o 8Va7@sNddlmZddlmZmZmZmZmZmZm Z m Z ddl m Z m Z mZddlmZmZddlmZddlmZddlmZmZmZmZddlmZdd lmZdd lm Z m!Z!dd l"m#Z#m$Z$dd lm%Z%m&Z&dd l'm(Z(m)Z)ddl*m+Z+ddl,m-Z.Gddde Z/Gddde/Z0Gddde/Z1Gddde/Z2Gddde/Z3Gddde/Z4Gddde/Z5ddZ6Gd d!d!e/Z7d"d#Z8d$d%Z9Gd&d'd'e7Z:Gd(d)d)e7Z;Gd*d+d+e7ZdAd2d3Z?Gd4d5d5e Z@Gd6d7d7e@ZAGd8d9d9e@ZBGd:d;d;e@ZCGdd?d?e ZEd@S)Bwraps)AddSpiIRationalWildcacheitsympify)FunctionArgumentIndexError_mexpand)fuzzy_or fuzzy_not)Pow) factorial)sincoscsccot)ceiling)Abs)explog)sqrtroot)reim)gammadigamma)hyper)spherical_bessel_fnc@s^eZdZdZeddZeddZeddZdd d Z d d Z d dZ ddZ ddZ dS) BesselBasea Abstract base class for Bessel-type functions. This class is meant to reduce code duplication. All Bessel-type functions can 1) be differentiated, with the derivatives expressed in terms of similar functions, and 2) be rewritten in terms of other Bessel-type functions. Here, Bessel-type functions are assumed to have one complex parameter. To use this base class, define class attributes ``_a`` and ``_b`` such that ``2*F_n' = -_a*F_{n+1} + b*F_{n-1}``. cC |jdS)z( The order of the Bessel-type function. rargsselfr)@/usr/lib/python3/dist-packages/sympy/functions/special/bessel.pyorder- zBesselBase.ordercCr$)z+ The argument of the Bessel-type function. r%r'r)r)r*argument2r,zBesselBase.argumentcCsdSNr)clsnuzr)r)r*eval7szBesselBase.evalcCsN|dkr t|||jd||jd|j|jd||jd|jSNr5r-)r _b __class__r+r._ar(argindexr)r)r*fdiff;s  zBesselBase.fdiffcCs*|j}|jdur||j|SdSNF)r.is_extended_negativer8r+ conjugater(r3r)r)r*_eval_conjugateAs zBesselBase._eval_conjugatecCsx|j|j}}||rdS|||sdS|||}|jr2t|ttt t t t fs-|j s2t|jStt|j |jgSr=)r+r.Zhas_eval_is_meromorphicsubs is_integer isinstancebesseljbesselihn1hn2jnynis_zerorZ is_infiniter)r(xar2r3Zz0r)r)r*rBFs    zBesselBase._eval_is_meromorphiccKs|j|j|j}}}|jr_|djr7|j |j||d|d|j|d||d||S|djr_d|j|d||d|||j|j||d|S|S)Nr-r5) r+r.r8Zis_real is_positiver9r7_eval_expand_func is_negative)r(hintsr2r3fr)r)r*rPSs & &zBesselBase._eval_expand_funccKsddlm}||S)Nr) besselsimp)Zsympy.simplify.simplifyrT)r(kwargsrTr)r)r*_eval_simplify^s zBesselBase._eval_simplifyNr5)__name__ __module__ __qualname____doc__propertyr+r. classmethodr4r<rArBrPrVr)r)r)r*r#s     r#cdeZdZdZejZejZeddZ ddZ ddZ dd Z dd d Z ddZdfdd ZZS)rFa3 Bessel function of the first kind. Explanation =========== The Bessel $J$ function of order $\nu$ is defined to be the function satisfying Bessel's differential equation .. math :: z^2 \frac{\mathrm{d}^2 w}{\mathrm{d}z^2} + z \frac{\mathrm{d}w}{\mathrm{d}z} + (z^2 - \nu^2) w = 0, with Laurent expansion .. math :: J_\nu(z) = z^\nu \left(\frac{1}{\Gamma(\nu + 1) 2^\nu} + O(z^2) \right), if $\nu$ is not a negative integer. If $\nu=-n \in \mathbb{Z}_{<0}$ *is* a negative integer, then the definition is .. math :: J_{-n}(z) = (-1)^n J_n(z). Examples ======== Create a Bessel function object: >>> from sympy import besselj, jn >>> from sympy.abc import z, n >>> b = besselj(n, z) Differentiate it: >>> b.diff(z) besselj(n - 1, z)/2 - besselj(n + 1, z)/2 Rewrite in terms of spherical Bessel functions: >>> b.rewrite(jn) sqrt(2)*sqrt(z)*jn(n - 1/2, z)/sqrt(pi) Access the parameter and argument: >>> b.order n >>> b.argument z See Also ======== bessely, besseli, besselk References ========== .. [1] Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 9", Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables .. [2] Luke, Y. L. (1969), The Special Functions and Their Approximations, Volume 1 .. [3] https://en.wikipedia.org/wiki/Bessel_function .. [4] http://functions.wolfram.com/Bessel-TypeFunctions/BesselJ/ cCsb|jr,|jr tjS|jr|jdust|jrtjSt|jr&|jdur&tjS|j r,tj S|tj us6|tj ur9tjS| rM||| | t|| S|jrp| r`tj| t| |S|t}|rpt|t||Sddlm}|jr||}||krt||Sn|\}}|dkrtd|t|tt||S||}||krt||SdSNFTr) unpolarifyr5)rLrOnerDrrOZerorQComplexInfinity is_imaginaryNaNInfinityNegativeInfinitycould_extract_minus_signrF