o 8Va9@sdZddlmZmZmZmZddlmZmZddl m Z ddl m Z ddl mZddlmZmZddlmZdd lmZmZGd d d eZGd d d eZGdddeZGdddeZdS)z Elliptic Integrals. )SpiIRational)FunctionArgumentIndexError)sign)atanh)sqrt)sintan)gamma)hypermeijergc@sXeZdZdZeddZdddZddZdd d Zd d Z ddZ ddZ ddZ dS) elliptic_kaM 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. Explanation =========== 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 >>> 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 cCs|jrttjS|tjurdttddttdddS|tjur'tjS|tjur=ttddddt dtS|tj tj t tj t tj tjfvrRtj SdS)N)is_zerorrHalfrr OneComplexInfinityZ NegativeOner InfinityNegativeInfinityrZero)clsmr L/usr/lib/python3/dist-packages/sympy/functions/special/elliptic_integrals.pyeval9s  $  " zelliptic_k.evalrcCs2|jd}t|d|t|d|d|S)Nrrr)args elliptic_er)selfargindexrr r r!fdiffGs (zelliptic_k.fdiffcCs0|jd}|jo |djdur||SdS)NrrFr#is_real is_positivefunc conjugater%rr r r!_eval_conjugateKs zelliptic_k._eval_conjugatercCs&ddlm}||tj|||dS)Nr hyperexpandnlogx)sympy.simplifyr0rewriter _eval_nseriesr%xr2r3cdirr0r r r!r6Ps zelliptic_k._eval_nseriescKs"ttjttjtjftjf|SN)rrrrrr%rkwargsr r r!_eval_rewrite_as_hyperTs"z!elliptic_k._eval_rewrite_as_hypercKs*ttjtjfgftjftjff| dSNr)rrrrr;r r r!_eval_rewrite_as_meijergWs*z#elliptic_k._eval_rewrite_as_meijergcCs|jd}|jr dSdS)NrT)r# is_infiniter-r r r! _eval_is_zeroZs zelliptic_k._eval_is_zerocGsNddlm}m}|d}|jd}|dtd|t|d|dtdfSNrIntegralDummytrr)sympyrDrEr#r r r)r%r#rDrErFrr r r!_eval_rewrite_as_Integral_s ,z$elliptic_k._eval_rewrite_as_IntegralNrr) __name__ __module__ __qualname____doc__ classmethodr"r'r.r6r=r?rArHr r r r!r s,    rc@s>eZdZdZeddZdddZddZd d Zd d Z d S) elliptic_fa 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}} Explanation =========== 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 >>> 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 cCsd|jrtjS|jr |Sd|t}|jr|t|S|tjtjfvr%tjS|r0t | | SdSr>) rrrr is_integerrrrcould_extract_minus_signrP)rzrkr r r!r"s  zelliptic_f.evalrcCs|j\}}td|t|d}|dkrd|S|dkrAt||d|d|t||d|td|dd||St||)Nrrr)r#r r r$rPr)r%r&rSrfmr r r!r's * zelliptic_f.fdiffcCs6|j\}}|jo |djdur|||SdS)NrFr(r%rSrr r r!r.s zelliptic_f._eval_conjugatecGsVddlm}m}|d}|jd|jd}}|dtd|t|d|d|fSrB)rGrDrEr#r r )r%r#rDrErFrSrr r r!rHs(z$elliptic_f._eval_rewrite_as_IntegralcCs,|j\}}|jr dS|jr|jrdSdSdS)NT)r#ris_extended_realr@rVr r r!rAs  zelliptic_f._eval_is_zeroNrI) rKrLrMrNrOr"r'r.rHrAr r r r!rPfs)    rPcsZeZdZdZedddZdddZdd Zdfd d Zd dZ ddZ ddZ Z S)r$a 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) Explanation =========== 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 >>> 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 NcCs|dur;||}}d|t}|jr|S|jrtjS|jr#|t|S|tjtjfvr.tjS| r9t| | SdS|jrBtdS|tj urJtj S|tjurTt tjS|tjur\tjS|tjurdtjSdSr>) rrrrrQr$rrrrRrr)rrrSrTr r r!r"s2        zelliptic_e.evalrcCst|jdkr/|j\}}|dkrtd|t|dS|dkr.t||t||d|Sn|jd}|dkrDt|t|d|St||)Nrrr)lenr#r r r$rPrr)r%r&rSrr r r!r's   zelliptic_e.fdiffcCstt|jdkr"|j\}}|jo|djdur |||SdS|jd}|jo.|djdur8||SdS)NrrFrrXr#r)r*r+r,rVr r r!r.s  zelliptic_e._eval_conjugatercsFddlm}t|jdkr||tj|||dStj|||dS)Nrr/rr1)r4r0rXr#r5rr6superr7 __class__r r!r6s zelliptic_e._eval_nseriescOs<t|dkr|d}tdttddtjftjf|SdS)Nrrrr)rXrrrrrrr%r#r<rr r r!r=#s $z!elliptic_e._eval_rewrite_as_hypercOsHt|dkr"|d}ttjtddfgftjftjff|  dSdS)Nrrrrr)rXrrrrrr]r r r!r?(s z#elliptic_e._eval_rewrite_as_meijergcGsfddlm}m}t|jdkrtd|jdfn|j\}}|d}|td|t|d|d|fS)NrrCrrrF)rGrDrErXr#rr r )r%r#rDrErSrrFr r r!rH.s*$z$elliptic_e._eval_rewrite_as_Integralr:rIrJ) rKrLrMrNrOr"r'r.r6r=r?rH __classcell__r r r[r!r$s/    r$c@s8eZdZdZed ddZddZd dd Zd d ZdS) elliptic_piaM 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) Explanation =========== 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 >>> 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 .. [3] http://functions.wolfram.com/EllipticIntegrals/EllipticPi NcCsH|dur|||}}}|jrt||S|tjur7t||td|t|dt|t||d|Sd|t}|j rG|t ||S|jr\t t|dt|t|dS||kr{t||t d||t|td|t|dS|tj tj fvrtjS|tj tj fvrtjS|rt || | S|jrt||S|jr|js|jr|jrtjSdSdS|jrt|S|tjurtjS|jrtdtd|S|tjkrtj t|dS||krt|d|S|tj tj fvrtjS|tj tj fvrtjS|jr t|S|jr|js|jr |jr"tjSdSdS)Nrr)rrPrrr r r r$rrQr_r rrrrRrWr@rrr)rr2rrSrTr r r!r"bsv   $    zelliptic_pi.evalcCst|jdkr2|j\}}}|jo|djdur.|jo|djdur0||||SdSdS|j\}}|||S)NrrFrY)r%r2rSrr r r!r.s  zelliptic_pi._eval_conjugatercCst|jdkr|j\}}}td|t|dd|t|d}}|dkr^t||||t||||d|t||||||td|d|d|||dS|dkrhd||S|dkrt|||dt||||td|d|d|d||SnE|j\}}|dkrt|||t|||d|t|||d|||dS|dkrt||dt||d||St||)Nrrr) rXr#r r r$rPr_rr)r%r&r2rSrrUfnr r r!r's@ .    & zelliptic_pi.fdiffcGsddlm}m}t|jdkr |jd|jdtd}}}n|j\}}}|d}|dd|t|dtd|t|d|d|fS)NrrCrrrF)rGrDrErXr#rr r )r%r#rDrEr2rrSrFr r r!rHs " <z%elliptic_pi._eval_rewrite_as_Integralr:rI) rKrLrMrNrOr"r.r'rHr r r r!r_5s, 1 r_N)rNZ sympy.corerrrrZsympy.core.functionrrZ$sympy.functions.elementary.complexesrZ%sympy.functions.elementary.hyperbolicr Z(sympy.functions.elementary.miscellaneousr Z(sympy.functions.elementary.trigonometricr r Z'sympy.functions.special.gamma_functionsr Zsympy.functions.special.hyperrrrrPr$r_r r r r!s    ZUz