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""" Elliptic Integrals. """ |
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from sympy.core import S, pi, I, Rational |
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from sympy.core.function import DefinedFunction, ArgumentIndexError |
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from sympy.core.symbol import Dummy,uniquely_named_symbol |
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from sympy.functions.elementary.complexes import sign |
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from sympy.functions.elementary.hyperbolic import atanh |
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from sympy.functions.elementary.miscellaneous import sqrt |
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from sympy.functions.elementary.trigonometric import sin, tan |
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from sympy.functions.special.gamma_functions import gamma |
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from sympy.functions.special.hyper import hyper, meijerg |
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class elliptic_k(DefinedFunction): |
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r""" |
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The complete elliptic integral of the first kind, defined by |
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.. math:: K(m) = F\left(\tfrac{\pi}{2}\middle| m\right) |
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where $F\left(z\middle| m\right)$ is the Legendre incomplete |
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elliptic integral of the first kind. |
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Explanation |
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=========== |
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The function $K(m)$ is a single-valued function on the complex |
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plane with branch cut along the interval $(1, \infty)$. |
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Note that our notation defines the incomplete elliptic integral |
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in terms of the parameter $m$ instead of the elliptic modulus |
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(eccentricity) $k$. |
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In this case, the parameter $m$ is defined as $m=k^2$. |
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Examples |
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======== |
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>>> from sympy import elliptic_k, I |
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>>> from sympy.abc import m |
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>>> elliptic_k(0) |
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pi/2 |
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>>> elliptic_k(1.0 + I) |
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1.50923695405127 + 0.625146415202697*I |
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>>> elliptic_k(m).series(n=3) |
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pi/2 + pi*m/8 + 9*pi*m**2/128 + O(m**3) |
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See Also |
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======== |
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elliptic_f |
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References |
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========== |
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.. [1] https://en.wikipedia.org/wiki/Elliptic_integrals |
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.. [2] https://functions.wolfram.com/EllipticIntegrals/EllipticK |
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""" |
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@classmethod |
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def eval(cls, m): |
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if m.is_zero: |
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return pi*S.Half |
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elif m is S.Half: |
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return 8*pi**Rational(3, 2)/gamma(Rational(-1, 4))**2 |
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elif m is S.One: |
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return S.ComplexInfinity |
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elif m is S.NegativeOne: |
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return gamma(Rational(1, 4))**2/(4*sqrt(2*pi)) |
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elif m in (S.Infinity, S.NegativeInfinity, I*S.Infinity, |
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I*S.NegativeInfinity, S.ComplexInfinity): |
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return S.Zero |
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def fdiff(self, argindex=1): |
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m = self.args[0] |
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return (elliptic_e(m) - (1 - m)*elliptic_k(m))/(2*m*(1 - m)) |
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def _eval_conjugate(self): |
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m = self.args[0] |
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if (m.is_real and (m - 1).is_positive) is False: |
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return self.func(m.conjugate()) |
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def _eval_nseries(self, x, n, logx, cdir=0): |
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from sympy.simplify import hyperexpand |
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return hyperexpand(self.rewrite(hyper)._eval_nseries(x, n=n, logx=logx)) |
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def _eval_rewrite_as_hyper(self, m, **kwargs): |
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return pi*S.Half*hyper((S.Half, S.Half), (S.One,), m) |
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def _eval_rewrite_as_meijerg(self, m, **kwargs): |
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return meijerg(((S.Half, S.Half), []), ((S.Zero,), (S.Zero,)), -m)/2 |
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def _eval_is_zero(self): |
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m = self.args[0] |
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if m.is_infinite: |
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return True |
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def _eval_rewrite_as_Integral(self, *args, **kwargs): |
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from sympy.integrals.integrals import Integral |
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t = Dummy(uniquely_named_symbol('t', args).name) |
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m = self.args[0] |
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return Integral(1/sqrt(1 - m*sin(t)**2), (t, 0, pi/2)) |
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class elliptic_f(DefinedFunction): |
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r""" |
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The Legendre incomplete elliptic integral of the first |
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kind, defined by |
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.. math:: F\left(z\middle| m\right) = |
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\int_0^z \frac{dt}{\sqrt{1 - m \sin^2 t}} |
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Explanation |
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=========== |
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This function reduces to a complete elliptic integral of |
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the first kind, $K(m)$, when $z = \pi/2$. |