NegativeOneextract_multiplicativelyrrGsympyr`extract_branch_factorrrr1r2r3Znewzr`nZnnur)r)r*r4s@     " z besselj.evalcKs4ddlm}ttt|dt||t |SNr) polar_liftr5)rkrprrrrGr(r2r3rUrpr)r)r*_eval_rewrite_as_besseli (z besselj._eval_rewrite_as_besselicKs<|jdurtt|t| |tt|t||SdSr=)rDrrbesselyrr(r2r3rUr)r)r*_eval_rewrite_as_bessely .z besselj._eval_rewrite_as_besselycK"td|tt|tj|jSNr5)rrrJrHalfr.rur)r)r*_eval_rewrite_as_jn"zbesselj._eval_rewrite_as_jnNrcCsN|j\}}||}||jvr||d|t|dS||||dSNr5r-r)r&as_leading_term free_symbolsrfuncrC)r(rMlogxcdirr2r3argr)r)r*_eval_as_leading_terms   zbesselj._eval_as_leading_termcC"|j\}}|jr |jrdSdSdSNTr&rDis_extended_realr(r2r3r)r)r*_eval_is_extended_real  zbesselj._eval_is_extended_realc s"ddlm}|j\}}z ||\}} Wn ttfy!|YSw| jrt|| } ||||} |d|||| } | t j urE| St | d|  } | |t |d}|g}td| ddD]}|| |||9}t ||  }||qet|| Stt|||||SNr)Orderr5r-)sympy.series.orderrr&leadterm ValueErrorNotImplementedErrorrOr _eval_nseriesremoveOrrbrrrangeappendrsuperrF)r(rMrnrrrr2r3_rnewnorttermskr8r)r*rs,      zbesselj._eval_nseriesNrr)rXrYrZr[rrar9r7r]r4rrrvr{rrr __classcell__r)r)rr*rFcsD $ rFcr^)rta_ Bessel function of the second kind. Explanation =========== The Bessel $Y$ function of order $\nu$ is defined as .. math :: Y_\nu(z) = \lim_{\mu \to \nu} \frac{J_\mu(z) \cos(\pi \mu) - J_{-\mu}(z)}{\sin(\pi \mu)}, where $J_\mu(z)$ is the Bessel function of the first kind. It is a solution to Bessel's equation, and linearly independent from $J_\nu$. Examples ======== >>> from sympy import bessely, yn >>> from sympy.abc import z, n >>> b = bessely(n, z) >>> b.diff(z) bessely(n - 1, z)/2 - bessely(n + 1, z)/2 >>> b.rewrite(yn) sqrt(2)*sqrt(z)*yn(n - 1/2, z)/sqrt(pi) See Also ======== besselj, besseli, besselk References ========== .. [1] http://functions.wolfram.com/Bessel-TypeFunctions/BesselY/ cCs~|jr|jr tjSt|jdurtjSt|jrtjS|tjus%|tjur(tjS|jr;| r=tj | t | |SdSdSr=) rLrrgrrcrerfrbrDrhrirtr0r)r)r*r42s z bessely.evalcKs<|jdurtt|tt|t||t| |SdSr=)rDrrrrFrur)r)r*_eval_rewrite_as_besseljBrwz bessely._eval_rewrite_as_besseljcK|j|j}|r |tSdSr/)rr&rewriterGr(r2r3rUZajr)r)r*rrF  z bessely._eval_rewrite_as_besselicKrxry)rrrKrrzr.rur)r)r*_eval_rewrite_as_ynKr|zbessely._eval_rewrite_as_ynNrc Cs|j\}}dtt|dt||}|djr(|d| t|dtntj}|d|tt|t|dtj }t |||g |} || j vrQ| S| |||dSr})r&rrrFrOrrrbr EulerGammarr~rrrCZcancel) r(rMrrr2r3Zterm_oneZterm_twoZ term_threerr)r)r*rNs .* zbessely._eval_as_leading_termcCrrr&rDrOrr)r)r*rYrzbessely._eval_is_extended_realc sddlm}|j\}}z ||\}} Wn ttfy!|YSw| jr|jrt|| } t ||} dt t |d|  ||||} gg} }||||}|d |||| }|tjurf|St|d| }|tjkr|| t|dt }| |td|dD]}||||d|9}t|| }| |q||t t|}|t|dtj}||td| ddD]*}|| |||9}t|| }|t||dt|d}||q| t| t|Stt| ||||Sr)rrr&rrrrOrDrrFrrrrrrbrrarrrr rrrrt)r(rMrnrrrr2r3rrrZbnrNbcrrrrrprr)r*r^sD    $        zbessely._eval_nseriesrr)rXrYrZr[rrar9r7r]r4rrrrrrrrr)r)rr*rts(   rtc@sJeZdZdZej ZejZeddZ ddZ ddZ dd Z d d Z d S) rGa Modified Bessel function of the first kind. Explanation =========== The Bessel $I$ function is a solution to the modified Bessel equation .. math :: z^2 \frac{\mathrm{d}^2 w}{\mathrm{d}z^2} + z \frac{\mathrm{d}w}{\mathrm{d}z} + (z^2 + \nu^2)^2 w = 0. It can be defined as .. math :: I_\nu(z) = i^{-\nu} J_\nu(iz), where $J_\nu(z)$ is the Bessel function of the first kind. Examples ======== >>> from sympy import besseli >>> from sympy.abc import z, n >>> besseli(n, z).diff(z) besseli(n - 1, z)/2 + besseli(n + 1, z)/2 See Also ======== besselj, bessely, besselk References ========== .. [1] http://functions.wolfram.com/Bessel-TypeFunctions/BesselI/ cCsb|jr,|jr tjS|jr|jdust|jrtjSt|jr&|jdur&tjS|j r,tj St |tj us:t |tj ur=tjS|rQ||| | t|| S|jrp|r^t| |S|t}|rpt| t|| Sddlm}|jr||}||krt||Sn|\}}|dkrtd|t|tt||S||}||krt||SdSr_)rLrrarDrrOrbrQrcrdrerrfrgrhrGrjrrFrkr`rlrrrmr)r)r*r4s@      " z besseli.evalcKs4ddlm}tt t|dt||t|Sro)rkrprrrrFrqr)r)r*rrsz besseli._eval_rewrite_as_besseljcKrr/rr&rrtrr)r)r*rvrz