o *ήcB@sddlmZddlmZddlmZddlmZddlm Z ddl m Z m Z m Z ddlmZmZddlmZmZmZdd lmZdd lmZmZdd lmZdd lmZdd lmZm Z m!Z!m"Z"ddl#m$Z$ddl%m&Z&m'Z'ddl(m)Z)m*Z*m+Z+ddl,m-Z-m.Z.m/Z/m0Z0m1Z1ddl2m3Z3m4Z4m5Z5ddl6m7Z7ddl8m9Z9ddl:m;Z;mGddde=Z?Gddde=Z@Gddde=ZAGd d!d!e=ZBGd"d#d#e=ZCd$d%ZDGd&d'd'e=ZEd(d)ZFd*d+ZGGd,d-d-eEZHGd.d/d/eEZIGd0d1d1eEZJGd2d3d3eJZKGd4d5d5eJZLdGd8d9ZMGd:d;d;e ZNGdd?d?eNZPGd@dAdAeNZQGdBdCdCeNZRGdDdEdEe ZSdFS)Hwraps)S)Add)cacheit)Expr)FunctionArgumentIndexError_mexpand)fuzzy_or fuzzy_not)RationalpiI)Pow)DummyWild)sympify) factorial)sincoscsccot)ceiling)explog)cbrtsqrtroot)Absreim polar_lift unpolarify)gammadigamma uppergamma)hyper)spherical_bessel_fn)mpworkprecc@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. rargsselfr1E/tmp/pip-target-vg8gfxp4/lib/python/sympy/functions/special/bessel.pyorder4 zBesselBase.ordercCr,)z+ The argument of the Bessel-type function. r-r/r1r1r2argument9r4zBesselBase.argumentcCsdSNr1clsnuzr1r1r2eval>szBesselBase.evalcCsN|dkr t|||jd||jd|j|jd||jd|jSNr=r5)r _b __class__r3r6_ar0argindexr1r1r2fdiffBs  zBesselBase.fdiffcCs*|j}|jdur||j|SdSNF)r6is_extended_negativer@r3 conjugater0r;r1r1r2_eval_conjugateHs zBesselBase._eval_conjugatecCsx|j|j}}||rdS|||sdS|||}|jr2t|ttt t t t fs-|j s2t|jStt|j |jgSrE)r3r6has_eval_is_meromorphicsubs is_integer isinstancebesseljbesselihn1hn2jnynis_zeror is_infiniter )r0xar:r;z0r1r1r2rKMs    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|SNr5r=) r3r6r@is_real is_positiverAr?_eval_expand_func is_negative)r0hintsr:r;fr1r1r2r]Zs & &zBesselBase._eval_expand_funccKsddlm}||S)Nr) besselsimp)sympy.simplify.simplifyra)r0kwargsrar1r1r2_eval_simplifyes zBesselBase._eval_simplifyNr=)__name__ __module__ __qualname____doc__propertyr3r6 classmethodr<rDrIrKr]rdr1r1r1r2r+$s     r+cdeZdZdZejZejZeddZ ddZ ddZ dd Z dd d Z ddZdfdd ZZS)rOa3 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/ cCsR|jr,|jr tjS|jr|jdust|jrtjSt|jr&|jdur&tjS|j r,tj S|tj tj fvr7tjS| rK||| | t|| S|jrn| r^tj| t| |S|t}|rnt|t||S|jrt|}||kr~t||Sn|\}}|dkrtd|t|tt||St|}||krt||SdSNFTrr=)rUrOnerMr r\Zeror^ComplexInfinity is_imaginaryNaNInfinityNegativeInfinitycould_extract_minus_signrO NegativeOneextract_multiplicativelyrrPr#extract_branch_factorrrr9r:r;newznnnur1r1r2r<s>    " z besselj.evalcKs(ttt|dt|tt |SNr=)rrrrPr"r0r:r;rcr1r1r2_eval_rewrite_as_besseli(z besselj._eval_rewrite_as_besselicKs<|jdurtt|t| |tt|t||SdSrE)rMrrbesselyrr~r1r1r2_eval_rewrite_as_bessely .z besselj._eval_rewrite_as_besselycK"td|tt|tj|jSr})rrrSrHalfr6r~r1r1r2_eval_rewrite_as_jn"zbesselj._eval_rewrite_as_jnNrcCsN|j\}}||}||jvr||d|t|dS||||dSNr=r5r)r.as_leading_term free_symbolsr$funcrL)r0rWlogxcdirr:r;argr1r1r2_eval_as_leading_terms   zbesselj._eval_as_leading_termcC"|j\}}|jr |jrdSdSdSNTr.rMis_extended_realr0r:r;r1r1r2_eval_is_extended_real  zbesselj._eval_is_extended_realc s"ddlm}|j\}}z ||\}} Wn ttfy!