: |
|
return 1/sympify(x) |
|
elif n < 0 or n % 2 == 0: |
|
return S.Zero |
|
else: |
|
x = sympify(x) |
|
|
|
B = bernoulli(n + 1) |
|
F = factorial(n + 1) |
|
|
|
return S.NegativeOne**((n + 1)//2)*2**(n + 1)*B/F*x**n |
|
|
|
def _eval_nseries(self, x, n, logx, cdir=0): |
|
i = self.args[0].limit(x, 0)/pi |
|
if i and i.is_Integer: |
|
return self.rewrite(cos)._eval_nseries(x, n=n, logx=logx) |
|
return self.rewrite(tan)._eval_nseries(x, n=n, logx=logx) |
|
|
|
def _eval_conjugate(self): |
|
return self.func(self.args[0].conjugate()) |
|
|
|
def as_real_imag(self, deep=True, **hints): |
|
re, im = self._as_real_imag(deep=deep, **hints) |
|
if im: |
|
from sympy.functions.elementary.hyperbolic import cosh, sinh |
|
denom = cos(2*re) - cosh(2*im) |
|
return (-sin(2*re)/denom, sinh(2*im)/denom) |
|
else: |
|
return (self.func(re), S.Zero) |
|
|
|
def _eval_rewrite_as_exp(self, arg, **kwargs): |
|
from sympy.functions.elementary.hyperbolic import HyperbolicFunction |
|
I = S.ImaginaryUnit |
|
if isinstance(arg, (TrigonometricFunction, HyperbolicFunction)): |
|
arg = arg.func(arg.args[0]).rewrite(exp, **kwargs) |
|
neg_exp, pos_exp = exp(-arg*I), exp(arg*I) |
|
return I*(pos_exp + neg_exp)/(pos_exp - neg_exp) |
|
|
|
def _eval_rewrite_as_Pow(self, arg, **kwargs): |
|
if isinstance(arg, log): |
|
I = S.ImaginaryUnit |
|
x = arg.args[0] |
|
return -I*(x**-I + x**I)/(x**-I - x**I) |
|
|
|
def _eval_rewrite_as_sin(self, x, **kwargs): |
|
return sin(2*x)/(2*(sin(x)**2)) |
|
|
|
def _eval_rewrite_as_cos(self, x, **kwargs): |
|
return cos(x)/cos(x - pi/2, evaluate=False) |
|
|
|
def _eval_rewrite_as_sincos(self, arg, **kwargs): |
|
return cos(arg)/sin(arg) |
|
|
|
def _eval_rewrite_as_tan(self, arg, **kwargs): |
|
return 1/tan(arg) |
|
|
|
def _eval_rewrite_as_sec(self, arg, **kwargs): |
|
cos_in_sec_form = cos(arg).rewrite(sec, **kwargs) |
|
sin_in_sec_form = sin(arg).rewrite(sec, **kwargs) |
|
return cos_in_sec_form/sin_in_sec_form |
|
|
|
def _eval_rewrite_as_csc(self, arg, **kwargs): |
|
cos_in_csc_form = cos(arg).rewrite(csc, **kwargs) |
|
sin_in_csc_form = sin(arg).rewrite(csc, **kwargs) |
|
return cos_in_csc_form/sin_in_csc_form |
|
|
|
def _eval_rewrite_as_pow(self, arg, **kwargs): |
|
y = self.rewrite(cos, **kwargs).rewrite(pow, **kwargs) |
|
if y.has(cos): |
|
return None |
|
return y |
|
|
|
def _eval_rewrite_as_sqrt(self, arg, **kwargs): |
|
y = self.rewrite(cos, **kwargs).rewrite(sqrt, **kwargs) |
|
if y.has(cos): |
|
return None |
|
return y |
|
|
|
def _eval_rewrite_as_besselj(self, arg, **kwargs): |
|
from sympy.functions.special.bessel import besselj |
|
return besselj(-S.Half, arg)/besselj(S.Half, arg) |
|
|
|
def _eval_as_leading_term(self, x, logx, cdir): |
|
from sympy.calculus.accumulationbounds import AccumBounds |
|
from sympy.functions.elementary.complexes import re |
|
arg = self.args[0] |
|
x0 = arg.subs(x, 0).cancel() |
|
n = 2*x0/pi |
|
if n.is_integer: |
|
lt = (arg - n*pi/2).as_leading_term(x) |
|
return 1/lt if n.is_even else -lt |
|
if x0 is S.ComplexInfinity: |
|
x0 = arg.limit(x, 0, dir='-' if re(cdir).is_negative else '+') |
|
if x0 in (S.Infinity, S.NegativeInfinity): |
|
return AccumBounds(S.NegativeInfinity, S.Infinity) |
|
return self.func(x0) if x0.is_finite else self |
|
|
|
def _eval_is_extended_real(self): |
|
return self.args[0].is_extended_real |
|
|
|
def _eval_expand_trig(self, **hints): |
|
arg = self.args[0] |
|
x = None |
|
if arg.is_Add: |
|
n = len(arg.args) |
|
CX = [] |
|
for x in arg.args: |
|
cx = cot(x, evaluate=False)._eval_expand_trig() |
|
CX.append(cx) |
|
|
|
Yg = numbered_symbols('Y') |
|
Y = [ next(Yg) for i in range(n) ] |
|
|
|
p = [0, 0] |
|
for i in range(n, -1, -1): |
|
p[(n - i) % 2] += symmetric_poly(i, Y)*(-1)**(((n - i) % 4)//2) |
|
return (p[0]/p[1]).subs(list(zip(Y, CX))) |
|
elif arg.is_Mul: |
|
coeff, terms = arg.as_coeff_Mul(rational=True) |
|
if coeff.is_Integer and coeff > 1: |
|
I = S.ImaginaryUnit |
|
z = Symbol('dummy', real=True) |
|
P = ((z + I)**coeff).expand() |
|
return (re(P)/im(P)).subs([(z, cot(terms))]) |
|
return cot(arg) |
|
|
|
def _eval_is_finite(self): |
|
arg = self.args[0] |
|
if arg.is_real and (arg/pi).is_integer is False: |
|
return True |
|
if arg.is_imaginary: |
|
return True |
|
|
|
def _eval_is_real(self): |
|
arg = self.args[0] |
|
if arg.is_real and (arg/pi).is_integer is False: |
|
return True |
|
|
|
def _eval_is_complex(self): |
|
arg = self.args[0] |
|
if arg.is_real and (arg/pi).is_integer is False: |
|
return True |
|
|
|
def _eval_is_zero(self): |
|
rest, pimult = _peeloff_pi(self.args[0]) |
|
if pimult and rest.is_zero: |
|
return (pimult - S.Half).is_integer |
|
|
|
def _eval_subs(self, old, new): |
|
arg = self.args[0] |
|
argnew = arg.subs(old, new) |
|
if arg != argnew and (argnew/pi).is_integer: |
|
return S.ComplexInfinity |
|
return cot(argnew) |
|
|
|
|
|
class ReciprocalTrigonometricFunction(TrigonometricFunction): |
|
"""Base class for reciprocal functions of trigonometric functions. """ |
|
|
|
_reciprocal_of = None |
|
_singularities = (S.ComplexInfinity,) |
|
|
|
|
|
|
|
|
|
|
|
|
|
_is_even: FuzzyBool = None |
|
_is_odd: FuzzyBool = None |
|
|
|
@classmethod |
|
def eval(cls, arg): |
|
if arg.could_extract_minus_sign(): |
|
if cls._is_even: |
|
return cls(-arg) |
|
if cls._is_odd: |
|
return -cls(-arg) |
|
|
|
pi_coeff = _pi_coeff(arg) |
|
if (pi_coeff is not None |
|
and not (2*pi_coeff).is_integer |
|
and pi_coeff.is_Rational): |
|
q = pi_coeff.q |
|
p = pi_coeff.p % (2*q) |
|
if p > q: |
|
narg = (pi_coeff - 1)*pi |
|
return -cls(narg) |
|
if 2*p > q: |
|
narg = (1 - pi_coeff)*pi |
|
if cls._is_odd: |
|
return cls(narg) |
|
elif cls._is_even: |
|
return -cls(narg) |
|
|
|
if hasattr(arg, 'inverse') and arg.inverse() == cls: |
|
return arg.args[0] |
|
|
|
t = cls._reciprocal_of.eval(arg) |
|
if t is None: |
|
return t |
|
elif any(isinstance(i, cos) for i in (t, -t)): |
|
return (1/t).rewrite(sec) |
|
elif any(isinstance(i, sin) for i in (t, -t)): |
|
return (1/t).rewrite(csc) |
|
else: |
|
return 1/t |
|
|
|
def _call_reciprocal(self, method_name, *args, **kwargs): |
|
|
|
o = self._reciprocal_of(self.args[0]) |
|
return getattr(o, method_name)(*args, **kwargs) |
|
|
|
def _calculate_reciprocal(self, method_name, *args, **kwargs): |
|
|
|
|
|
t = self._call_reciprocal(method_name, *args, **kwargs) |
|
return 1/t if t is not None else t |
|
|
|
def _rewrite_reciprocal(self, method_name, arg): |
|
|
|
|
|
t = self._call_reciprocal(method_name, arg) |
|
if t is not None and t != self._reciprocal_of(arg): |
|
return 1/t |
|
|
|
def _period(self, symbol): |
|
f = expand_mul(self.args[0]) |
|
return self._reciprocal_of(f).period(symbol) |
|
|
|
def fdiff(self, argindex=1): |
|
return -self._calculate_reciprocal("fdiff", argindex)/self**2 |
|
|
|
def _eval_rewrite_as_exp(self, arg, **kwargs): |
|
return self._rewrite_reciprocal("_eval_rewrite_as_exp", arg) |
|
|
|
def _eval_rewrite_as_Pow(self, arg, **kwargs): |
|
return self._rewrite_reciprocal("_eval_rewrite_as_Pow", arg) |
|
|
|
def _eval_rewrite_as_sin(self, arg, **kwargs): |
|
return self._rewrite_reciprocal("_eval_rewrite_as_sin", arg) |
|
|
|
def _eval_rewrite_as_cos(self, arg, **kwargs): |
|
return self._rewrite_reciprocal("_eval_rewrite_as_cos", arg) |
|
|
|
def _eval_rewrite_as_tan(self, arg, **kwargs): |
|
return self._rewrite_reciprocal("_eval_rewrite_as_tan", arg) |
|
|
|
def _eval_rewrite_as_pow(self, arg, **kwargs): |
|
return self._rewrite_reciprocal("_eval_rewrite_as_pow", arg) |
|
|
|
