ó ¡¼™\c@ sdZddlmZmZddlmZmZmZddlm Z m Z ddl m Z ddl mZddlmZddlmZmZdd lmZdd lmZmZd e fd „ƒYZd e fd„ƒYZde fd„ƒYZde fd„ƒYZdS(s Elliptic integrals. iÿÿÿÿ(tprint_functiontdivision(tStpitI(tFunctiontArgumentIndexError(tsign(tatanh(tsqrt(tsinttan(tgamma(thypertmeijergt elliptic_kcB sVeZdZed„ƒZdd„Zd„Zd„Zd„Zd„Z d„Z RS( s0 The complete elliptic integral of the first kind, defined by .. math:: K(m) = F\left(\tfrac{\pi}{2}\middle| m\right) where `F\left(z\middle| m\right)` is the Legendre incomplete elliptic integral of the first kind. The function `K(m)` is a single-valued function on the complex plane with branch cut along the interval `(1, \infty)`. Note that our notation defines the incomplete elliptic integral in terms of the parameter `m` instead of the elliptic modulus (eccentricity) `k`. In this case, the parameter `m` is defined as `m=k^2`. Examples ======== >>> from sympy import elliptic_k, I, pi >>> from sympy.abc import m >>> elliptic_k(0) pi/2 >>> elliptic_k(1.0 + I) 1.50923695405127 + 0.625146415202697*I >>> elliptic_k(m).series(n=3) pi/2 + pi*m/8 + 9*pi*m**2/128 + O(m**3) See Also ======== elliptic_f References ========== .. [1] https://en.wikipedia.org/wiki/Elliptic_integrals .. [2] http://functions.wolfram.com/EllipticIntegrals/EllipticK cC sá|tjkrtdS|tjkrUdttdƒdttdƒ dƒdS|tjkrktjS|tjkr¤ttdƒdƒddtdtƒS|tj tj t tj t tj tjfkrÝtjSdS(Niiiii( RtZeroRtHalfR tOnetComplexInfinityt NegativeOneR tInfinitytNegativeInfinityR(tclstm((sIlib/python2.7/site-packages/sympy/functions/special/elliptic_integrals.pyteval8s/*icC s9|jd}t|ƒd|t|ƒd|d|S(Niii(targst elliptic_eR(tselftargindexR((sIlib/python2.7/site-packages/sympy/functions/special/elliptic_integrals.pytfdiffFs cC s@|jd}|jo |djtkr<|j|jƒƒSdS(Nii(Rtis_realt is_positivetFalsetfunct conjugate(RR((sIlib/python2.7/site-packages/sympy/functions/special/elliptic_integrals.pyt_eval_conjugateJs cC s8ddlm}||jtƒj|d|d|ƒƒS(Niÿÿÿÿ(t hyperexpandtntlogx(tsympy.simplifyR%trewriteR t _eval_nseries(RtxR&R'R%((sIlib/python2.7/site-packages/sympy/functions/special/elliptic_integrals.pyR*OscK s*tdttjtjftjf|ƒS(Ni(RR RRR(RRtkwargs((sIlib/python2.7/site-packages/sympy/functions/special/elliptic_integrals.pyt_eval_rewrite_as_hyperSscK s9ttjtjfgftjftjff| ƒdS(Ni(RRRR(RRR,((sIlib/python2.7/site-packages/sympy/functions/special/elliptic_integrals.pyt_eval_rewrite_as_meijergVscC s)ddlj}|j|jdjƒƒS(Niÿÿÿÿi(tsage.alltallt elliptic_kcRt_sage_(Rtsage((sIlib/python2.7/site-packages/sympy/functions/special/elliptic_integrals.pyR2Ys( t__name__t __module__t__doc__t classmethodRRR$R*R-R.R2(((sIlib/python2.7/site-packages/sympy/functions/special/elliptic_integrals.pyRs(     t elliptic_fcB s2eZdZed„ƒZdd„Zd„ZRS(sÛ The Legendre incomplete elliptic integral of the first kind, defined by .. math:: F\left(z\middle| m\right) = \int_0^z \frac{dt}{\sqrt{1 - m \sin^2 t}} This function reduces to a complete elliptic integral of the first kind, `K(m)`, when `z = \pi/2`. Note that our notation defines the incomplete elliptic integral in terms of the parameter `m` instead of the elliptic modulus (eccentricity) `k`. In this case, the parameter `m` is defined as `m=k^2`. Examples ======== >>> from sympy import elliptic_f, I, O >>> from sympy.abc import z, m >>> elliptic_f(z, m).series(z) z + z**5*(3*m**2/40 - m/30) + m*z**3/6 + O(z**6) >>> elliptic_f(3.0 + I/2, 1.0 + I) 2.909449841483 + 1.74720545502474*I See Also ======== elliptic_k References ========== .. [1] https://en.wikipedia.org/wiki/Elliptic_integrals .. [2] http://functions.wolfram.com/EllipticIntegrals/EllipticF cC s€d|t}|jr|S|jr+tjS|jrB|t|ƒS|tjtjfkratjS|jƒr|t | |ƒ SdS(Ni( Rtis_zeroRRt is_integerRRRtcould_extract_minus_signR8(RtzRtk((sIlib/python2.7/site-packages/sympy/functions/special/elliptic_integrals.pyR…s    icC s°|j\}}td|t|ƒdƒ}|dkrAd|S|dkrt||ƒd|d|t||ƒd|td|ƒdd||St||ƒ‚dS(Niii(RR R RR8R(RRR<Rtfm((sIlib/python2.7/site-packages/sympy/functions/special/elliptic_integrals.pyR“s  1cC