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Note that our notation defines the incomplete elliptic integral |
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in terms of the parameter $m$ instead of the elliptic modulus |
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(eccentricity) $k$. |
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In this case, the parameter $m$ is defined as $m=k^2$. |
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Examples |
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======== |
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>>> from sympy import elliptic_f, I |
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>>> from sympy.abc import z, m |
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>>> elliptic_f(z, m).series(z) |
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z + z**5*(3*m**2/40 - m/30) + m*z**3/6 + O(z**6) |
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>>> elliptic_f(3.0 + I/2, 1.0 + I) |
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2.909449841483 + 1.74720545502474*I |
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See Also |
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======== |
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elliptic_k |
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References |
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========== |
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.. [1] https://en.wikipedia.org/wiki/Elliptic_integrals |
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.. [2] https://functions.wolfram.com/EllipticIntegrals/EllipticF |
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""" |
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@classmethod |
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def eval(cls, z, m): |
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if z.is_zero: |
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return S.Zero |
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if m.is_zero: |
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return z |
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k = 2*z/pi |
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if k.is_integer: |
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return k*elliptic_k(m) |
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elif m in (S.Infinity, S.NegativeInfinity): |
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return S.Zero |
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elif z.could_extract_minus_sign(): |
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return -elliptic_f(-z, m) |
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def fdiff(self, argindex=1): |
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z, m = self.args |
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fm = sqrt(1 - m*sin(z)**2) |
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if argindex == 1: |
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return 1/fm |
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elif argindex == 2: |
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return (elliptic_e(z, m)/(2*m*(1 - m)) - elliptic_f(z, m)/(2*m) - |
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sin(2*z)/(4*(1 - m)*fm)) |
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raise ArgumentIndexError(self, argindex) |
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def _eval_conjugate(self): |
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z, m = self.args |
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if (m.is_real and (m - 1).is_positive) is False: |
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return self.func(z.conjugate(), m.conjugate()) |
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def _eval_rewrite_as_Integral(self, *args, **kwargs): |
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from sympy.integrals.integrals import Integral |
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t = Dummy(uniquely_named_symbol('t', args).name) |
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z, m = self.args[0], self.args[1] |
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return Integral(1/(sqrt(1 - m*sin(t)**2)), (t, 0, z)) |
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def _eval_is_zero(self): |
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z, m = self.args |
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if z.is_zero: |
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return True |
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if m.is_extended_real and m.is_infinite: |
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return True |
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class elliptic_e(DefinedFunction): |
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r""" |
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Called with two arguments $z$ and $m$, evaluates the |
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incomplete elliptic integral of the second kind, defined by |
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.. math:: E\left(z\middle| m\right) = \int_0^z \sqrt{1 - m \sin^2 t} dt |
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Called with a single argument $m$, evaluates the Legendre complete |
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elliptic integral of the second kind |
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.. math:: E(m) = E\left(\tfrac{\pi}{2}\middle| m\right) |
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Explanation |
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=========== |
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The function $E(m)$ is a single-valued function on the complex |
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plane with branch cut along the interval $(1, \infty)$. |
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Note that our notation defines the incomplete elliptic integral |
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in terms of the parameter $m$ instead of the elliptic modulus |
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(eccentricity) $k$. |
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In this case, the parameter $m$ is defined as $m=k^2$. |
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Examples |
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======== |
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>>> from sympy import elliptic_e, I |
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>>> from sympy.abc import z, m |
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>>> elliptic_e(z, m).series(z) |
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z + z**5*(-m**2/40 + m/30) - m*z**3/6 + O(z**6) |
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>>> elliptic_e(m).series(n=4) |
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pi/2 - pi*m/8 - 3*pi*m**2/128 - 5*pi*m**3/512 + O(m**4) |
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>>> elliptic_e(1 + I, 2 - I/2).n() |
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1.55203744279187 + 0.290764986058437*I |
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>>> elliptic_e(0) |
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pi/2 |
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>>> elliptic_e(2.0 - I) |
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0.991052601328069 + 0.81879421395609*I |
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References |
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========== |
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.. [1] https://en.wikipedia.org/wiki/Elliptic_integrals |
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.. [2] https://functions.wolfram.com/EllipticIntegrals/EllipticE2 |
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.. [3] https://functions.wolfram.com/EllipticIntegrals/EllipticE |
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""" |