def _eval_rewrite_as_sqrt(self, arg, **kwargs): |
|
return self._rewrite_reciprocal("_eval_rewrite_as_sqrt", arg) |
|
|
|
def _eval_conjugate(self): |
|
return self.func(self.args[0].conjugate()) |
|
|
|
def as_real_imag(self, deep=True, **hints): |
|
return (1/self._reciprocal_of(self.args[0])).as_real_imag(deep, |
|
**hints) |
|
|
|
def _eval_expand_trig(self, **hints): |
|
return self._calculate_reciprocal("_eval_expand_trig", **hints) |
|
|
|
def _eval_is_extended_real(self): |
|
return self._reciprocal_of(self.args[0])._eval_is_extended_real() |
|
|
|
def _eval_as_leading_term(self, x, logx, cdir): |
|
return (1/self._reciprocal_of(self.args[0]))._eval_as_leading_term(x, logx=logx, cdir=cdir) |
|
|
|
def _eval_is_finite(self): |
|
return (1/self._reciprocal_of(self.args[0])).is_finite |
|
|
|
def _eval_nseries(self, x, n, logx, cdir=0): |
|
return (1/self._reciprocal_of(self.args[0]))._eval_nseries(x, n, logx) |
|
|
|
|
|
class sec(ReciprocalTrigonometricFunction): |
|
""" |
|
The secant function. |
|
|
|
Returns the secant of x (measured in radians). |
|
|
|
Explanation |
|
=========== |
|
|
|
See :class:`sin` for notes about automatic evaluation. |
|
|
|
Examples |
|
======== |
|
|
|
>>> from sympy import sec |
|
>>> from sympy.abc import x |
|
>>> sec(x**2).diff(x) |
|
2*x*tan(x**2)*sec(x**2) |
|
>>> sec(1).diff(x) |
|
0 |
|
|
|
See Also |
|
======== |
|
|
|
sin, csc, cos, tan, cot |
|
asin, acsc, acos, asec, atan, acot, atan2 |
|
|
|
References |
|
========== |
|
|
|
.. [1] https://en.wikipedia.org/wiki/Trigonometric_functions |
|
.. [2] https://dlmf.nist.gov/4.14 |
|
.. [3] https://functions.wolfram.com/ElementaryFunctions/Sec |
|
|
|
""" |
|
|
|
_reciprocal_of = cos |
|
_is_even = True |
|
|
|
def period(self, symbol=None): |
|
return self._period(symbol) |
|
|
|
def _eval_rewrite_as_cot(self, arg, **kwargs): |
|
cot_half_sq = cot(arg/2)**2 |
|
return (cot_half_sq + 1)/(cot_half_sq - 1) |
|
|
|
def _eval_rewrite_as_cos(self, arg, **kwargs): |
|
return (1/cos(arg)) |
|
|
|
def _eval_rewrite_as_sincos(self, arg, **kwargs): |
|
return sin(arg)/(cos(arg)*sin(arg)) |
|
|
|
def _eval_rewrite_as_sin(self, arg, **kwargs): |
|
return (1/cos(arg).rewrite(sin, **kwargs)) |
|
|
|
def _eval_rewrite_as_tan(self, arg, **kwargs): |
|
return (1/cos(arg).rewrite(tan, **kwargs)) |
|
|
|
def _eval_rewrite_as_csc(self, arg, **kwargs): |
|
return csc(pi/2 - arg, evaluate=False) |
|
|
|
def fdiff(self, argindex=1): |
|
if argindex == 1: |
|
return tan(self.args[0])*sec(self.args[0]) |
|
else: |
|
raise ArgumentIndexError(self, argindex) |
|
|
|
def _eval_rewrite_as_besselj(self, arg, **kwargs): |
|
from sympy.functions.special.bessel import besselj |
|
return Piecewise( |
|
(1/(sqrt(pi*arg)/(sqrt(2))*besselj(-S.Half, arg)), Ne(arg, 0)), |
|
(1, True) |
|
) |
|
|
|
def _eval_is_complex(self): |
|
arg = self.args[0] |
|
|
|
if arg.is_complex and (arg/pi - S.Half).is_integer is False: |
|
return True |
|
|
|
@staticmethod |
|
@cacheit |
|
def taylor_term(n, x, *previous_terms): |
|
|
|
|
|
if n < 0 or n % 2 == 1: |
|
return S.Zero |
|
else: |
|
x = sympify(x) |
|
k = n//2 |
|
return S.NegativeOne**k*euler(2*k)/factorial(2*k)*x**(2*k) |
|
|
|
def _eval_as_leading_term(self, x, logx, cdir): |
|
from sympy.calculus.accumulationbounds import AccumBounds |
|
from sympy.functions.elementary.complexes import re |
|
arg = self.args[0] |
|
x0 = arg.subs(x, 0).cancel() |
|
n = (x0 + pi/2)/pi |
|
if n.is_integer: |
|
lt = (arg - n*pi + pi/2).as_leading_term(x) |
|
return (S.NegativeOne**n)/lt |
|
if x0 is S.ComplexInfinity: |
|
x0 = arg.limit(x, 0, dir='-' if re(cdir).is_negative else '+') |
|
if x0 in (S.Infinity, S.NegativeInfinity): |
|
return AccumBounds(S.NegativeInfinity, S.Infinity) |
|
return self.func(x0) if x0.is_finite else self |
|
|
|
|
|
class csc(ReciprocalTrigonometricFunction): |
|
""" |
|
The cosecant function. |
|
|
|
Returns the cosecant of x (measured in radians). |
|
|
|
Explanation |
|
=========== |
|
|
|
See :func:`sin` for notes about automatic evaluation. |
|
|
|
Examples |
|
======== |
|
|
|
>>> from sympy import csc |
|
>>> from sympy.abc import x |
|
>>> csc(x**2).diff(x) |
|
-2*x*cot(x**2)*csc(x**2) |
|
>>> csc(1).diff(x) |
|
0 |
|
|
|
See Also |
|
======== |
|
|
|
sin, cos, sec, tan, cot |
|
asin, acsc, acos, asec, atan, acot, atan2 |
|
|
|
References |
|
========== |
|
|
|
.. [1] https://en.wikipedia.org/wiki/Trigonometric_functions |
|
.. [2] https://dlmf.nist.gov/4.14 |
|
.. [3] https://functions.wolfram.com/ElementaryFunctions/Csc |
|
|
|
""" |
|
|
|
_reciprocal_of = sin |
|
_is_odd = True |
|
|
|
def period(self, symbol=None): |
|
return self._period(symbol) |
|
|
|
def _eval_rewrite_as_sin(self, arg, **kwargs): |
|
return (1/sin(arg)) |
|
|
|
def _eval_rewrite_as_sincos(self, arg, **kwargs): |
|
return cos(arg)/(sin(arg)*cos(arg)) |
|
|
|
def _eval_rewrite_as_cot(self, arg, **kwargs): |
|
cot_half = cot(arg/2) |
|
return (1 + cot_half**2)/(2*cot_half) |
|
|
|
def _eval_rewrite_as_cos(self, arg, **kwargs): |
|
return 1/sin(arg).rewrite(cos, **kwargs) |
|
|
|
def _eval_rewrite_as_sec(self, arg, **kwargs): |
|
return sec(pi/2 - arg, evaluate=False) |
|
|
|
def _eval_rewrite_as_tan(self, arg, **kwargs): |
|
return (1/sin(arg).rewrite(tan, **kwargs)) |
|
|
|
def _eval_rewrite_as_besselj(self, arg, **kwargs): |
|
from sympy.functions.special.bessel import besselj |
|
return sqrt(2/pi)*(1/(sqrt(arg)*besselj(S.Half, arg))) |
|
|
|
def fdiff(self, argindex=1): |
|
if argindex == 1: |
|
return -cot(self.args[0])*csc(self.args[0]) |
|
else: |
|
raise ArgumentIndexError(self, argindex) |
|
|
|
def _eval_is_complex(self): |
|
arg = self.args[0] |
|
if arg.is_real and (arg/pi).is_integer is False: |
|
return True |
|
|
|
@staticmethod |
|
@cacheit |
|
def taylor_term(n, x, *previous_terms): |
|
if n == 0: |
|
return 1/sympify(x) |
|
elif n < 0 or n % 2 == 0: |
|
return S.Zero |
|
else: |
|
x = sympify(x) |
|
k = n//2 + 1 |
|
return (S.NegativeOne**(k - 1)*2*(2**(2*k - 1) - 1)* |
|
bernoulli(2*k)*x**(2*k - 1)/factorial(2*k)) |
|
|
|
def _eval_as_leading_term(self, x, logx, cdir): |
|
from sympy.calculus.accumulationbounds import AccumBounds |
|
from sympy.functions.elementary.complexes import re |
|
arg = self.args[0] |
|
x0 = arg.subs(x, 0).cancel() |
|
n = x0/pi |
|
if n.is_integer: |
|
lt = (arg - n*pi).as_leading_term(x) |
|
return (S.NegativeOne**n)/lt |
|
if x0 is S.ComplexInfinity: |
|
x0 = arg.limit(x, 0, dir='-' if re(cdir).is_negative else '+') |
|
if x0 in (S.Infinity, S.NegativeInfinity): |
|
return AccumBounds(S.NegativeInfinity, S.Infinity) |
|
return self.func(x0) if x0.is_finite else self |
|
|
|
|
|
class sinc(DefinedFunction): |
|
r""" |
|
Represents an unnormalized sinc function: |
|
|
|
.. math:: |
|
|
|
\operatorname{sinc}(x) = |
|
\begin{cases} |
|
\frac{\sin x}{x} & \qquad x \neq 0 \\ |
|
1 & \qquad x = 0 |
|
\end{cases} |
|
|
|
Examples |
|
======== |
|
|
|
>>> from sympy import sinc, oo, jn |
|
>>> from sympy.abc import x |
|
>>> sinc(x) |
|
sinc(x) |
|
|
|
* Automated Evaluation |
|
|
|
>>> sinc(0) |
|
1 |
|
>>> sinc(oo) |
|
0 |
|
|
|
* Differentiation |
|
|
|
>>> sinc(x).diff() |
|
cos(x)/x - sin(x)/x**2 |
|
|
|
* Series Expansion |
|
|
|
>>> sinc(x).series() |
|
1 - x**2/6 + x**4/120 + O(x**6) |
|
|
|
* As zero'th order spherical Bessel Function |
|
|
|
>>> sinc(x).rewrite(jn) |
|
jn(0, x) |
|
|
|
See also |
|
======== |
|
|
|
sin |
|
|
|
References |
|
========== |
|
|
|
.. [1] https://en.wikipedia.org/wiki/Sinc_function |
|
|
|
""" |
|
_singularities = (S.ComplexInfinity,) |
|
|
|