besseli._eval_rewrite_as_besselycKs|j|jtSr/)rr&rrJrur)r)r*r{szbesseli._eval_rewrite_as_jncCrrrrr)r)r*rrzbesseli._eval_is_extended_realN)rXrYrZr[rrar9r7r]r4rrvr{rr)r)r)r*rGs' $ rGc@sReZdZdZejZej ZeddZ ddZ ddZ dd Z d d Z d d ZdS)besselka Modified Bessel function of the second kind. Explanation =========== The Bessel $K$ function of order $\nu$ is defined as .. math :: K_\nu(z) = \lim_{\mu \to \nu} \frac{\pi}{2} \frac{I_{-\mu}(z) -I_\mu(z)}{\sin(\pi \mu)}, where $I_\mu(z)$ is the modified Bessel function of the first kind. It is a solution of the modified Bessel equation, and linearly independent from $Y_\nu$. Examples ======== >>> from sympy import besselk >>> from sympy.abc import z, n >>> besselk(n, z).diff(z) -besselk(n - 1, z)/2 - besselk(n + 1, z)/2 See Also ======== besselj, besseli, bessely References ========== .. [1] http://functions.wolfram.com/Bessel-TypeFunctions/BesselK/ cCsz|jr|jr tjSt|jdurtjSt|jrtjS|tjttjttjfvr,tjS|j r9| r;t | |SdSdSr=) rLrrfrrcrerrgrbrDrhrr0r)r)r*r4s  z besselk.evalcKs8|jdurttt|t| |t||dSdS)NFr5)rDrrrGrur)r)r*rr%s *z besselk._eval_rewrite_as_besselicKrr/)rrr&rrF)r(r2r3rUZair)r)r*r)rz besselk._eval_rewrite_as_besseljcKrr/rrr)r)r*rv.rz besselk._eval_rewrite_as_besselycKrr/)rvr&rrK)r(r2r3rUZayr)r)r*r3rzbesselk._eval_rewrite_as_yncCrrrrr)r)r*r8rzbesselk._eval_is_extended_realN)rXrYrZr[rrar9r7r]r4rrrrvrrr)r)r)r*rs%  rc@$eZdZdZejZejZddZdS)hankel1a Hankel function of the first kind. Explanation =========== This function is defined as .. math :: H_\nu^{(1)} = J_\nu(z) + iY_\nu(z), where $J_\nu(z)$ is the Bessel function of the first kind, and $Y_\nu(z)$ is the Bessel function of the second kind. It is a solution to Bessel's equation. Examples ======== >>> from sympy import hankel1 >>> from sympy.abc import z, n >>> hankel1(n, z).diff(z) hankel1(n - 1, z)/2 - hankel1(n + 1, z)/2 See Also ======== hankel2, besselj, bessely References ========== .. [1] http://functions.wolfram.com/Bessel-TypeFunctions/HankelH1/ cC(|j}|jdurt|j|SdSr=)r.r>hankel2r+r?r@r)r)r*rAf zhankel1._eval_conjugateN rXrYrZr[rrar9r7rAr)r)r)r*r>s $ rc@r)ra Hankel function of the second kind. Explanation =========== This function is defined as .. math :: H_\nu^{(2)} = J_\nu(z) - iY_\nu(z), where $J_\nu(z)$ is the Bessel function of the first kind, and $Y_\nu(z)$ is the Bessel function of the second kind. It is a solution to Bessel's equation, and linearly independent from $H_\nu^{(1)}$. Examples ======== >>> from sympy import hankel2 >>> from sympy.abc import z, n >>> hankel2(n, z).diff(z) hankel2(n - 1, z)/2 - hankel2(n + 1, z)/2 See Also ======== hankel1, besselj, bessely References ========== .. [1] http://functions.wolfram.com/Bessel-TypeFunctions/HankelH2/ cCrr=)r.r>rr+r?r@r)r)r*rArzhankel2._eval_conjugateNrr)r)r)r*rls % rcstfdd}|S)Ncs|jr |||SdSr/)rDrfnr)r*gs zassume_integer_order..gr)rrr)rr*assume_integer_ordersrc@s*eZdZdZddZddZd ddZd S) SphericalBesselBasea- Base class for spherical Bessel functions. These are thin wrappers around ordinary Bessel functions, since spherical Bessel functions differ from the ordinary ones just by a slight change in order. To use this class, define the ``_eval_evalf()`` and ``_expand()`` methods. cKstd)z@ Expand self into a polynomial. Nu is guaranteed to be Integer. Z expansionrr(rRr)r)r*_expandszSphericalBesselBase._expandcKs|jjr |jdi|S|SNr))r+ is_Integerrrr)r)r*rPsz%SphericalBesselBase._eval_expand_funcr5cCs:|dkr t||||jd|j||jd|jSr6)r r8r+r.r:r)r)r*r<s  zSphericalBesselBase.fdiffNrW)rXrYrZr[rrPr<r)r)r)r*rs  rcCs6t||t|d|dt| d|t|SNr-rrrrnr3r)r)r*_jns6rcCs6d|dt| d|t|t||t|Srrrr)r)r*_yns6rc@sDeZdZdZeddZddZddZdd Zd d Z d d Z dS)rJa Spherical Bessel function of the first kind. Explanation =========== This function is a solution to the spherical Bessel equation .. math :: z^2 \frac{\mathrm{d}^2 w}{\mathrm{d}z^2} + 2z \frac{\mathrm{d}w}{\mathrm{d}z} + (z^2 - \nu(\nu + 1)) w = 0. It can be defined as .. math :: j_\nu(z) = \sqrt{\frac{\pi}{2z}} J_{\nu + \frac{1}{2}}(z), where $J_\nu(z)$ is the Bessel function of the first kind. The spherical Bessel functions of integral order are calculated using the formula: .. math:: j_n(z) = f_n(z) \sin{z} + (-1)^{n+1} f_{-n-1}(z) \cos{z}, where the coefficients $f_n(z)$ are available as :func:`sympy.polys.orthopolys.spherical_bessel_fn`. Examples ======== >>> from sympy import Symbol, jn, sin, cos, expand_func, besselj, bessely >>> z = Symbol("z") >>> nu = Symbol("nu", integer=True) >>> print(expand_func(jn(0, z))) sin(z)/z >>> expand_func(jn(1, z)) == sin(z)/z**2 - cos(z)/z True >>> expand_func(jn(3, z)) (-6/z**2 + 15/z**4)*sin(z) + (1/z - 15/z**3)*cos(z) >>> jn(nu, z).rewrite(besselj) sqrt(2)*sqrt(pi)*sqrt(1/z)*besselj(nu + 1/2, z)/2 >>> jn(nu, z).rewrite(bessely) (-1)**nu*sqrt(2)*sqrt(pi)*sqrt(1/z)*bessely(-nu - 1/2, z)/2 >>> jn(2, 5.2+0.3j).evalf(20) 0.099419756723640344491 - 0.054525080242173562897*I See Also ======== besselj, bessely, besselk, yn References ========== .. [1] http://dlmf.nist.gov/10.47 cCsD|jr|jr tjS|jr|jrtjStjS|tjtjfvr tjSdSr/) rLrrarDrOrbrcrgrfr0r)r)r*r4szjn.evalcK ttd|t|tj|SryrrrFrrzrur)r)r*rs zjn._eval_rewrite_as_besseljcKs*d|ttd|t| tj|S)Nrr5rrrtrrzrur)r)r*rvs*zjn._eval_rewrite_as_besselycKsd|t| d|Sr)rKrur)r)r*rszjn._eval_rewrite_as_yncKt|j|jSr/)rr+r.rr)r)r*rz jn._expandcC|jjr |t|SdSr/r+rrrF _eval_evalfr(precr)r)r*rzjn._eval_evalfN) rXrYrZr[r]r4rrvrrrr)r)r)r*rJs9   rJc@s@eZdZdZeddZeddZddZdd Zd d Z d S) rKa Spherical Bessel function of the second kind. Explanation =========== This function is another solution to the spherical Bessel equation, and linearly independent from $j_n$. It can be defined as .. math :: y_\nu(z) = \sqrt{\frac{\pi}{2z}} Y_{\nu + \frac{1}{2}}(z), where $Y_\nu(z)$ is the Bessel function of the second kind. For integral orders $n$, $y_n$ is calculated using the formula: .. math:: y_n(z) = (-1)^{n+1} j_{-n-1}(z) Examples ======== >>> from sympy import Symbol, yn, sin, cos, expand_func, besselj, bessely >>> z = Symbol("z") >>> nu = Symbol("nu", integer=True) >>> print(expand_func(yn(0, z))) -cos(z)/z >>> expand_func(yn(1, z)) == -cos(z)/z**2-sin(z)/z True >>> yn(nu, z).rewrite(besselj) (-1)**(nu + 1)*sqrt(2)*sqrt(pi)*sqrt(1/z)*besselj(-nu - 1/2, z)/2 >>> yn(nu, z).rewrite(bessely) sqrt(2)*sqrt(pi)*sqrt(1/z)*bessely(nu + 1/2, z)/2 >>> yn(2, 5.2+0.3j).evalf(20) 0.18525034196069722536 + 0.014895573969924817587*I See Also ======== besselj, bessely, besselk, jn References ========== .. [1] http://dlmf.nist.gov/10.47 cKs.d|dttd|t| tj|S)Nrr-r5rrur)r)r*rOs.zyn._eval_rewrite_as_besseljcKrryrrur)r)r*rvSs zyn._eval_rewrite_as_besselycKsd|dt| d|Sr)rJrur)r)r*r{Wszyn._eval_rewrite_as_jncKrr/)rr+r.rr)r)r*rZrz yn._expandcCrr/)r+rrrtrrr)r)r*r]rzyn._eval_evalfN) rXrYrZr[rrrvr{rrr)r)r)r*rK s.   rKc@sLeZdZeddZeddZddZddZd d Zd d Z d dZ dS)SphericalHankelBasecKsL|j}ttd|t|tj||td|dt| tj|S)Nr5rr-)_hankel_kind_signrrrFrrzrr(r2r3rUhksr)r)r*rds$z,SphericalHankelBase._eval_rewrite_as_besseljcKsH|j}ttd|d|t| tj||tt|tj|S)Nr5r)rrrrtrrzrrr)r)r*rvms&z,SphericalHankelBase._eval_rewrite_as_besselycKs(|j}t||t|tt||Sr/)rrJrrKrrr)r)r*rv"z'SphericalHankelBase._eval_rewrite_as_yncKs(|j}t|||tt||tSr/)rrJrrKrrr)r)r*r{zrz'SphericalHankelBase._eval_rewrite_as_jncKsF|jjr |jdi|S|j}|j}|j}t|||tt||Sr)r+rrr.rrJrrK)r(rRr2r3rr)r)r*rP~s z%SphericalHankelBase._eval_expand_funccKs2|j}|j}|j}t|||tt||Sr/)r+r.rrrrexpand)r(rRrnr3rr)r)r*rs zSphericalHankelBase._expandcCrr/rrr)r)r*rrzSphericalHankelBase._eval_evalfN) rXrYrZrrrvrr{rPrrr)r)r)r*rbs   rc@s"eZdZdZejZeddZdS)rHa Spherical Hankel function of the first kind. Explanation =========== This function is defined as .. math:: h_\nu^(1)(z) = j_\nu(z) + i y_\nu(z), where $j_\nu(z)$ and $y_\nu(z)$ are the spherical Bessel function of the first and second