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@classmethod |
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def eval(cls, m, z=None): |
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if z is not None: |
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z, m = m, z |
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k = 2*z/pi |
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if m.is_zero: |
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return z |
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if z.is_zero: |
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return S.Zero |
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elif k.is_integer: |
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return k*elliptic_e(m) |
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elif m in (S.Infinity, S.NegativeInfinity): |
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return S.ComplexInfinity |
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elif z.could_extract_minus_sign(): |
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return -elliptic_e(-z, m) |
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else: |
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if m.is_zero: |
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return pi/2 |
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elif m is S.One: |
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return S.One |
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elif m is S.Infinity: |
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return I*S.Infinity |
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elif m is S.NegativeInfinity: |
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return S.Infinity |
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elif m is S.ComplexInfinity: |
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return S.ComplexInfinity |
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def fdiff(self, argindex=1): |
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if len(self.args) == 2: |
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z, m = self.args |
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if argindex == 1: |
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return sqrt(1 - m*sin(z)**2) |
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elif argindex == 2: |
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return (elliptic_e(z, m) - elliptic_f(z, m))/(2*m) |
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else: |
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m = self.args[0] |
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if argindex == 1: |
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return (elliptic_e(m) - elliptic_k(m))/(2*m) |
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raise ArgumentIndexError(self, argindex) |
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def _eval_conjugate(self): |
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if len(self.args) == 2: |
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z, m = self.args |
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if (m.is_real and (m - 1).is_positive) is False: |
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return self.func(z.conjugate(), m.conjugate()) |
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else: |
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m = self.args[0] |
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if (m.is_real and (m - 1).is_positive) is False: |
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return self.func(m.conjugate()) |
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def _eval_nseries(self, x, n, logx, cdir=0): |
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from sympy.simplify import hyperexpand |
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if len(self.args) == 1: |
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return hyperexpand(self.rewrite(hyper)._eval_nseries(x, n=n, logx=logx)) |
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return super()._eval_nseries(x, n=n, logx=logx) |
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def _eval_rewrite_as_hyper(self, *args, **kwargs): |
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if len(args) == 1: |
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m = args[0] |
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return (pi/2)*hyper((Rational(-1, 2), S.Half), (S.One,), m) |
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def _eval_rewrite_as_meijerg(self, *args, **kwargs): |
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if len(args) == 1: |
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m = args[0] |
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return -meijerg(((S.Half, Rational(3, 2)), []), \ |
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((S.Zero,), (S.Zero,)), -m)/4 |
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def _eval_rewrite_as_Integral(self, *args, **kwargs): |
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from sympy.integrals.integrals import Integral |
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z, m = (pi/2, self.args[0]) if len(self.args) == 1 else self.args |
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t = Dummy(uniquely_named_symbol('t', args).name) |
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return Integral(sqrt(1 - m*sin(t)**2), (t, 0, z)) |
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class elliptic_pi(DefinedFunction): |
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r""" |
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Called with three arguments $n$, $z$ and $m$, evaluates the |
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Legendre incomplete elliptic integral of the third kind, defined by |
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.. math:: \Pi\left(n; z\middle| m\right) = \int_0^z \frac{dt} |
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{\left(1 - n \sin^2 t\right) \sqrt{1 - m \sin^2 t}} |
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Called with two arguments $n$ and $m$, evaluates the complete |
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elliptic integral of the third kind: |
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.. math:: \Pi\left(n\middle| m\right) = |
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\Pi\left(n; \tfrac{\pi}{2}\middle| m\right) |
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Explanation |
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=========== |
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Note that our notation defines the incomplete elliptic integral |
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in terms of the parameter $m$ instead of the elliptic modulus |
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(eccentricity) $k$. |
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In this case, the parameter $m$ is defined as $m=k^2$. |
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Examples |
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======== |
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>>> from sympy import elliptic_pi, I |
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>>> from sympy.abc import z, n, m |
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>>> elliptic_pi(n, z, m).series(z, n=4) |
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z + z**3*(m/6 + n/3) + O(z**4) |
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>>> elliptic_pi(0.5 + I, 1.0 - I, 1.2) |
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2.50232379629182 - 0.760939574180767*I |
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>>> elliptic_pi(0, 0) |
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pi/2 |
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>>> elliptic_pi(1.0 - I/3, 2.0 + I) |