def fdiff(self, argindex=1): |
|
x = self.args[0] |
|
if argindex == 1: |
|
|
|
|
|
|
|
|
|
|
|
|
|
return cos(x)/x - sin(x)/x**2 |
|
else: |
|
raise ArgumentIndexError(self, argindex) |
|
|
|
@classmethod |
|
def eval(cls, arg): |
|
if arg.is_zero: |
|
return S.One |
|
if arg.is_Number: |
|
if arg in [S.Infinity, S.NegativeInfinity]: |
|
return S.Zero |
|
elif arg is S.NaN: |
|
return S.NaN |
|
|
|
if arg is S.ComplexInfinity: |
|
return S.NaN |
|
|
|
if arg.could_extract_minus_sign(): |
|
return cls(-arg) |
|
|
|
pi_coeff = _pi_coeff(arg) |
|
if pi_coeff is not None: |
|
if pi_coeff.is_integer: |
|
if fuzzy_not(arg.is_zero): |
|
return S.Zero |
|
elif (2*pi_coeff).is_integer: |
|
return S.NegativeOne**(pi_coeff - S.Half)/arg |
|
|
|
def _eval_nseries(self, x, n, logx, cdir=0): |
|
x = self.args[0] |
|
return (sin(x)/x)._eval_nseries(x, n, logx) |
|
|
|
def _eval_rewrite_as_jn(self, arg, **kwargs): |
|
from sympy.functions.special.bessel import jn |
|
return jn(0, arg) |
|
|
|
def _eval_rewrite_as_sin(self, arg, **kwargs): |
|
return Piecewise((sin(arg)/arg, Ne(arg, S.Zero)), (S.One, S.true)) |
|
|
|
def _eval_is_zero(self): |
|
if self.args[0].is_infinite: |
|
return True |
|
rest, pi_mult = _peeloff_pi(self.args[0]) |
|
if rest.is_zero: |
|
return fuzzy_and([pi_mult.is_integer, pi_mult.is_nonzero]) |
|
if rest.is_Number and pi_mult.is_integer: |
|
return False |
|
|
|
def _eval_is_real(self): |
|
if self.args[0].is_extended_real or self.args[0].is_imaginary: |
|
return True |
|
|
|
_eval_is_finite = _eval_is_real |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
class InverseTrigonometricFunction(DefinedFunction): |
|
"""Base class for inverse trigonometric functions.""" |
|
_singularities: tuple[Expr, ...] = (S.One, S.NegativeOne, S.Zero, S.ComplexInfinity) |
|
|
|
@staticmethod |
|
@cacheit |
|
def _asin_table(): |
|
|
|
|
|
return { |
|
sqrt(3)/2: pi/3, |
|
sqrt(2)/2: pi/4, |
|
1/sqrt(2): pi/4, |
|
sqrt((5 - sqrt(5))/8): pi/5, |
|
sqrt(2)*sqrt(5 - sqrt(5))/4: pi/5, |
|
sqrt((5 + sqrt(5))/8): pi*Rational(2, 5), |
|
sqrt(2)*sqrt(5 + sqrt(5))/4: pi*Rational(2, 5), |
|
S.Half: pi/6, |
|
sqrt(2 - sqrt(2))/2: pi/8, |
|
sqrt(S.Half - sqrt(2)/4): pi/8, |
|
sqrt(2 + sqrt(2))/2: pi*Rational(3, 8), |
|
sqrt(S.Half + sqrt(2)/4): pi*Rational(3, 8), |
|
(sqrt(5) - 1)/4: pi/10, |
|
(1 - sqrt(5))/4: -pi/10, |
|
(sqrt(5) + 1)/4: pi*Rational(3, 10), |
|
sqrt(6)/4 - sqrt(2)/4: pi/12, |
|
-sqrt(6)/4 + sqrt(2)/4: -pi/12, |
|
(sqrt(3) - 1)/sqrt(8): pi/12, |
|
(1 - sqrt(3))/sqrt(8): -pi/12, |
|
sqrt(6)/4 + sqrt(2)/4: pi*Rational(5, 12), |
|
(1 + sqrt(3))/sqrt(8): pi*Rational(5, 12) |
|
} |
|
|
|
|
|
@staticmethod |
|
@cacheit |
|
def _atan_table(): |
|
|
|
|
|
return { |
|
sqrt(3)/3: pi/6, |
|
1/sqrt(3): pi/6, |
|
sqrt(3): pi/3, |
|
sqrt(2) - 1: pi/8, |
|
1 - sqrt(2): -pi/8, |
|
1 + sqrt(2): pi*Rational(3, 8), |
|
sqrt(5 - 2*sqrt(5)): pi/5, |
|
sqrt(5 + 2*sqrt(5)): pi*Rational(2, 5), |
|
sqrt(1 - 2*sqrt(5)/5): pi/10, |
|
sqrt(1 + 2*sqrt(5)/5): pi*Rational(3, 10), |
|
2 - sqrt(3): pi/12, |
|
-2 + sqrt(3): -pi/12, |
|
2 + sqrt(3): pi*Rational(5, 12) |
|
} |
|
|
|
@staticmethod |
|
@cacheit |
|
def _acsc_table(): |
|
|
|
|
|
return { |
|
2*sqrt(3)/3: pi/3, |
|
sqrt(2): pi/4, |
|
sqrt(2 + 2*sqrt(5)/5): pi/5, |
|
1/sqrt(Rational(5, 8) - sqrt(5)/8): pi/5, |
|
sqrt(2 - 2*sqrt(5)/5): pi*Rational(2, 5), |
|
1/sqrt(Rational(5, 8) + sqrt(5)/8): pi*Rational(2, 5), |
|
2: pi/6, |
|
sqrt(4 + 2*sqrt(2)): pi/8, |
|
2/sqrt(2 - sqrt(2)): pi/8, |
|
sqrt(4 - 2*sqrt(2)): pi*Rational(3, 8), |
|
2/sqrt(2 + sqrt(2)): pi*Rational(3, 8), |
|
1 + sqrt(5): pi/10, |
|
sqrt(5) - 1: pi*Rational(3, 10), |
|
-(sqrt(5) - 1): pi*Rational(-3, 10), |
|
sqrt(6) + sqrt(2): pi/12, |
|
sqrt(6) - sqrt(2): pi*Rational(5, 12), |
|
-(sqrt(6) - sqrt(2)): pi*Rational(-5, 12) |
|
} |
|
|
|
|
|
class asin(InverseTrigonometricFunction): |
|
r""" |
|
The inverse sine function. |
|
|
|
Returns the arcsine of x in radians. |
|
|
|
Explanation |
|
=========== |
|
|
|
``asin(x)`` will evaluate automatically in the cases |
|
$x \in \{\infty, -\infty, 0, 1, -1\}$ and for some instances when the |
|
result is a rational multiple of $\pi$ (see the ``eval`` class method). |
|
|
|
A purely imaginary argument will lead to an asinh expression. |
|
|
|
Examples |
|
======== |
|
|
|
>>> from sympy import asin, oo |
|
>>> asin(1) |
|
pi/2 |
|
>>> asin(-1) |
|
-pi/2 |
|
>>> asin(-oo) |
|
oo*I |
|
>>> asin(oo) |
|
-oo*I |
|
|
|
See Also |
|
======== |
|
|
|
sin, csc, cos, sec, tan, cot |
|
acsc, acos, asec, atan, acot, atan2 |
|
|
|
References |
|
========== |
|
|
|
.. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions |
|
.. [2] https://dlmf.nist.gov/4.23 |
|
.. [3] https://functions.wolfram.com/ElementaryFunctions/ArcSin |
|
|
|
""" |
|
|
|
def fdiff(self, argindex=1): |
|
if argindex == 1: |
|
return 1/sqrt(1 - self.args[0]**2) |
|
else: |
|
raise ArgumentIndexError(self, argindex) |
|
|
|
def _eval_is_rational(self): |
|
s = self.func(*self.args) |
|
if s.func == self.func: |
|
if s.args[0].is_rational: |
|
return False |
|
else: |
|
return s.is_rational |
|
|
|
def _eval_is_positive(self): |
|
return self._eval_is_extended_real() and self.args[0].is_positive |
|
|
|
def _eval_is_negative(self): |
|
return self._eval_is_extended_real() and self.args[0].is_negative |
|
|
|
@classmethod |
|
def eval(cls, arg): |
|
if arg.is_Number: |
|
if arg is S.NaN: |
|
return S.NaN |
|
elif arg is S.Infinity: |
|
return S.NegativeInfinity*S.ImaginaryUnit |
|
elif arg is S.NegativeInfinity: |
|
return S.Infinity*S.ImaginaryUnit |
|
elif arg.is_zero: |
|
return S.Zero |
|
elif arg is S.One: |
|
return pi/2 |
|
elif arg is S.NegativeOne: |
|
return -pi/2 |
|
|
|
if arg is S.ComplexInfinity: |
|
return S.ComplexInfinity |
|
|
|
if arg.could_extract_minus_sign(): |
|
return -cls(-arg) |
|
|
|
if arg.is_number: |
|
asin_table = cls._asin_table() |
|
if arg in asin_table: |
|
return asin_table[arg] |
|
|
|
i_coeff = _imaginary_unit_as_coefficient(arg) |
|
if i_coeff is not None: |
|
from sympy.functions.elementary.hyperbolic import asinh |
|
return S.ImaginaryUnit*asinh(i_coeff) |
|
|
|
if arg.is_zero: |
|
return S.Zero |
|
|
|
if isinstance(arg, sin): |
|
ang = arg.args[0] |
|
if ang.is_comparable: |
|
ang %= 2*pi |
|
if ang > pi: |
|
ang = pi - ang |
|
|
|
|
|
if ang > pi/2: |
|
ang = pi - ang |
|
if ang < -pi/2: |
|
ang = -pi - ang |
|
|
|
return ang |
|
|
|
if isinstance(arg, cos): |
|
ang = arg.args[0] |
|
if ang.is_comparable: |
|
return pi/2 - acos(arg) |
|
|
|
@staticmethod |
|
@cacheit |
|
def taylor_term(n, x, *previous_terms): |
|
if n < 0 or n % 2 == 0: |
|
return S.Zero |
|
else: |
|
x = sympify(x) |
|
if len(previous_terms) >= 2 and n > 2: |
|
p = previous_terms[-2] |
|
return p*(n - 2)**2/(n*(n - 1))*x**2 |
|
else: |
|
k = (n - 1) // 2 |
|
R = RisingFactorial(S.Half, k) |
|
F = factorial(k) |
|
return R/F*x**n/n |
|
|
|
def _eval_as_leading_term(self, x, logx, cdir): |
|
arg = self.args[0] |
|
x0 = arg.subs(x, 0).cancel() |
|
if x0 is S.NaN: |
|
return self.func(arg.as_leading_term(x)) |
|
if x0.is_zero: |
|
return arg.as_leading_term(x) |
|
|
|
|
|
if x0 in (-S.One, S.One, S.ComplexInfinity): |
|
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir).expand() |
|
|
|
if (1 - x0**2).is_negative: |
|
ndir = arg.dir(x, cdir if cdir else 1) |
|
if im(ndir).is_negative: |
|
if x0.is_negative: |
|
return -pi - self.func(x0) |
|
elif im(ndir).is_positive: |
|
if x0.is_positive: |
|
return pi - self.func(x0) |
|
else: |
|
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir).expand() |