kinds. For integral orders $n$, $h_n^(1)$ is calculated using the formula: .. math:: h_n^(1)(z) = j_{n}(z) + i (-1)^{n+1} j_{-n-1}(z) Examples ======== >>> from sympy import Symbol, hn1, hankel1, expand_func, yn, jn >>> z = Symbol("z") >>> nu = Symbol("nu", integer=True) >>> print(expand_func(hn1(nu, z))) jn(nu, z) + I*yn(nu, z) >>> print(expand_func(hn1(0, z))) sin(z)/z - I*cos(z)/z >>> print(expand_func(hn1(1, z))) -I*sin(z)/z - cos(z)/z + sin(z)/z**2 - I*cos(z)/z**2 >>> hn1(nu, z).rewrite(jn) (-1)**(nu + 1)*I*jn(-nu - 1, z) + jn(nu, z) >>> hn1(nu, z).rewrite(yn) (-1)**nu*yn(-nu - 1, z) + I*yn(nu, z) >>> hn1(nu, z).rewrite(hankel1) sqrt(2)*sqrt(pi)*sqrt(1/z)*hankel1(nu, z)/2 See Also ======== hn2, jn, yn, hankel1, hankel2 References ========== .. [1] http://dlmf.nist.gov/10.47 cKttd|t||Sry)rrrrur)r)r*_eval_rewrite_as_hankel1zhn1._eval_rewrite_as_hankel1N) rXrYrZr[rrarrrr)r)r)r*rHs 0rHc@s$eZdZdZej ZeddZdS)rIa Spherical Hankel function of the second kind. Explanation =========== This function is defined as .. math:: h_\nu^(2)(z) = j_\nu(z) - i y_\nu(z), where $j_\nu(z)$ and $y_\nu(z)$ are the spherical Bessel function of the first and second kinds. For integral orders $n$, $h_n^(2)$ is calculated using the formula: .. math:: h_n^(2)(z) = j_{n} - i (-1)^{n+1} j_{-n-1}(z) Examples ======== >>> from sympy import Symbol, hn2, hankel2, expand_func, jn, yn >>> z = Symbol("z") >>> nu = Symbol("nu", integer=True) >>> print(expand_func(hn2(nu, z))) jn(nu, z) - I*yn(nu, z) >>> print(expand_func(hn2(0, z))) sin(z)/z + I*cos(z)/z >>> print(expand_func(hn2(1, z))) I*sin(z)/z - cos(z)/z + sin(z)/z**2 + I*cos(z)/z**2 >>> hn2(nu, z).rewrite(hankel2) sqrt(2)*sqrt(pi)*sqrt(1/z)*hankel2(nu, z)/2 >>> hn2(nu, z).rewrite(jn) -(-1)**(nu + 1)*I*jn(-nu - 1, z) + jn(nu, z) >>> hn2(nu, z).rewrite(yn) (-1)**nu*yn(-nu - 1, z) - I*yn(nu, z) See Also ======== hn1, jn, yn, hankel1, hankel2 References ========== .. [1] http://dlmf.nist.gov/10.47 cKrry)rrrrur)r)r*_eval_rewrite_as_hankel2rzhn2._eval_rewrite_as_hankel2N) rXrYrZr[rrarrrr)r)r)r*rIs 0rIrkc sddlm}dkr1ddlmddlm}ddlm||fddtd |d DSd kradd l m zdd l m fd d}Wnt y`ddl mfdd}Ynwtdfdd}|}|||}|g} t|d D]} ||||}| |q~| S)a Zeros of the spherical Bessel function of the first kind. Explanation =========== This returns an array of zeros of $jn$ up to the $k$-th zero. * method = "sympy": uses `mpmath.besseljzero `_ * method = "scipy": uses the `SciPy's sph_jn `_ and `newton `_ to find all roots, which is faster than computing the zeros using a general numerical solver, but it requires SciPy and only works with low precision floating point numbers. (The function used with method="sympy" is a recent addition to mpmath; before that a general solver was used.) Examples ======== >>> from sympy import jn_zeros >>> jn_zeros(2, 4, dps=5) [5.7635, 9.095, 12.323, 15.515] See Also ======== jn, yn, besselj, besselk, bessely Parameters ========== n : integer order of Bessel function k : integer number of zeros to return r)rrk) besseljzero) dps_to_precExprcs0g|]}tdt|qS)g?) _from_mpmathr _to_mpmathint).0l)rrrnrr)r* ?szjn_zeros..r-scipy)newton) spherical_jncs |Sr/r)rM)rnrr)r*Fs zjn_zeros..)sph_jncs|ddS)Nrrr)r)rnrr)r*rIsUnknown method.csdkr ||}|Std)Nrrr)rSrMr)methodrr)r*solverMs zjn_zeros..solver)ZmathrmpmathrZmpmath.libmp.libmpfrrkrrZscipy.optimizerZ scipy.specialr ImportErrorrrr) rnrrZdpsrrrSrrrootsir))rrrrnrrrrr*jn_zeros s6 -          rc@s4eZdZdZddZddZd ddZd d d Zd S) AiryBasezg Abstract base class for Airy functions. This class is meant to reduce code duplication. cCs||jdSr)rr&r?r'r)r)r*rAhszAiryBase._eval_conjugatecCs |jdjSr)r&rr'r)r)r*rks zAiryBase._eval_is_extended_realTcKsL|jd}|}|j}||||d}t||||d}||fS)Nrr5)r&r?rr)r(deeprRr3ZzcrSuvr)r)r* as_real_imagns zAiryBase.as_real_imagcKs&|jdd|i|\}}||tjS)Nrr))rrZ ImaginaryUnit)r(rrRZre_partZim_partr)r)r*_eval_expand_complexvszAiryBase._eval_expand_complexN)T)rXrYrZr[rArrrr)r)r)r*r`s  