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3.29136443417283 + 0.32555634906645*I |
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References |
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========== |
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.. [1] https://en.wikipedia.org/wiki/Elliptic_integrals |
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.. [2] https://functions.wolfram.com/EllipticIntegrals/EllipticPi3 |
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.. [3] https://functions.wolfram.com/EllipticIntegrals/EllipticPi |
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""" |
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@classmethod |
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def eval(cls, n, m, z=None): |
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if z is not None: |
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z, m = m, z |
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if n.is_zero: |
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return elliptic_f(z, m) |
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elif n is S.One: |
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return (elliptic_f(z, m) + |
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(sqrt(1 - m*sin(z)**2)*tan(z) - |
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elliptic_e(z, m))/(1 - m)) |
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k = 2*z/pi |
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if k.is_integer: |
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return k*elliptic_pi(n, m) |
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elif m.is_zero: |
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return atanh(sqrt(n - 1)*tan(z))/sqrt(n - 1) |
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elif n == m: |
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return (elliptic_f(z, n) - elliptic_pi(1, z, n) + |
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tan(z)/sqrt(1 - n*sin(z)**2)) |
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elif n in (S.Infinity, S.NegativeInfinity): |
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return S.Zero |
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elif m in (S.Infinity, S.NegativeInfinity): |
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return S.Zero |
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elif z.could_extract_minus_sign(): |
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return -elliptic_pi(n, -z, m) |
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if n.is_zero: |
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return elliptic_f(z, m) |
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if m.is_extended_real and m.is_infinite or \ |
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n.is_extended_real and n.is_infinite: |
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return S.Zero |
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else: |
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if n.is_zero: |
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return elliptic_k(m) |
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elif n is S.One: |
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return S.ComplexInfinity |
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elif m.is_zero: |
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return pi/(2*sqrt(1 - n)) |
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elif m == S.One: |
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return S.NegativeInfinity/sign(n - 1) |
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elif n == m: |
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return elliptic_e(n)/(1 - n) |
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elif n in (S.Infinity, S.NegativeInfinity): |
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return S.Zero |
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elif m in (S.Infinity, S.NegativeInfinity): |
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return S.Zero |
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if n.is_zero: |
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return elliptic_k(m) |
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if m.is_extended_real and m.is_infinite or \ |
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n.is_extended_real and n.is_infinite: |
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return S.Zero |
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def _eval_conjugate(self): |
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if len(self.args) == 3: |
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n, z, m = self.args |
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if (n.is_real and (n - 1).is_positive) is False and \ |
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(m.is_real and (m - 1).is_positive) is False: |
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return self.func(n.conjugate(), z.conjugate(), m.conjugate()) |
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else: |
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n, m = self.args |
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return self.func(n.conjugate(), m.conjugate()) |
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def fdiff(self, argindex=1): |
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if len(self.args) == 3: |
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n, z, m = self.args |
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fm, fn = sqrt(1 - m*sin(z)**2), 1 - n*sin(z)**2 |
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if argindex == 1: |
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return (elliptic_e(z, m) + (m - n)*elliptic_f(z, m)/n + |
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(n**2 - m)*elliptic_pi(n, z, m)/n - |
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n*fm*sin(2*z)/(2*fn))/(2*(m - n)*(n - 1)) |
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elif argindex == 2: |
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return 1/(fm*fn) |
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elif argindex == 3: |
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return (elliptic_e(z, m)/(m - 1) + |
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elliptic_pi(n, z, m) - |
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m*sin(2*z)/(2*(m - 1)*fm))/(2*(n - m)) |
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else: |
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n, m = self.args |
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if argindex == 1: |
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return (elliptic_e(m) + (m - n)*elliptic_k(m)/n + |
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(n**2 - m)*elliptic_pi(n, m)/n)/(2*(m - n)*(n - 1)) |
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elif argindex == 2: |
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return (elliptic_e(m)/(m - 1) + elliptic_pi(n, m))/(2*(n - m)) |
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raise ArgumentIndexError(self, argindex) |
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def _eval_rewrite_as_Integral(self, *args, **kwargs): |
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from sympy.integrals.integrals import Integral |
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if len(self.args) == 2: |
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n, m, z = self.args[0], self.args[1], pi/2 |
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else: |
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n, z, m = self.args |
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t = Dummy(uniquely_named_symbol('t', args).name) |
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return Integral(1/((1 - n*sin(t)**2)*sqrt(1 - m*sin(t)**2)), (t, 0, z)) |
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|