|
return self.func(x0) |
|
|
|
def _eval_nseries(self, x, n, logx, cdir=0): |
|
from sympy.series.order import O |
|
arg0 = self.args[0].subs(x, 0) |
|
|
|
if arg0 is S.One: |
|
t = Dummy('t', positive=True) |
|
ser = asin(S.One - t**2).rewrite(log).nseries(t, 0, 2*n) |
|
arg1 = S.One - self.args[0] |
|
f = arg1.as_leading_term(x) |
|
g = (arg1 - f)/ f |
|
if not g.is_meromorphic(x, 0): |
|
return O(1) if n == 0 else pi/2 + O(sqrt(x)) |
|
res1 = sqrt(S.One + g)._eval_nseries(x, n=n, logx=logx) |
|
res = (res1.removeO()*sqrt(f)).expand() |
|
return ser.removeO().subs(t, res).expand().powsimp() + O(x**n, x) |
|
|
|
if arg0 is S.NegativeOne: |
|
t = Dummy('t', positive=True) |
|
ser = asin(S.NegativeOne + t**2).rewrite(log).nseries(t, 0, 2*n) |
|
arg1 = S.One + self.args[0] |
|
f = arg1.as_leading_term(x) |
|
g = (arg1 - f)/ f |
|
if not g.is_meromorphic(x, 0): |
|
return O(1) if n == 0 else -pi/2 + O(sqrt(x)) |
|
res1 = sqrt(S.One + g)._eval_nseries(x, n=n, logx=logx) |
|
res = (res1.removeO()*sqrt(f)).expand() |
|
return ser.removeO().subs(t, res).expand().powsimp() + O(x**n, x) |
|
|
|
res = super()._eval_nseries(x, n=n, logx=logx) |
|
if arg0 is S.ComplexInfinity: |
|
return res |
|
|
|
if (1 - arg0**2).is_negative: |
|
ndir = self.args[0].dir(x, cdir if cdir else 1) |
|
if im(ndir).is_negative: |
|
if arg0.is_negative: |
|
return -pi - res |
|
elif im(ndir).is_positive: |
|
if arg0.is_positive: |
|
return pi - res |
|
else: |
|
return self.rewrite(log)._eval_nseries(x, n, logx=logx, cdir=cdir) |
|
return res |
|
|
|
def _eval_rewrite_as_acos(self, x, **kwargs): |
|
return pi/2 - acos(x) |
|
|
|
def _eval_rewrite_as_atan(self, x, **kwargs): |
|
return 2*atan(x/(1 + sqrt(1 - x**2))) |
|
|
|
def _eval_rewrite_as_log(self, x, **kwargs): |
|
return -S.ImaginaryUnit*log(S.ImaginaryUnit*x + sqrt(1 - x**2)) |
|
|
|
_eval_rewrite_as_tractable = _eval_rewrite_as_log |
|
|
|
def _eval_rewrite_as_acot(self, arg, **kwargs): |
|
return 2*acot((1 + sqrt(1 - arg**2))/arg) |
|
|
|
def _eval_rewrite_as_asec(self, arg, **kwargs): |
|
return pi/2 - asec(1/arg) |
|
|
|
def _eval_rewrite_as_acsc(self, arg, **kwargs): |
|
return acsc(1/arg) |
|
|
|
def _eval_is_extended_real(self): |
|
x = self.args[0] |
|
return x.is_extended_real and (1 - abs(x)).is_nonnegative |
|
|
|
def inverse(self, argindex=1): |
|
""" |
|
Returns the inverse of this function. |
|
""" |
|
return sin |
|
|
|
|
|
class acos(InverseTrigonometricFunction): |
|
r""" |
|
The inverse cosine function. |
|
|
|
Explanation |
|
=========== |
|
|
|
Returns the arc cosine of x (measured in radians). |
|
|
|
``acos(x)`` will evaluate automatically in the cases |
|
$x \in \{\infty, -\infty, 0, 1, -1\}$ and for some instances when |
|
the result is a rational multiple of $\pi$ (see the eval class method). |
|
|
|
``acos(zoo)`` evaluates to ``zoo`` |
|
(see note in :class:`sympy.functions.elementary.trigonometric.asec`) |
|
|
|
A purely imaginary argument will be rewritten to asinh. |
|
|
|
Examples |
|
======== |
|
|
|
>>> from sympy import acos, oo |
|
>>> acos(1) |
|
0 |
|
>>> acos(0) |
|
pi/2 |
|
>>> acos(oo) |
|
oo*I |
|
|
|
See Also |
|
======== |
|
|
|
sin, csc, cos, sec, tan, cot |
|
asin, acsc, asec, atan, acot, atan2 |
|
|
|
References |
|
========== |
|
|
|
.. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions |
|
.. [2] https://dlmf.nist.gov/4.23 |
|
.. [3] https://functions.wolfram.com/ElementaryFunctions/ArcCos |
|
|
|
""" |
|
|
|
def fdiff(self, argindex=1): |
|
if argindex == 1: |
|
return -1/sqrt(1 - self.args[0]**2) |
|
else: |
|
raise ArgumentIndexError(self, argindex) |
|
|
|
def _eval_is_rational(self): |
|
s = self.func(*self.args) |
|
if s.func == self.func: |
|
if s.args[0].is_rational: |
|
return False |
|
else: |
|
return s.is_rational |
|
|
|
@classmethod |
|
def eval(cls, arg): |
|
if arg.is_Number: |
|
if arg is S.NaN: |
|
return S.NaN |
|
elif arg is S.Infinity: |
|
return S.Infinity*S.ImaginaryUnit |
|
elif arg is S.NegativeInfinity: |
|
return S.NegativeInfinity*S.ImaginaryUnit |
|
elif arg.is_zero: |
|
return pi/2 |
|
elif arg is S.One: |
|
return S.Zero |
|
elif arg is S.NegativeOne: |
|
return pi |
|
|
|
if arg is S.ComplexInfinity: |
|
return S.ComplexInfinity |
|
|
|
if arg.is_number: |
|
asin_table = cls._asin_table() |
|
if arg in asin_table: |
|
return pi/2 - asin_table[arg] |
|
elif -arg in asin_table: |
|
return pi/2 + asin_table[-arg] |
|
|
|
i_coeff = _imaginary_unit_as_coefficient(arg) |
|
if i_coeff is not None: |
|
return pi/2 - asin(arg) |
|
|
|
if arg.is_Mul and len(arg.args) == 2 and arg.args[0] == -1: |
|
narg = arg.args[1] |
|
minus = True |
|
else: |
|
narg = arg |
|
minus = False |
|
|
|
if isinstance(narg, cos): |
|
|
|
ang = narg.args[0] |
|
if ang.is_comparable: |
|
if minus: |
|
ang = pi - ang |
|
ang %= 2*pi |
|
if ang > pi: |
|
ang = 2*pi - ang |
|
return ang |
|
|
|
if isinstance(narg, sin): |
|
ang = narg.args[0] |
|
if ang.is_comparable: |
|
if minus: |
|
return pi/2 + asin(narg) |
|
return pi/2 - asin(narg) |
|
|
|
@staticmethod |
|
@cacheit |
|
def taylor_term(n, x, *previous_terms): |
|
if n == 0: |
|
return pi/2 |
|
elif n < 0 or n % 2 == 0: |
|
return S.Zero |
|
else: |
|
x = sympify(x) |
|
if len(previous_terms) >= 2 and n > 2: |
|
p = previous_terms[-2] |
|
return p*(n - 2)**2/(n*(n - 1))*x**2 |
|
else: |
|
k = (n - 1) // 2 |
|
R = RisingFactorial(S.Half, k) |
|
F = factorial(k) |
|
return -R/F*x**n/n |
|
|
|
def _eval_as_leading_term(self, x, logx, cdir): |
|
arg = self.args[0] |
|
x0 = arg.subs(x, 0).cancel() |
|
if x0 is S.NaN: |
|
return self.func(arg.as_leading_term(x)) |
|
|
|
if x0 == 1: |
|
return sqrt(2)*sqrt((S.One - arg).as_leading_term(x)) |
|
if x0 in (-S.One, S.ComplexInfinity): |
|
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir) |
|
|
|
if (1 - x0**2).is_negative: |
|
ndir = arg.dir(x, cdir if cdir else 1) |
|
if im(ndir).is_negative: |
|
if x0.is_negative: |
|
return 2*pi - self.func(x0) |
|
elif im(ndir).is_positive: |
|
if x0.is_positive: |
|
return -self.func(x0) |
|
else: |
|
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir).expand() |
|
return self.func(x0) |
|
|
|
def _eval_is_extended_real(self): |
|
x = self.args[0] |
|
return x.is_extended_real and (1 - abs(x)).is_nonnegative |
|
|
|
def _eval_is_nonnegative(self): |
|
return self._eval_is_extended_real() |
|
|
|
def _eval_nseries(self, x, n, logx, cdir=0): |
|
from sympy.series.order import O |
|
arg0 = self.args[0].subs(x, 0) |
|
|
|
if arg0 is S.One: |
|
t = Dummy('t', positive=True) |
|
ser = acos(S.One - t**2).rewrite(log).nseries(t, 0, 2*n) |
|
arg1 = S.One - self.args[0] |
|
f = arg1.as_leading_term(x) |
|
g = (arg1 - f)/ f |
|
if not g.is_meromorphic(x, 0): |
|
return O(1) if n == 0 else O(sqrt(x)) |
|
res1 = sqrt(S.One + g)._eval_nseries(x, n=n, logx=logx) |
|
res = (res1.removeO()*sqrt(f)).expand() |
|
return ser.removeO().subs(t, res).expand().powsimp() + O(x**n, x) |
|
|
|
if arg0 is S.NegativeOne: |
|
t = Dummy('t', positive=True) |
|
ser = acos(S.NegativeOne + t**2).rewrite(log).nseries(t, 0, 2*n) |
|
arg1 = S.One + self.args[0] |
|
f = arg1.as_leading_term(x) |
|
g = (arg1 - f)/ f |
|
if not g.is_meromorphic(x, 0): |
|
return O(1) if n == 0 else pi + O(sqrt(x)) |
|
res1 = sqrt(S.One + g)._eval_nseries(x, n=n, logx=logx) |
|
res = (res1.removeO()*sqrt(f)).expand() |
|
return ser.removeO().subs(t, res).expand().powsimp() + O(x**n, x) |
|
|
|
res = super()._eval_nseries(x, n=n, logx=logx) |
|
if arg0 is S.ComplexInfinity: |
|
return res |
|
|
|
if (1 - arg0**2).is_negative: |