rc@^eZdZdZdZdZeddZdddZe e dd Z d d Z d d Z ddZddZdS)airyaia The Airy function $\operatorname{Ai}$ of the first kind. Explanation =========== The Airy function $\operatorname{Ai}(z)$ is defined to be the function satisfying Airy's differential equation .. math:: \frac{\mathrm{d}^2 w(z)}{\mathrm{d}z^2} - z w(z) = 0. Equivalently, for real $z$ .. math:: \operatorname{Ai}(z) := \frac{1}{\pi} \int_0^\infty \cos\left(\frac{t^3}{3} + z t\right) \mathrm{d}t. Examples ======== Create an Airy function object: >>> from sympy import airyai >>> from sympy.abc import z >>> airyai(z) airyai(z) Several special values are known: >>> airyai(0) 3**(1/3)/(3*gamma(2/3)) >>> from sympy import oo >>> airyai(oo) 0 >>> airyai(-oo) 0 The Airy function obeys the mirror symmetry: >>> from sympy import conjugate >>> conjugate(airyai(z)) airyai(conjugate(z)) Differentiation with respect to $z$ is supported: >>> from sympy import diff >>> diff(airyai(z), z) airyaiprime(z) >>> diff(airyai(z), z, 2) z*airyai(z) Series expansion is also supported: >>> from sympy import series >>> series(airyai(z), z, 0, 3) 3**(5/6)*gamma(1/3)/(6*pi) - 3**(1/6)*z*gamma(2/3)/(2*pi) + O(z**3) We can numerically evaluate the Airy function to arbitrary precision on the whole complex plane: >>> airyai(-2).evalf(50) 0.22740742820168557599192443603787379946077222541710 Rewrite $\operatorname{Ai}(z)$ in terms of hypergeometric functions: >>> from sympy import hyper >>> airyai(z).rewrite(hyper) -3**(2/3)*z*hyper((), (4/3,), z**3/9)/(3*gamma(1/3)) + 3**(1/3)*hyper((), (2/3,), z**3/9)/(3*gamma(2/3)) See Also ======== airybi: Airy function of the second kind. airyaiprime: Derivative of the Airy function of the first kind. airybiprime: Derivative of the Airy function of the second kind. References ========== .. [1] https://en.wikipedia.org/wiki/Airy_function .. [2] http://dlmf.nist.gov/9 .. [3] http://www.encyclopediaofmath.org/index.php/Airy_functions .. [4] http://mathworld.wolfram.com/AiryFunctions.html r-TcCs|jr/|tjur tjS|tjurtjS|tjurtjS|jr/tjdtddt tddS|jrCtjdtddt tddSdS)Nr5) is_NumberrrerfrbrgrLrarrr1rr)r)r*r4s   ""z airyai.evalcC |dkr t|jdSt||Nr-r) airyaiprimer&r r:r)r)r*r< z airyai.fdiffcGsF|dkrtjSt|}t|dkrp|d}dtdd|| dtdd||dtt|tddtddt|t|dtddtt|tddtddt|dt|dtdd|Stj dtddtt|tj tdtdt|tj tdt|t dd||S)Nrr-rrr5) rrbr lenrrrrrrarrnrMZprevious_termsrr)r)r* taylor_terms" X@Jzairyai.taylor_termcKs`tdd}tdd}t| tdd}t|jr.|t| t| ||t|||SdSNr-rr5rrrrQrrFr(r3rUotttrNr)r)r*r   ,zairyai._eval_rewrite_as_besseljcKstdd}tdd}t|tdd}t|jr,|t|t| ||t|||S|t||t| |||t|| t|||SrrrrrOrrGr r)r)r*rrs   *<zairyai._eval_rewrite_as_besselicKs~tjdtddttdd}|tddttdd}|tgtddg|dd|tgtddg|ddS)Nrr5r- r)rrarrrr!r(r3rUZpf1Zpf2r)r)r*_eval_rewrite_as_hyper s"@zairyai._eval_rewrite_as_hyperc Ks"|jd}|j}t|dkr|}td|gd}td|gd}td|gd}td|gd}||||||} | dur| |}d|jr| |}| |}| |}|||||||||} ||||||} tj| tj t | | tj t dt | SdSdSdS Nrr-r)Zexcludedmrnr) r&rrpopr matchrDrrzrarrairybi r(rRrZsymbsr3rrrrnMZpfZnewargr)r)r*rP*   $2zairyai._eval_expand_funcNr-rXrYrZr[nargs unbranchedr]r4r< staticmethodr rrrrrrPr)r)r)r*r{sX     rc@r)ra The Airy function $\operatorname{Bi}$ of the second kind. Explanation =========== The Airy function $\operatorname{Bi}(z)$ is defined to be the function satisfying Airy's differential equation .. math:: \frac{\mathrm{d}^2 w(z)}{\mathrm{d}z^2} - z w(z) = 0. Equivalently, for real $z$ .. math:: \operatorname{Bi}(z) := \frac{1}{\pi} \int_0^\infty \exp\left(-\frac{t^3}{3} + z t\right) + \sin\left(\frac{t^3}{3} + z t\right) \mathrm{d}t. Examples ======== Create an Airy function object: >>> from sympy import airybi >>> from sympy.abc import z >>> airybi(z) airybi(z) Several special values are known: >>> airybi(0) 3**(5/6)/(3*gamma(2/3)) >>> from sympy import oo >>> airybi(oo) oo >>> airybi(-oo) 0 The Airy function obeys the