446 lines
14 KiB
Python
446 lines
14 KiB
Python
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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 Function, ArgumentIndexError
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from sympy.core.symbol import Dummy
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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(Function):
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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):
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from sympy.integrals.integrals import Integral
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t = Dummy('t')
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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(Function):
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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):
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from sympy.integrals.integrals import Integral
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t = Dummy('t')
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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(Function):
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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):
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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('t')
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return Integral(sqrt(1 - m*sin(t)**2), (t, 0, z))
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class elliptic_pi(Function):
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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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n, z, m = n, 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:
|
||
|
|
return (elliptic_f(z, m) +
|
||
|
|
(sqrt(1 - m*sin(z)**2)*tan(z) -
|
||
|
|
elliptic_e(z, m))/(1 - m))
|
||
|
|
k = 2*z/pi
|
||
|
|
if k.is_integer:
|
||
|
|
return k*elliptic_pi(n, m)
|
||
|
|
elif m.is_zero:
|
||
|
|
return atanh(sqrt(n - 1)*tan(z))/sqrt(n - 1)
|
||
|
|
elif n == m:
|
||
|
|
return (elliptic_f(z, n) - elliptic_pi(1, z, n) +
|
||
|
|
tan(z)/sqrt(1 - n*sin(z)**2))
|
||
|
|
elif n in (S.Infinity, S.NegativeInfinity):
|
||
|
|
return S.Zero
|
||
|
|
elif m in (S.Infinity, S.NegativeInfinity):
|
||
|
|
return S.Zero
|
||
|
|
elif z.could_extract_minus_sign():
|
||
|
|
return -elliptic_pi(n, -z, m)
|
||
|
|
if n.is_zero:
|
||
|
|
return elliptic_f(z, m)
|
||
|
|
if m.is_extended_real and m.is_infinite or \
|
||
|
|
n.is_extended_real and n.is_infinite:
|
||
|
|
return S.Zero
|
||
|
|
else:
|
||
|
|
if n.is_zero:
|
||
|
|
return elliptic_k(m)
|
||
|
|
elif n is S.One:
|
||
|
|
return S.ComplexInfinity
|
||
|
|
elif m.is_zero:
|
||
|
|
return pi/(2*sqrt(1 - n))
|
||
|
|
elif m == S.One:
|
||
|
|
return S.NegativeInfinity/sign(n - 1)
|
||
|
|
elif n == m:
|
||
|
|
return elliptic_e(n)/(1 - n)
|
||
|
|
elif n in (S.Infinity, S.NegativeInfinity):
|
||
|
|
return S.Zero
|
||
|
|
elif m in (S.Infinity, S.NegativeInfinity):
|
||
|
|
return S.Zero
|
||
|
|
if n.is_zero:
|
||
|
|
return elliptic_k(m)
|
||
|
|
if m.is_extended_real and m.is_infinite or \
|
||
|
|
n.is_extended_real and n.is_infinite:
|
||
|
|
return S.Zero
|
||
|
|
|
||
|
|
def _eval_conjugate(self):
|
||
|
|
if len(self.args) == 3:
|
||
|
|
n, z, m = self.args
|
||
|
|
if (n.is_real and (n - 1).is_positive) is False and \
|
||
|
|
(m.is_real and (m - 1).is_positive) is False:
|
||
|
|
return self.func(n.conjugate(), z.conjugate(), m.conjugate())
|
||
|
|
else:
|
||
|
|
n, m = self.args
|
||
|
|
return self.func(n.conjugate(), m.conjugate())
|
||
|
|
|
||
|
|
def fdiff(self, argindex=1):
|
||
|
|
if len(self.args) == 3:
|
||
|
|
n, z, m = self.args
|
||
|
|
fm, fn = sqrt(1 - m*sin(z)**2), 1 - n*sin(z)**2
|
||
|
|
if argindex == 1:
|
||
|
|
return (elliptic_e(z, m) + (m - n)*elliptic_f(z, m)/n +
|
||
|
|
(n**2 - m)*elliptic_pi(n, z, m)/n -
|
||
|
|
n*fm*sin(2*z)/(2*fn))/(2*(m - n)*(n - 1))
|
||
|
|
elif argindex == 2:
|
||
|
|
return 1/(fm*fn)
|
||
|
|
elif argindex == 3:
|
||
|
|
return (elliptic_e(z, m)/(m - 1) +
|
||
|
|
elliptic_pi(n, z, m) -
|
||
|
|
m*sin(2*z)/(2*(m - 1)*fm))/(2*(n - m))
|
||
|
|
else:
|
||
|
|
n, m = self.args
|
||
|
|
if argindex == 1:
|
||
|
|
return (elliptic_e(m) + (m - n)*elliptic_k(m)/n +
|
||
|
|
(n**2 - m)*elliptic_pi(n, m)/n)/(2*(m - n)*(n - 1))
|
||
|
|
elif argindex == 2:
|
||
|
|
return (elliptic_e(m)/(m - 1) + elliptic_pi(n, m))/(2*(n - m))
|
||
|
|
raise ArgumentIndexError(self, argindex)
|
||
|
|
|
||
|
|
def _eval_rewrite_as_Integral(self, *args):
|
||
|
|
from sympy.integrals.integrals import Integral
|
||
|
|
if len(self.args) == 2:
|
||
|
|
n, m, z = self.args[0], self.args[1], pi/2
|
||
|
|
else:
|
||
|
|
n, z, m = self.args
|
||
|
|
t = Dummy('t')
|
||
|
|
return Integral(1/((1 - n*sin(t)**2)*sqrt(1 - m*sin(t)**2)), (t, 0, z))
|