|
ndir = self.args[0].dir(x, cdir if cdir else 1) |
|
if im(ndir).is_negative: |
|
if arg0.is_negative: |
|
return 2*pi - res |
|
elif im(ndir).is_positive: |
|
if arg0.is_positive: |
|
return -res |
|
else: |
|
return self.rewrite(log)._eval_nseries(x, n, logx=logx, cdir=cdir) |
|
return res |
|
|
|
def _eval_rewrite_as_log(self, x, **kwargs): |
|
return pi/2 + S.ImaginaryUnit*\ |
|
log(S.ImaginaryUnit*x + sqrt(1 - x**2)) |
|
|
|
_eval_rewrite_as_tractable = _eval_rewrite_as_log |
|
|
|
def _eval_rewrite_as_asin(self, x, **kwargs): |
|
return pi/2 - asin(x) |
|
|
|
def _eval_rewrite_as_atan(self, x, **kwargs): |
|
return atan(sqrt(1 - x**2)/x) + (pi/2)*(1 - x*sqrt(1/x**2)) |
|
|
|
def inverse(self, argindex=1): |
|
""" |
|
Returns the inverse of this function. |
|
""" |
|
return cos |
|
|
|
def _eval_rewrite_as_acot(self, arg, **kwargs): |
|
return pi/2 - 2*acot((1 + sqrt(1 - arg**2))/arg) |
|
|
|
def _eval_rewrite_as_asec(self, arg, **kwargs): |
|
return asec(1/arg) |
|
|
|
def _eval_rewrite_as_acsc(self, arg, **kwargs): |
|
return pi/2 - acsc(1/arg) |
|
|
|
def _eval_conjugate(self): |
|
z = self.args[0] |
|
r = self.func(self.args[0].conjugate()) |
|
if z.is_extended_real is False: |
|
return r |
|
elif z.is_extended_real and (z + 1).is_nonnegative and (z - 1).is_nonpositive: |
|
return r |
|
|
|
|
|
class atan(InverseTrigonometricFunction): |
|
r""" |
|
The inverse tangent function. |
|
|
|
Returns the arc tangent of x (measured in radians). |
|
|
|
Explanation |
|
=========== |
|
|
|
``atan(x)`` will evaluate automatically in the cases |
|
$x \in \{\infty, -\infty, 0, 1, -1\}$ and for some instances when the |
|
result is a rational multiple of $\pi$ (see the eval class method). |
|
|
|
Examples |
|
======== |
|
|
|
>>> from sympy import atan, oo |
|
>>> atan(0) |
|
0 |
|
>>> atan(1) |
|
pi/4 |
|
>>> atan(oo) |
|
pi/2 |
|
|
|
See Also |
|
======== |
|
|
|
sin, csc, cos, sec, tan, cot |
|
asin, acsc, acos, asec, acot, atan2 |
|
|
|
References |
|
========== |
|
|
|
.. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions |
|
.. [2] https://dlmf.nist.gov/4.23 |
|
.. [3] https://functions.wolfram.com/ElementaryFunctions/ArcTan |
|
|
|
""" |
|
|
|
args: tuple[Expr] |
|
|
|
_singularities = (S.ImaginaryUnit, -S.ImaginaryUnit) |
|
|
|
def fdiff(self, argindex=1): |
|
if argindex == 1: |
|
return 1/(1 + self.args[0]**2) |
|
else: |
|
raise ArgumentIndexError(self, argindex) |
|
|
|
def _eval_is_rational(self): |
|
s = self.func(*self.args) |
|
if s.func == self.func: |
|
if s.args[0].is_rational: |
|
return False |
|
else: |
|
return s.is_rational |
|
|
|
def _eval_is_positive(self): |
|
return self.args[0].is_extended_positive |
|
|
|
def _eval_is_nonnegative(self): |
|
return self.args[0].is_extended_nonnegative |
|
|
|
def _eval_is_zero(self): |
|
return self.args[0].is_zero |
|
|
|
def _eval_is_real(self): |
|
return self.args[0].is_extended_real |
|
|
|
@classmethod |
|
def eval(cls, arg): |
|
if arg.is_Number: |
|
if arg is S.NaN: |
|
return S.NaN |
|
elif arg is S.Infinity: |
|
return pi/2 |
|
elif arg is S.NegativeInfinity: |
|
return -pi/2 |
|
elif arg.is_zero: |
|
return S.Zero |
|
elif arg is S.One: |
|
return pi/4 |
|
elif arg is S.NegativeOne: |
|
return -pi/4 |
|
|
|
if arg is S.ComplexInfinity: |
|
from sympy.calculus.accumulationbounds import AccumBounds |
|
return AccumBounds(-pi/2, pi/2) |
|
|
|
if arg.could_extract_minus_sign(): |
|
return -cls(-arg) |
|
|
|
if arg.is_number: |
|
atan_table = cls._atan_table() |
|
if arg in atan_table: |
|
return atan_table[arg] |
|
|
|
i_coeff = _imaginary_unit_as_coefficient(arg) |
|
if i_coeff is not None: |
|
from sympy.functions.elementary.hyperbolic import atanh |
|
return S.ImaginaryUnit*atanh(i_coeff) |
|
|
|
if arg.is_zero: |
|
return S.Zero |
|
|
|
if isinstance(arg, tan): |
|
ang = arg.args[0] |
|
if ang.is_comparable: |
|
ang %= pi |
|
if ang > pi/2: |
|
ang -= pi |
|
|
|
return ang |
|
|
|
if isinstance(arg, cot): |
|
ang = arg.args[0] |
|
if ang.is_comparable: |
|
ang = pi/2 - acot(arg) |
|
if ang > pi/2: |
|
ang -= pi |
|
return ang |
|
|
|
@staticmethod |
|
@cacheit |
|
def taylor_term(n, x, *previous_terms): |
|
if n < 0 or n % 2 == 0: |
|
return S.Zero |
|
else: |
|
x = sympify(x) |
|
return S.NegativeOne**((n - 1)//2)*x**n/n |
|
|
|
def _eval_as_leading_term(self, x, logx, cdir): |
|
arg = self.args[0] |
|
x0 = arg.subs(x, 0).cancel() |
|
if x0 is S.NaN: |
|
return self.func(arg.as_leading_term(x)) |
|
if x0.is_zero: |
|
return arg.as_leading_term(x) |
|
|
|
if x0 in (-S.ImaginaryUnit, S.ImaginaryUnit, S.ComplexInfinity): |
|
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir).expand() |
|
|
|
if (1 + x0**2).is_negative: |
|
ndir = arg.dir(x, cdir if cdir else 1) |
|
if re(ndir).is_negative: |
|
if im(x0).is_positive: |
|
return self.func(x0) - pi |
|
elif re(ndir).is_positive: |
|
if im(x0).is_negative: |
|
return self.func(x0) + pi |
|
else: |
|
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir).expand() |
|
return self.func(x0) |
|
|
|
def _eval_nseries(self, x, n, logx, cdir=0): |
|
arg0 = self.args[0].subs(x, 0) |
|
|
|
|
|
if arg0 in (S.ImaginaryUnit, S.NegativeOne*S.ImaginaryUnit): |
|
return self.rewrite(log)._eval_nseries(x, n, logx=logx, cdir=cdir) |
|
|
|
res = super()._eval_nseries(x, n=n, logx=logx) |
|
ndir = self.args[0].dir(x, cdir if cdir else 1) |
|
if arg0 is S.ComplexInfinity: |
|
if re(ndir) > 0: |
|
return res - pi |
|
return res |
|
|
|
if (1 + arg0**2).is_negative: |
|
if re(ndir).is_negative: |
|
if im(arg0).is_positive: |
|
return res - pi |
|
elif re(ndir).is_positive: |
|
if im(arg0).is_negative: |
|
return res + pi |
|
else: |
|
return self.rewrite(log)._eval_nseries(x, n, logx=logx, cdir=cdir) |
|
return res |
|
|
|
def _eval_rewrite_as_log(self, x, **kwargs): |
|
return S.ImaginaryUnit/2*(log(S.One - S.ImaginaryUnit*x) |
|
- log(S.One + S.ImaginaryUnit*x)) |
|
|
|
_eval_rewrite_as_tractable = _eval_rewrite_as_log |
|
|
|
def _eval_aseries(self, n, args0, x, logx): |
|
if args0[0] in [S.Infinity, S.NegativeInfinity]: |
|
return (pi/2 - atan(1/self.args[0]))._eval_nseries(x, n, logx) |
|
else: |
|
return super()._eval_aseries(n, args0, x, logx) |
|
|
|
def inverse(self, argindex=1): |
|
""" |
|
Returns the inverse of this function. |
|
""" |
|
return tan |
|
|
|
def _eval_rewrite_as_asin(self, arg, **kwargs): |
|
return sqrt(arg**2)/arg*(pi/2 - asin(1/sqrt(1 + arg**2))) |
|
|
|
def _eval_rewrite_as_acos(self, arg, **kwargs): |
|
return sqrt(arg**2)/arg*acos(1/sqrt(1 + arg**2)) |
|
|
|
def _eval_rewrite_as_acot(self, arg, **kwargs): |
|
return acot(1/arg) |
|
|
|
def _eval_rewrite_as_asec(self, arg, **kwargs): |
|
return sqrt(arg**2)/arg*asec(sqrt(1 + arg**2)) |
|
|
|
def _eval_rewrite_as_acsc(self, arg, **kwargs): |
|
return sqrt(arg**2)/arg*(pi/2 - acsc(sqrt(1 + arg**2))) |
|
|
|
|
|
class acot(InverseTrigonometricFunction): |
|
r""" |
|
The inverse cotangent function. |
|
|
|
Returns the arc cotangent of x (measured in radians). |
|
|
|
Explanation |
|
=========== |
|
|
|
``acot(x)`` will evaluate automatically in the cases |
|
$x \in \{\infty, -\infty, \tilde{\infty}, 0, 1, -1\}$ |
|
and for some instances when the result is a rational multiple of $\pi$ |
|
(see the eval class method). |
|
|
|
A purely imaginary argument will lead to an ``acoth`` expression. |
|
|
|
``acot(x)`` has a branch cut along $(-i, i)$, hence it is discontinuous |
|
at 0. Its range for real $x$ is $(-\frac{\pi}{2}, \frac{\pi}{2}]$. |
|
|
|
Examples |
|
======== |
|
|
|
>>> from sympy import acot, sqrt |