mirror symmetry: >>> from sympy import conjugate >>> conjugate(airybi(z)) airybi(conjugate(z)) Differentiation with respect to $z$ is supported: >>> from sympy import diff >>> diff(airybi(z), z) airybiprime(z) >>> diff(airybi(z), z, 2) z*airybi(z) Series expansion is also supported: >>> from sympy import series >>> series(airybi(z), z, 0, 3) 3**(1/3)*gamma(1/3)/(2*pi) + 3**(2/3)*z*gamma(2/3)/(2*pi) + O(z**3) We can numerically evaluate the Airy function to arbitrary precision on the whole complex plane: >>> airybi(-2).evalf(50) -0.41230258795639848808323405461146104203453483447240 Rewrite $\operatorname{Bi}(z)$ in terms of hypergeometric functions: >>> from sympy import hyper >>> airybi(z).rewrite(hyper) 3**(1/6)*z*hyper((), (4/3,), z**3/9)/gamma(1/3) + 3**(5/6)*hyper((), (2/3,), z**3/9)/(3*gamma(2/3)) See Also ======== airyai: Airy function of the first kind. airyaiprime: Derivative of the Airy function of the first kind. airybiprime: Derivative of the Airy function of the second kind. References ========== .. [1] https://en.wikipedia.org/wiki/Airy_function .. [2] http://dlmf.nist.gov/9 .. [3] http://www.encyclopediaofmath.org/index.php/Airy_functions .. [4] http://mathworld.wolfram.com/AiryFunctions.html r-TcCs|jr/|tjur tjS|tjurtjS|tjurtjS|jr/tjdtddt tddS|jrCtjdtddt tddSdS)Nrr-r5) rrrerfrgrbrLrarrrr)r)r*r4s   ""z airybi.evalcCrr) airybiprimer&r r:r)r)r*r<rz airybi.fdiffcGs|dkrtjSt|}t|dkr\|d}dtdd|ttdt|tjtdt |tjtd|tjtt dt|tj tdt |dtd|Stjt ddtt |tjtdttdt|tjtdt |t dd||S)Nrr-rrr5r)rrbr rrrrrrarrrzrrrr)r)r*rs H>Jzairybi.taylor_termcKs`tdd}tdd}t| tdd}t|jr.t| dt| ||t|||SdSrrr r)r)r*rr zairybi._eval_rewrite_as_besseljcKstdd}tdd}t|tdd}t|jr.t|tdt| ||t|||St||}t|| }t||t| ||||t|||Srr r(r3rUr r rNrrr)r)r*rrs   .  2zairybi._eval_rewrite_as_besselicKsztjtddttdd}|tddttdd}|tgtddg|dd|tgtddg|ddS)Nrrr5r-rr)rrarrrr!rr)r)r*rs@zairybi._eval_rewrite_as_hyperc Ks"|jd}|j}t|dkr|}td|gd}td|gd}td|gd}td|gd}||||||} | dur| |}d|jr| |}| |}| |}|||||||||} ||||||} tjt dtj | t | tj | t | SdSdSdSr) r&rrrr rrDrrzrrarrrr)r)r*rPrzairybi._eval_expand_funcNrrr)r)r)r*r'sZ     rc@VeZdZdZdZdZeddZdddZdd Z d d Z d d Z ddZ ddZ dS)ra+ The derivative $\operatorname{Ai}^\prime$ of the Airy function of the first kind. Explanation =========== The Airy function $\operatorname{Ai}^\prime(z)$ is defined to be the function .. math:: \operatorname{Ai}^\prime(z) := \frac{\mathrm{d} \operatorname{Ai}(z)}{\mathrm{d} z}. Examples ======== Create an Airy function object: >>> from sympy import airyaiprime >>> from sympy.abc import z >>> airyaiprime(z) airyaiprime(z) Several special values are known: >>> airyaiprime(0) -3**(2/3)/(3*gamma(1/3)) >>> from sympy import oo >>> airyaiprime(oo) 0 The Airy function obeys the mirror symmetry: >>> from sympy import conjugate >>> conjugate(airyaiprime(z)) airyaiprime(conjugate(z)) Differentiation with respect to $z$ is supported: >>> from sympy import diff >>> diff(airyaiprime(z), z) z*airyai(z) >>> diff(airyaiprime(z), z, 2) z*airyaiprime(z) + airyai(z) Series expansion is also supported: >>> from sympy import series >>> series(airyaiprime(z), z, 0, 3) -3**(2/3)/(3*gamma(1/3)) + 3**(1/3)*z**2/(6*gamma(2/3)) + O(z**3) We can numerically evaluate the Airy function to arbitrary precision on the whole complex plane: >>> airyaiprime(-2).evalf(50) 0.61825902074169104140626429133247528291577794512415 Rewrite $\operatorname{Ai}^\prime(z)$ in terms of hypergeometric functions: >>> from sympy import hyper >>> airyaiprime(z).rewrite(hyper) 3**(1/3)*z**2*hyper((), (5/3,), z**3/9)/(6*gamma(2/3)) - 3**(2/3)*hyper((), (1/3,), z**3/9)/(3*gamma(1/3)) See Also ======== airyai: Airy function of the first kind. airybi: Airy function of the second kind. airybiprime: Derivative of the Airy function of the second kind. References ========== .. [1] https://en.wikipedia.org/wiki/Airy_function .. [2] http://dlmf.nist.gov/9 .. [3] http://www.encyclopediaofmath.org/index.php/Airy_functions .. [4] http://mathworld.wolfram.com/AiryFunctions.html r-TcCsR|jr|tjur tjS|tjurtjS|jr'tjdtddttddSdS)Nrr-) rrrerfrbrLrirrrr)r)r*r4-s  "zairyaiprime.evalcC*|dkr|jdt|jdSt||r)r&rr r:r)r)r*r<8 zairyaiprime.fdiffcCnddlm}m}ddlm}|jd|}|||j|dd}Wdn1s,wY|||SNr)mpworkprecrr-)Z derivative) rr'r(rkrr&rrrr(rr'r(rr3resr)r)r*r>   zairyaiprime._eval_evalfcKsPtdd}t| tdd}t|jr&|dt| ||t|||SdSNr5r)rrrrQrFr(r3rUr rNr)r)r*rFs  &z$airyaiprime._eval_rewrite_as_besseljcKstdd}tdd}|t|tdd}t|jr(|dt||t| |St|tdd}t||}t|| }||d|t||||t| ||Sr)rrrrOrGr!r)r)r*rrLs     2z$airyaiprime._eval_rewrite_as_besselicKs|dddtddttdd}dtddttdd}|tgtddg|dd|tgtddg|ddS)Nr5rr-r)rrrr!rr)r)r*rXs(@z"airyaiprime._eval_rewrite_as_hyperc Ks"|jd}|j}t|dkr|}td|gd}td|gd}td|gd}td|gd}||||||} | dur| |}d|jr| |}| |}| |}|||||||||} ||||||} tj| tj t | | tj t dt | SdSdSdSr) r&rrrr rrDrrzrarrr rr)r)r*rP]*   $2zairyaiprime._eval_expand_funcNrrXrYrZr[rrr]r4r<rrrrrrPr)r)r)r*rsQ   rc@r")r a< The derivative $\operatorname{Bi}^\prime$ of the Airy function of the first kind. Explanation =========== The Airy function $\operatorname{Bi}^\prime(z)$ is defined to be the function .. math:: \operatorname{Bi}^\prime(z) := \frac{\mathrm{d} \operatorname{Bi}(z)}{\mathrm{d} z}. Examples ======== Create an Airy function object: >>> from sympy import airybiprime >>> from sympy.abc import z >>> airybiprime(z) airybiprime(z) Several special values are known: >>> airybiprime(0) 3**(1/6)/gamma(1/3) >>> from sympy import oo >>> airybiprime(oo) oo >>> airybiprime(-oo) 0 The Airy function obeys the mirror symmetry: >>> from sympy import conjugate >>> conjugate(airybiprime(z)) airybiprime(conjugate(z)) Differentiation with respect to $z$ is supported: >>> from sympy import diff >>> diff(airybiprime(z), z) z*airybi(z) >>> diff(airybiprime(z), z, 2) z*airybiprime(z) + airybi(z) Series expansion is also supported: >>> from sympy import series >>> series(airybiprime(z), z, 0, 3) 3**(1/6)/gamma(1/3) + 3**(5/6)*z**2/(6*gamma(2/3)) + O(z**3) We can numerically evaluate the Airy function to arbitrary precision on the whole complex plane: >>> airybiprime(-2).evalf(50) 0.27879516692116952268509756941098324140300059345163 Rewrite $\operatorname{Bi}^\prime(z)$ in terms of hypergeometric functions: >>> from sympy import hyper >>> airybiprime(z).rewrite(hyper) 3**(5/6)*z**2*hyper((), (5/3,), z**3/9)/(6*gamma(2/3)) + 3**(1/6)*hyper((), (1/3,), z**3/9)/gamma(1/3) See Also ======== airyai: Airy function of the first kind. airybi: Airy function of the second kind. airyaiprime: Derivative of the Airy function of the first kind. References ========== .. [1] https://en.wikipedia.org/wiki/Airy_function .. [2] http://dlmf.nist.gov/9 .. [3] http://www.encyclopediaofmath.org/index.php/Airy_functions .. 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Explanation =========== The Marcum Q-function is defined by the meromorphic continuation of .. math:: Q_m(a, b) = a^{- m + 1} \int_{b}^{\infty} x^{m} e^{- \frac{a^{2}}{2} - \frac{x^{2}}{2}} I_{m - 1}\left(a x\right)\, dx Examples ======== >>> from sympy import marcumq >>> from sympy.abc import m, a, b >>> marcumq(m, a, b) marcumq(m, a, b) Special values: >>> marcumq(m, 0, b) uppergamma(m, b**2/2)/gamma(m) >>> marcumq(0, 0, 0) 0 >>> marcumq(0, a, 0) 1 - exp(-a**2/2) >>> marcumq(1, a, a) 1/2 + exp(-a**2)*besseli(0, a**2)/2 >>> marcumq(2, a, a) 1/2 + exp(-a**2)*besseli(0, a**2)/2 + exp(-a**2)*besseli(1, a**2) Differentiation with respect to $a$ and $b$ is supported: >>> from sympy import diff >>> diff(marcumq(m, a, b), a) a*(-marcumq(m, a, b) + marcumq(m + 1, a, b)) >>> diff(marcumq(m, a, b), b) -a**(1 - m)*b**m*exp(-a**2/2 - b**2/2)*besseli(m - 1, a*b) References ========== .. [1] https://en.wikipedia.org/wiki/Marcum_Q-function .. 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