sK|j\}}|jo"|djtkrG|j|jƒ|jƒƒSdS(Ni(RRR R!R"R#(RR<R((sIlib/python2.7/site-packages/sympy/functions/special/elliptic_integrals.pyR$s(R4R5R6R7RRR$(((sIlib/python2.7/site-packages/sympy/functions/special/elliptic_integrals.pyR8^s% RcB sPeZdZedd„ƒZdd„Zd„Zd„Zd„Z d„Z RS( s” Called with two arguments `z` and `m`, evaluates the incomplete elliptic integral of the second kind, defined by .. math:: E\left(z\middle| m\right) = \int_0^z \sqrt{1 - m \sin^2 t} dt Called with a single argument `m`, evaluates the Legendre complete elliptic integral of the second kind .. math:: E(m) = E\left(\tfrac{\pi}{2}\middle| m\right) The function `E(m)` is a single-valued function on the complex plane with branch cut along the interval `(1, \infty)`. Note that our notation defines the incomplete elliptic integral in terms of the parameter `m` instead of the elliptic modulus (eccentricity) `k`. In this case, the parameter `m` is defined as `m=k^2`. Examples ======== >>> from sympy import elliptic_e, I, pi, O >>> from sympy.abc import z, m >>> elliptic_e(z, m).series(z) z + z**5*(-m**2/40 + m/30) - m*z**3/6 + O(z**6) >>> elliptic_e(m).series(n=4) pi/2 - pi*m/8 - 3*pi*m**2/128 - 5*pi*m**3/512 + O(m**4) >>> elliptic_e(1 + I, 2 - I/2).n() 1.55203744279187 + 0.290764986058437*I >>> elliptic_e(0) pi/2 >>> elliptic_e(2.0 - I) 0.991052601328069 + 0.81879421395609*I References ========== .. [1] https://en.wikipedia.org/wiki/Elliptic_integrals .. [2] http://functions.wolfram.com/EllipticIntegrals/EllipticE2 .. [3] http://functions.wolfram.com/EllipticIntegrals/EllipticE cC s |dk r˜||}}d|t}|jr4|S|jrDtjS|jr[|t|ƒS|tjtjfkrztj S|j ƒrt| |ƒ Snm|jr©tdS|tj kr¿tj S|tjkrÙt tjS|tjkrïtjS|tj krtj SdS(Ni( tNoneRR9RRR:RRRRR;RR(RRR<R=((sIlib/python2.7/site-packages/sympy/functions/special/elliptic_integrals.pyRÐs.        icC sÅt|jƒdkr}|j\}}|dkrLtd|t|ƒdƒS|dkr²t||ƒt||ƒd|Sn5|jd}|dkr²t|ƒt|ƒd|St||ƒ‚dS(Niii(tlenRR R RR8RR(RRR<R((sIlib/python2.7/site-packages/sympy/functions/special/elliptic_integrals.pyRës  %  cC sŸt|jƒdkr_|j\}}|jo7|djtkr›|j|jƒ|jƒƒSn<|jd}|jo|djtkr›|j|jƒƒSdS(Niii(R@RRR R!R"R#(RR<R((sIlib/python2.7/site-packages/sympy/functions/special/elliptic_integrals.pyR$øs cC soddlm}t|jƒdkrM||jtƒj|d|d|ƒƒStt|ƒj|d|d|ƒS(Niÿÿÿÿ(R%iR&R'( R(R%R@RR)R R*tsuperR(RR+R&R'R%((sIlib/python2.7/site-packages/sympy/functions/special/elliptic_integrals.pyR*s(cO sKt|ƒdkrG|d}tdttj tjftjf|ƒSdS(Niii(R@RR RRR(RRR,R((sIlib/python2.7/site-packages/sympy/functions/special/elliptic_integrals.pyR-s cO sat|ƒdkr]|d}ttjtdƒdfgftjftjff| ƒ dSdS(Niiiii(R@RRRR(RRR,R((sIlib/python2.7/site-packages/sympy/functions/special/elliptic_integrals.pyR. s N( R4R5R6R7R?RRR$R*R-R.(((sIlib/python2.7/site-packages/sympy/functions/special/elliptic_integrals.pyR£s+  t elliptic_picB s5eZdZedd„ƒZd„Zdd„ZRS(s6 Called with three arguments `n`, `z` and `m`, evaluates the Legendre incomplete elliptic integral of the third kind, defined by .. math:: \Pi\left(n; z\middle| m\right) = \int_0^z \frac{dt} {\left(1 - n \sin^2 t\right) \sqrt{1 - m \sin^2 t}} Called with two arguments `n` and `m`, evaluates the complete elliptic integral of the third kind: .. math:: \Pi\left(n\middle| m\right) = \Pi\left(n; \tfrac{\pi}{2}\middle| m\right) Note that our notation defines the incomplete elliptic integral in terms of the parameter `m` instead of the elliptic modulus (eccentricity) `k`. In this case, the parameter `m` is defined as `m=k^2`. Examples ======== >>> from sympy import elliptic_pi, I, pi, O, S >>> from sympy.abc import z, n, m >>> elliptic_pi(n, z, m).series(z, n=4) z + z**3*(m/6 + n/3) + O(z**4) >>> elliptic_pi(0.5 + I, 1.0 - I, 1.2) 2.50232379629182 - 0.760939574180767*I >>> elliptic_pi(0, 0) pi/2 >>> elliptic_pi(1.0 - I/3, 2.0 + I) 3.29136443417283 + 0.32555634906645*I References ========== .. [1] https://en.wikipedia.org/wiki/Elliptic_integrals .. [2] http://functions.wolfram.com/EllipticIntegrals/EllipticPi3 .. 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