|
>>> acot(0) |
|
pi/2 |
|
>>> acot(1) |
|
pi/4 |
|
>>> acot(sqrt(3) - 2) |
|
-5*pi/12 |
|
|
|
See Also |
|
======== |
|
|
|
sin, csc, cos, sec, tan, cot |
|
asin, acsc, acos, asec, atan, atan2 |
|
|
|
References |
|
========== |
|
|
|
.. [1] https://dlmf.nist.gov/4.23 |
|
.. [2] https://functions.wolfram.com/ElementaryFunctions/ArcCot |
|
|
|
""" |
|
_singularities = (S.ImaginaryUnit, -S.ImaginaryUnit) |
|
|
|
def fdiff(self, argindex=1): |
|
if argindex == 1: |
|
return -1/(1 + self.args[0]**2) |
|
else: |
|
raise ArgumentIndexError(self, argindex) |
|
|
|
def _eval_is_rational(self): |
|
s = self.func(*self.args) |
|
if s.func == self.func: |
|
if s.args[0].is_rational: |
|
return False |
|
else: |
|
return s.is_rational |
|
|
|
def _eval_is_positive(self): |
|
return self.args[0].is_nonnegative |
|
|
|
def _eval_is_negative(self): |
|
return self.args[0].is_negative |
|
|
|
def _eval_is_extended_real(self): |
|
return self.args[0].is_extended_real |
|
|
|
@classmethod |
|
def eval(cls, arg): |
|
if arg.is_Number: |
|
if arg is S.NaN: |
|
return S.NaN |
|
elif arg is S.Infinity: |
|
return S.Zero |
|
elif arg is S.NegativeInfinity: |
|
return S.Zero |
|
elif arg.is_zero: |
|
return pi/ 2 |
|
elif arg is S.One: |
|
return pi/4 |
|
elif arg is S.NegativeOne: |
|
return -pi/4 |
|
|
|
if arg is S.ComplexInfinity: |
|
return S.Zero |
|
|
|
if arg.could_extract_minus_sign(): |
|
return -cls(-arg) |
|
|
|
if arg.is_number: |
|
atan_table = cls._atan_table() |
|
if arg in atan_table: |
|
ang = pi/2 - atan_table[arg] |
|
if ang > pi/2: |
|
ang -= pi |
|
return ang |
|
|
|
i_coeff = _imaginary_unit_as_coefficient(arg) |
|
if i_coeff is not None: |
|
from sympy.functions.elementary.hyperbolic import acoth |
|
return -S.ImaginaryUnit*acoth(i_coeff) |
|
|
|
if arg.is_zero: |
|
return pi*S.Half |
|
|
|
if isinstance(arg, cot): |
|
ang = arg.args[0] |
|
if ang.is_comparable: |
|
ang %= pi |
|
if ang > pi/2: |
|
ang -= pi |
|
return ang |
|
|
|
if isinstance(arg, tan): |
|
ang = arg.args[0] |
|
if ang.is_comparable: |
|
ang = pi/2 - atan(arg) |
|
if ang > pi/2: |
|
ang -= pi |
|
return ang |
|
|
|
@staticmethod |
|
@cacheit |
|
def taylor_term(n, x, *previous_terms): |
|
if n == 0: |
|
return pi/2 |
|
elif n < 0 or n % 2 == 0: |
|
return S.Zero |
|
else: |
|
x = sympify(x) |
|
return S.NegativeOne**((n + 1)//2)*x**n/n |
|
|
|
def _eval_as_leading_term(self, x, logx, cdir): |
|
arg = self.args[0] |
|
x0 = arg.subs(x, 0).cancel() |
|
if x0 is S.NaN: |
|
return self.func(arg.as_leading_term(x)) |
|
if x0 is S.ComplexInfinity: |
|
return (1/arg).as_leading_term(x) |
|
|
|
if x0 in (-S.ImaginaryUnit, S.ImaginaryUnit, S.Zero): |
|
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir).expand() |
|
|
|
if x0.is_imaginary and (1 + x0**2).is_positive: |
|
ndir = arg.dir(x, cdir if cdir else 1) |
|
if re(ndir).is_positive: |
|
if im(x0).is_positive: |
|
return self.func(x0) + pi |
|
elif re(ndir).is_negative: |
|
if im(x0).is_negative: |
|
return self.func(x0) - pi |
|
else: |
|
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir).expand() |
|
return self.func(x0) |
|
|
|
def _eval_nseries(self, x, n, logx, cdir=0): |
|
arg0 = self.args[0].subs(x, 0) |
|
|
|
|
|
if arg0 in (S.ImaginaryUnit, S.NegativeOne*S.ImaginaryUnit): |
|
return self.rewrite(log)._eval_nseries(x, n, logx=logx, cdir=cdir) |
|
|
|
res = super()._eval_nseries(x, n=n, logx=logx) |
|
if arg0 is S.ComplexInfinity: |
|
return res |
|
ndir = self.args[0].dir(x, cdir if cdir else 1) |
|
if arg0.is_zero: |
|
if re(ndir) < 0: |
|
return res - pi |
|
return res |
|
|
|
if arg0.is_imaginary and (1 + arg0**2).is_positive: |
|
if re(ndir).is_positive: |
|
if im(arg0).is_positive: |
|
return res + pi |
|
elif re(ndir).is_negative: |
|
if im(arg0).is_negative: |
|
return res - pi |
|
else: |
|
return self.rewrite(log)._eval_nseries(x, n, logx=logx, cdir=cdir) |
|
return res |
|
|
|
def _eval_aseries(self, n, args0, x, logx): |
|
if args0[0] in [S.Infinity, S.NegativeInfinity]: |
|
return atan(1/self.args[0])._eval_nseries(x, n, logx) |
|
else: |
|
return super()._eval_aseries(n, args0, x, logx) |
|
|
|
def _eval_rewrite_as_log(self, x, **kwargs): |
|
return S.ImaginaryUnit/2*(log(1 - S.ImaginaryUnit/x) |
|
- log(1 + S.ImaginaryUnit/x)) |
|
|
|
_eval_rewrite_as_tractable = _eval_rewrite_as_log |
|
|
|
def inverse(self, argindex=1): |
|
""" |
|
Returns the inverse of this function. |
|
""" |
|
return cot |
|
|
|
def _eval_rewrite_as_asin(self, arg, **kwargs): |
|
return (arg*sqrt(1/arg**2)* |
|
(pi/2 - asin(sqrt(-arg**2)/sqrt(-arg**2 - 1)))) |
|
|
|
def _eval_rewrite_as_acos(self, arg, **kwargs): |
|
return arg*sqrt(1/arg**2)*acos(sqrt(-arg**2)/sqrt(-arg**2 - 1)) |
|
|
|
def _eval_rewrite_as_atan(self, arg, **kwargs): |
|
return atan(1/arg) |
|
|
|
def _eval_rewrite_as_asec(self, arg, **kwargs): |
|
return arg*sqrt(1/arg**2)*asec(sqrt((1 + arg**2)/arg**2)) |
|
|
|
def _eval_rewrite_as_acsc(self, arg, **kwargs): |
|
return arg*sqrt(1/arg**2)*(pi/2 - acsc(sqrt((1 + arg**2)/arg**2))) |
|
|
|
|
|
class asec(InverseTrigonometricFunction): |
|
r""" |
|
The inverse secant function. |
|
|
|
Returns the arc secant of x (measured in radians). |
|
|
|
Explanation |
|
=========== |
|
|
|
``asec(x)`` will evaluate automatically in the cases |
|
$x \in \{\infty, -\infty, 0, 1, -1\}$ and for some instances when the |
|
result is a rational multiple of $\pi$ (see the eval class method). |
|
|
|
``asec(x)`` has branch cut in the interval $[-1, 1]$. For complex arguments, |
|
it can be defined [4]_ as |
|
|
|
.. math:: |
|
\operatorname{sec^{-1}}(z) = -i\frac{\log\left(\sqrt{1 - z^2} + 1\right)}{z} |
|
|
|
At ``x = 0``, for positive branch cut, the limit evaluates to ``zoo``. For |
|
negative branch cut, the limit |
|
|
|
.. math:: |
|
\lim_{z \to 0}-i\frac{\log\left(-\sqrt{1 - z^2} + 1\right)}{z} |
|
|
|
simplifies to :math:`-i\log\left(z/2 + O\left(z^3\right)\right)` which |
|
ultimately evaluates to ``zoo``. |
|
|
|
As ``acos(x) = asec(1/x)``, a similar argument can be given for |
|
``acos(x)``. |
|
|
|
Examples |
|
======== |
|
|
|
>>> from sympy import asec, oo |
|
>>> asec(1) |
|
0 |
|
>>> asec(-1) |
|
pi |
|
>>> asec(0) |
|
zoo |
|
>>> asec(-oo) |
|
pi/2 |
|
|
|
See Also |
|
======== |
|
|
|
sin, csc, cos, sec, tan, cot |
|
asin, acsc, acos, atan, acot, atan2 |
|
|
|
References |
|
========== |
|
|
|
.. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions |
|
.. [2] https://dlmf.nist.gov/4.23 |
|
.. [3] https://functions.wolfram.com/ElementaryFunctions/ArcSec |
|
.. [4] https://reference.wolfram.com/language/ref/ArcSec.html |
|
|
|
""" |
|
|
|
@classmethod |
|
def eval(cls, arg): |
|
if arg.is_zero: |
|
return S.ComplexInfinity |
|
if arg.is_Number: |
|
if arg is S.NaN: |
|
return S.NaN |
|
elif arg is S.One: |
|
return S.Zero |
|
elif arg is S.NegativeOne: |
|
return pi |
|
if arg in [S.Infinity, S.NegativeInfinity, S.ComplexInfinity]: |
|
return pi/2 |
|
|
|
if arg.is_number: |
|
acsc_table = cls._acsc_table() |
|
if arg in acsc_table: |
|
return pi/2 - acsc_table[arg] |
|
elif -arg in acsc_table: |
|
return pi/2 + acsc_table[-arg] |
|
|
|
if arg.is_infinite: |
|
return pi/2 |
|
|
|
if arg.is_Mul and len(arg.args) == 2 and arg.args[0] == -1: |
|
narg = arg.args[1] |
|
minus = True |
|
else: |
|
narg = arg |
|
minus = False |
|
|
|
if isinstance(narg, sec): |
|
|
|
ang = narg.args[0] |
|
if ang.is_comparable: |
|
if minus: |
|
ang = pi - ang |
|
ang %= 2*pi |
|
if ang > pi: |
|
ang = 2*pi - ang |
|
return ang |
|
|
|
if isinstance(narg, csc): |
|
ang = narg.args[0] |
|
if ang.is_comparable: |
|
if minus: |
|
pi/2 + acsc(narg) |
|
return pi/2 - acsc(narg) |
|
|
|
def fdiff(self, argindex=1): |
|
if argindex == 1: |
|
return 1/(self.args[0]**2*sqrt(1 - 1/self.args[0]**2)) |
|
else: |
|
raise ArgumentIndexError(self, argindex) |
|
|
|
def inverse(self, argindex=1): |
|
""" |
|
Returns the inverse of this function. |
|
""" |
|
return sec |
|
|
|
@staticmethod |
|
@cacheit |
|
def taylor_term(n, x, *previous_terms): |
|
if n == 0: |
|
return S.ImaginaryUnit*log(2 / x) |
|
elif n < 0 or n % 2 == 1: |
|
return S.Zero |
|
else: |
|
x = sympify(x) |
|
if len(previous_terms) > 2 and n > 2: |
|
p = previous_terms[-2] |
|
return p * ((n - 1)*(n-2)) * x**2/(4 * (n//2)**2) |
|
else: |
|
k = n // 2 |
|
R = RisingFactorial(S.Half, k) * n |
|
F = factorial(k) * n // 2 * n // 2 |
|
return -S.ImaginaryUnit * R / F * x**n / 4 |
|
|
|
def _eval_as_leading_term(self, x, logx, cdir): |
|
arg = self.args[0] |
|
x0 = arg.subs(x, 0).cancel() |
|
if x0 is S.NaN: |
|
return self.func(arg.as_leading_term(x)) |
|
|
|
if x0 == 1: |
|
return sqrt(2)*sqrt((arg - S.One).as_leading_term(x)) |
|
if x0 in (-S.One, S.Zero): |
|
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir) |
|
|
|
if x0.is_real and (1 - x0**2).is_positive: |
|
ndir = arg.dir(x, cdir if cdir else 1) |
|
if im(ndir).is_negative: |
|
if x0.is_positive: |
|
return -self.func(x0) |
|
elif im(ndir).is_positive: |
|
if x0.is_negative: |
|
return 2*pi - self.func(x0) |
|
else: |
|
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir).expand() |
|
return self.func(x0) |
|
|
|
def _eval_nseries(self, x, n, logx, cdir=0): |
|
from sympy.series.order import O |
|
arg0 = self.args[0].subs(x, 0) |
|
|
|
if arg0 is S.One: |
|
t = Dummy('t', positive=True) |
|
ser = asec(S.One + t**2).rewrite(log).nseries(t, 0, 2*n) |
|
arg1 = S.NegativeOne + self.args[0] |
|
f = arg1.as_leading_term(x) |
|
g = (arg1 - f)/ f |
|
res1 = sqrt(S.One + g)._eval_nseries(x, n=n, logx=logx) |
|
res = (res1.removeO()*sqrt(f)).expand() |
|
return ser.removeO().subs(t, res).expand().powsimp() + O(x**n, x) |
|
|
|
if arg0 is S.NegativeOne: |
|
t = Dummy('t', positive=True) |
|
ser = asec(S.NegativeOne - t**2).rewrite(log).nseries(t, 0, 2*n) |
|
arg1 = S.NegativeOne - self.args[0] |
|
f = arg1.as_leading_term(x) |
|
g = (arg1 - f)/ f |
|
res1 = sqrt(S.One + g)._eval_nseries(x, n=n, logx=logx) |
|
res = (res1.removeO()*sqrt(f)).expand() |
|
return ser.removeO().subs(t, res).expand().powsimp() + O(x**n, x) |
|
|
|
res = super()._eval_nseries(x, n=n, logx=logx) |
|
if arg0 is S.ComplexInfinity: |
|
return res |
|
|
|
if arg0.is_real and (1 - arg0**2).is_positive: |
|
ndir = self.args[0].dir(x, cdir if cdir else 1) |
|
if im(ndir).is_negative: |
|
if arg0.is_positive: |
|
return -res |
|
elif im(ndir).is_positive: |
|
if arg0.is_negative: |
|
return 2*pi - res |
|
else: |
|
return self.rewrite(log)._eval_nseries(x, n, logx=logx, cdir=cdir) |
|
return res |
|
|
|
def _eval_is_extended_real(self): |
|
x = self.args[0] |
|
if x.is_extended_real is False: |
|
return False |
|
return fuzzy_or(((x - 1).is_nonnegative, (-x - 1).is_nonnegative)) |
|
|
|
def _eval_rewrite_as_log(self, arg, **kwargs): |
|
return pi/2 + S.ImaginaryUnit*log(S.ImaginaryUnit/arg + sqrt(1 - 1/arg**2)) |
|
|
|
_eval_rewrite_as_tractable = _eval_rewrite_as_log |
|
|
|
def _eval_rewrite_as_asin(self, arg, **kwargs): |
|
return pi/2 - asin(1/arg) |
|
|
|
def _eval_rewrite_as_acos(self, arg, **kwargs): |
|
return acos(1/arg) |
|
|
|
def _eval_rewrite_as_atan(self, x, **kwargs): |
|
sx2x = sqrt(x**2)/x |
|
return pi/2*(1 - sx2x) + sx2x*atan(sqrt(x**2 - 1)) |
|
|
|
def _eval_rewrite_as_acot(self, x, **kwargs): |
|
sx2x = sqrt(x**2)/x |
|
return pi/2*(1 - sx2x) + sx2x*acot(1/sqrt(x**2 - 1)) |
|
|
|
def _eval_rewrite_as_acsc(self, arg, **kwargs): |
|
return pi/2 - acsc(arg) |
|
|
|
|
|
class acsc(InverseTrigonometricFunction): |
|
r""" |
|
The inverse cosecant function. |
|
|
|
Returns the arc cosecant of x (measured in radians). |
|
|
|
Explanation |
|
=========== |
|
|
|
``acsc(x)`` will evaluate automatically in the cases |
|
$x \in \{\infty, -\infty, 0, 1, -1\}$` and for some instances when the |
|
result is a rational multiple of $\pi$ (see the ``eval`` class method). |
|
|
|
Examples |
|
======== |
|
|
|
>>> from sympy import acsc, oo |
|
>>> acsc(1) |
|
pi/2 |
|
>>> acsc(-1) |
|
-pi/2 |
|
>>> acsc(oo) |
|
0 |
|
>>> acsc(-oo) == acsc(oo) |
|
True |
|
>>> acsc(0) |
|
zoo |
|
|
|
See Also |
|
======== |
|
|
|
sin, csc, cos, sec, tan, cot |
|
asin, acos, asec, atan, acot, atan2 |
|
|
|
References |
|
========== |
|
|
|
.. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions |
|
.. [2] https://dlmf.nist.gov/4.23 |
|
.. [3] https://functions.wolfram.com/ElementaryFunctions/ArcCsc |
|
|
|
""" |
|
|
|
@classmethod |
|
def eval(cls, arg): |
|
if arg.is_zero: |
|
return S.ComplexInfinity |
|
if arg.is_Number: |
|
if arg is S.NaN: |
|
return S.NaN |
|
elif arg is S.One: |
|
return pi/2 |
|
elif arg is S.NegativeOne: |
|
return -pi/2 |
|
if arg in [S.Infinity, S.NegativeInfinity, S.ComplexInfinity]: |
|
return S.Zero |
|
|
|
if arg.could_extract_minus_sign(): |
|
return -cls(-arg) |
|
|
|
if arg.is_infinite: |
|
return S.Zero |
|
|
|
if arg.is_number: |
|
acsc_table = cls._acsc_table() |
|
if arg in acsc_table: |
|
return acsc_table[arg] |
|
|
|
if isinstance(arg, csc): |
|
ang = arg.args[0] |
|
if ang.is_comparable: |
|
ang %= 2*pi |
|
if ang > pi: |
|
ang = pi - ang |
|
|
|
|
|
if ang > pi/2: |
|
ang = pi - ang |
|
if ang < -pi/2: |
|
ang = -pi - ang |
|
|
|
return ang |
|
|
|
if isinstance(arg, sec): |
|
ang = arg.args[0] |
|
if ang.is_comparable: |
|
return pi/2 - asec(arg) |
|
|
|
def fdiff(self, argindex=1): |
|
if argindex == 1: |
|
return -1/(self.args[0]**2*sqrt(1 - 1/self.args[0]**2)) |
|
else: |
|
raise ArgumentIndexError(self, argindex) |
|
|
|
def inverse(self, argindex=1): |
|
""" |
|
Returns the inverse of this function. |
|
""" |
|
return csc |
|
|
|
@staticmethod |
|
@cacheit |
|
def taylor_term(n, x, *previous_terms): |
|
if n == 0: |
|
return pi/2 - S.ImaginaryUnit*log(2) + S.ImaginaryUnit*log(x) |
|
elif n < 0 or n % 2 == 1: |
|
return S.Zero |
|
else: |
|
x = sympify(x) |
|
if len(previous_terms) > 2 and n > 2: |
|
p = previous_terms[-2] |
|
return p * ((n - 1)*(n-2)) * x**2/(4 * (n//2)**2) |
|
else: |
|
k = n // 2 |
|
R = RisingFactorial(S.Half, k) * n |
|
F = factorial(k) * n // 2 * n // 2 |
|
return S.ImaginaryUnit * R / F * x**n / 4 |
|
|
|
def _eval_as_leading_term(self, x, logx, cdir): |
|
arg = self.args[0] |
|
x0 = arg.subs(x, 0).cancel() |
|
if x0 is S.NaN: |
|
return self.func(arg.as_leading_term(x)) |
|
|
|
if x0 in (-S.One, S.One, S.Zero): |
|
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir).expand() |
|
if x0 is S.ComplexInfinity: |
|
return (1/arg).as_leading_term(x) |
|
|
|
if x0.is_real and (1 - x0**2).is_positive: |
|
ndir = arg.dir(x, cdir if cdir else 1) |
|
if im(ndir).is_negative: |
|
if x0.is_positive: |
|
return pi - self.func(x0) |
|
elif im(ndir).is_positive: |
|
if x0.is_negative: |
|
return -pi - self.func(x0) |
|
else: |
|
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir).expand() |
|
return self.func(x0) |
|
|
|
def _eval_nseries(self, x, n, logx, cdir=0): |
|
from sympy.series.order import O |
|
arg0 = self.args[0].subs(x, 0) |
|
|
|
if arg0 is S.One: |
|
t = Dummy('t', positive=True) |
|
ser = acsc(S.One + t**2).rewrite(log).nseries(t, 0, 2*n) |
|
arg1 = S.NegativeOne + self.args[0] |
|
f = arg1.as_leading_term(x) |
|
g = (arg1 - f)/ f |
|
res1 = sqrt(S.One + g)._eval_nseries(x, n=n, logx=logx) |
|
res = (res1.removeO()*sqrt(f)).expand() |
|
return ser.removeO().subs(t, res).expand().powsimp() + O(x**n, x) |
|
|
|
if arg0 is S.NegativeOne: |
|
t = Dummy('t', positive=True) |
|
ser = acsc(S.NegativeOne - t**2).rewrite(log).nseries(t, 0, 2*n) |
|
arg1 = S.NegativeOne - self.args[0] |
|
f = arg1.as_leading_term(x) |
|
g = (arg1 - f)/ f |
|
res1 = sqrt(S.One + g)._eval_nseries(x, n=n, logx=logx) |
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res = (res1.removeO()*sqrt(f)).expand() |
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return ser.removeO().subs(t, res).expand().powsimp() + O(x**n, x) |
|
|
|
res = super()._eval_nseries(x, n=n, logx=logx) |
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if arg0 is S.ComplexInfinity: |
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return res |
|
|
|
if arg0.is_real and (1 - arg0**2).is_positive: |
|
ndir = self.args[0].dir(x, cdir if cdir else 1) |
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if im(ndir).is_negative: |
|
if arg0.is_positive: |
|
return pi - res |
|
elif im(ndir).is_positive: |
|
if arg0.is_negative: |
|
return -pi - res |
|
else: |
|
return self.rewrite(log)._eval_nseries(x, n, logx=logx, cdir=cdir) |
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return res |
|
|
|
def _eval_rewrite_as_log(self, arg, **kwargs): |
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return -S.ImaginaryUnit*log(S.ImaginaryUnit/arg + sqrt(1 - 1/arg**2)) |
|
|
|
_eval_rewrite_as_tractable = _eval_rewrite_as_log |
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|
|
def _eval_rewrite_as_asin(self, arg, **kwargs): |
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return asin(1/arg) |
|
|
|
def _eval_rewrite_as_acos(self, arg, **kwargs): |
|
return pi/2 - acos(1/arg) |
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|
|
def _eval_rewrite_as_atan(self, x, **kwargs): |
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return sqrt(x**2)/x*(pi/2 - atan(sqrt(x**2 - 1))) |
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|
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def _eval_rewrite_as_acot(self, arg, **kwargs): |
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return sqrt(arg**2)/arg*(pi/2 - acot(1/sqrt(arg**2 - 1))) |
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|
|
def _eval_rewrite_as_asec(self, arg, **kwargs): |
|
return pi/2 - asec(arg) |
|
|
|
|
|
class atan2(InverseTrigonometricFunction): |
|
r""" |
|
The function ``atan2(y, x)`` computes `\operatorname{atan}(y/x)` taking |
|
two arguments `y` and `x`. Signs of both `y` and `x` are considered to |
|
determine the appropriate quadrant of `\operatorname{atan}(y/x)`. |
|
The range is `(-\pi, \pi]`. The complete definition reads as follows: |
|
|
|
.. math:: |
|
|
|
\operatorname{atan2}(y, x) = |
|
\begin{cases} |
|
\arctan\left(\frac y x\right) & \qquad x > 0 \\ |
|
\arctan\left(\frac y x\right) + \pi& \qquad y \ge 0, x < 0 \\ |
|
\arctan\left(\frac y x\right) - \pi& \qquad y < 0, x < 0 \\ |
|
+\frac{\pi}{2} & \qquad y > 0, x = 0 \\ |
|
-\frac{\pi}{2} & \qquad y < 0, x = 0 \\ |
|
\text{undefined} & \qquad y = 0, x = 0 |
|
\end{cases} |
|
|
|
Attention: Note the role reversal of both arguments. The `y`-coordinate |
|
is the first argument and the `x`-coordinate the second. |
|
|
|
If either `x` or `y` is complex: |
|
|
|
.. math:: |
|
|
|
\operatorname{atan2}(y, x) = |
|
-i\log\left(\frac{x + iy}{\sqrt{x^2 + y^2}}\right) |
|
|
|
Examples |
|
======== |
|
|
|
Going counter-clock wise around the origin we find the |
|
following angles: |
|
|
|
>>> from sympy import atan2 |
|
>>> atan2(0, 1) |
|
0 |
|
>>> atan2(1, 1) |
|
pi/4 |
|
>>> atan2(1, 0) |
|
pi/2 |
|
>>> atan2(1, -1) |
|
3*pi/4 |
|
>>> atan2(0, -1) |
|
pi |
|
>>> atan2(-1, -1) |
|
-3*pi/4 |
|
>>> atan2(-1, 0) |
|
-pi/2 |
|
>>> atan2(-1, 1) |
|
-pi/4 |
|
|
|
which are all correct. Compare this to the results of the ordinary |
|
`\operatorname{atan}` function for the point `(x, y) = (-1, 1)` |
|
|
|
>>> from sympy import atan, S |
|
>>> atan(S(1)/-1) |
|
-pi/4 |
|
>>> atan2(1, -1) |
|
3*pi/4 |
|
|
|
where only the `\operatorname{atan2}` function returns what we expect. |
|
We can differentiate the function with respect to both arguments: |
|
|
|
>>> from sympy import diff |
|
>>> from sympy.abc import x, y |
|
>>> diff(atan2(y, x), x) |
|
-y/(x**2 + y**2) |
|
|
|
>>> diff(atan2(y, x), y) |
|
x/(x**2 + y**2) |
|
|
|
We can express the `\operatorname{atan2}` function in terms of |
|
complex logarithms: |
|
|
|
>>> from sympy import log |
|
>>> atan2(y, x).rewrite(log) |
|
-I*log((x + I*y)/sqrt(x**2 + y**2)) |
|
|
|
and in terms of `\operatorname(atan)`: |
|
|
|
>>> from sympy import atan |
|
>>> atan2(y, x).rewrite(atan) |
|
Piecewise((2*atan(y/(x + sqrt(x**2 + y**2))), Ne(y, 0)), (pi, re(x) < 0), (0, Ne(x, 0)), (nan, True)) |
|
|
|
but note that this form is undefined on the negative real axis. |
|
|
|
See Also |
|
======== |
|
|
|
sin, csc, cos, sec, tan, cot |
|
asin, acsc, acos, asec, atan, acot |
|
|
|
References |
|
========== |
|
|
|
.. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions |
|
.. [2] https://en.wikipedia.org/wiki/Atan2 |
|
.. [3] https://functions.wolfram.com/ElementaryFunctions/ArcTan2 |
|
|
|
""" |
|
|
|
@classmethod |
|
def eval(cls, y, x): |
|
from sympy.functions.special.delta_functions import Heaviside |
|
if x is S.NegativeInfinity: |
|
if y.is_zero: |
|
|
|
return pi |
|
return 2*pi*(Heaviside(re(y))) - pi |
|
elif x is S.Infinity: |
|
return S.Zero |
|
elif x.is_imaginary and y.is_imaginary and x.is_number and y.is_number: |
|
x = im(x) |
|
y = im(y) |
|
|
|
if x.is_extended_real and y.is_extended_real: |
|
if x.is_positive: |
|
return atan(y/x) |
|
elif x.is_negative: |
|
if y.is_negative: |
|
return atan(y/x) - pi |
|
elif y.is_nonnegative: |
|
return atan(y/x) + pi |
|
elif x.is_zero: |
|
if y.is_positive: |
|
return pi/2 |
|
elif y.is_negative: |
|
return -pi/2 |
|
elif y.is_zero: |
|
return S.NaN |
|
if y.is_zero: |
|
if x.is_extended_nonzero: |
|
return pi*(S.One - Heaviside(x)) |
|
if x.is_number: |
|
return Piecewise((pi, re(x) < 0), |
|
(0, Ne(x, 0)), |
|
(S.NaN, True)) |
|
if x.is_number and y.is_number: |
|
return -S.ImaginaryUnit*log( |
|
(x + S.ImaginaryUnit*y)/sqrt(x**2 + y**2)) |
|
|
|
def _eval_rewrite_as_log(self, y, x, **kwargs): |
|
return -S.ImaginaryUnit*log((x + S.ImaginaryUnit*y)/sqrt(x**2 + y**2)) |
|
|
|
def _eval_rewrite_as_atan(self, y, x, **kwargs): |
|
return Piecewise((2*atan(y/(x + sqrt(x**2 + y**2))), Ne(y, 0)), |
|
(pi, re(x) < 0), |
|
(0, Ne(x, 0)), |
|
(S.NaN, True)) |
|
|
|
def _eval_rewrite_as_arg(self, y, x, **kwargs): |
|
if x.is_extended_real and y.is_extended_real: |
|
return arg_f(x + y*S.ImaginaryUnit) |
|
n = x + S.ImaginaryUnit*y |
|
d = x**2 + y**2 |
|
return arg_f(n/sqrt(d)) - S.ImaginaryUnit*log(abs(n)/sqrt(abs(d))) |
|
|
|
def _eval_is_extended_real(self): |
|
return self.args[0].is_extended_real and self.args[1].is_extended_real |
|
|
|
def _eval_conjugate(self): |
|
return self.func(self.args[0].conjugate(), self.args[1].conjugate()) |
|
|
|
def fdiff(self, argindex): |
|
y, x = self.args |
|
if argindex == 1: |
|
|
|
return x/(x**2 + y**2) |
|
elif argindex == 2: |
|
|
|
return -y/(x**2 + y**2) |
|
else: |
|
raise ArgumentIndexError(self, argindex) |
|
|
|
def _eval_evalf(self, prec): |
|
y, x = self.args |
|
if x.is_extended_real and y.is_extended_real: |
|
return super()._eval_evalf(prec) |
|
|