|YSw| jrt|| } ||||} |d|||| } | t j urE| St | d|  } | |t |d}|g}td| ddD]}|| |||9}t ||  }||qet|| Stt|||||SNr)Orderr=r5)sympy.series.orderrr.leadterm ValueErrorNotImplementedErrorr\r _eval_nseriesremoveOrror r$rangeappendrsuperrO)r0rWr{rrrr:r;_rnewnorttermskr@r1r2rs,      zbesselj._eval_nseriesNrr)rfrgrhrirrnrAr?rkr<rrrrrr __classcell__r1r1rr2rOjsD # rOcrl)ra_ 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/ cCsz|jr|jr tjSt|jdurtjSt|jrtjS|tjtjfvr&tjS|jr9| r;tj | t | |SdSdSrE) rUrrtr rprrrsrorMrurvrr8r1r1r2r<7s z bessely.evalcKs<|jdurtt|tt|t||t| |SdSrE)rMrrrrOr~r1r1r2_eval_rewrite_as_besseljGrz bessely._eval_rewrite_as_besseljcK|j|j}|r |tSdSr7)rr.rewriterPr0r:r;rcajr1r1r2rK  z bessely._eval_rewrite_as_besselicKrr})rrrTrrr6r~r1r1r2_eval_rewrite_as_ynPrzbessely._eval_rewrite_as_ynNrc Cs|j\}}dtt|dt||}|jr'|d|  t|dtntj}|d| tt|t|dtj }t |||gj ||d} || j vsV|| j vrX| S| |||dS)Nr=r5)rr)r.rrrOr\rrror% EulerGammarrrrrLcancel) r0rWrrr:r;term_oneterm_two term_threerr1r1r2rSs ,,zbessely._eval_as_leading_termcCrrr.rMr\rr1r1r2r^rzbessely._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.rrrr\rMrrOrrrrrror rnrrrr%rrrr)r0rWr{rrrr:r;rrrbnrXbcrrrrrprr1r2rcsD    $        zbessely._eval_nseriesrr)rfrgrhrirrnrAr?rkr<rrrrrrrr1r1rr2r s(   rc@sJeZdZdZej ZejZeddZ ddZ ddZ dd Z d d Z d S) rPa 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/ cCsN|jr,|jr tjS|jr|jdust|jrtjSt|jr&|jdur&tjS|j r,tj St |tj tj fvr9tjS|rM||| | t|| S|jrl|rZt| |S|t}|rlt| t|| S|jr}t|}||kr|t||Sn|\}}|dkrtd|t|tt||St|}||krt||SdSrm)rUrrnrMr r\ror^rprqrrr!rsrtrurPrwrrOr#rxrrryr1r1r2r<s>     " z besseli.evalcKs(tt t|dt|tt|Sr})rrrrOr"r~r1r1r2rrz besseli._eval_rewrite_as_besseljcKrr7rr.rrrr1r1r2rrz besseli._eval_rewrite_as_besselycKs|j|jtSr7)rr.rrSr~r1r1r2rszbesseli._eval_rewrite_as_jncCrrrrr1r1r2rrzbesseli._eval_is_extended_realN)rfrgrhrirrnrAr?rkr<rrrrr1r1r1r2rPs' # rPc@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 | |SdSdSrE) rUrrsr rprrrrtrorMrurr8r1r1r2r<s  z besselk.evalcKs8|jdurttt|t| |t||dSdS)NFr=)rMrrrPr~r1r1r2r(s *z besselk._eval_rewrite_as_besselicKrr7)rr.rrO)r0r:r;rcair1r1r2r,rz besselk._eval_rewrite_as_besseljcKrr7rrr1r1r2r1rz besselk._eval_rewrite_as_besselycKrr7)rr.rrT)r0r:r;rcayr1r1r2r6rzbesselk._eval_rewrite_as_yncCrrrrr1r1r2r;rzbesselk._eval_is_extended_realN)rfrgrhrirrnrAr?rkr<rrrrrr1r1r1r2rs%  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|SdSrE)r6rFhankel2r3rGrHr1r1r2rIi zhankel1._eval_conjugateN rfrgrhrirrnrAr?rIr1r1r1r2rAs $ 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/ cCrrE)r6rFrr3rGrHr1r1r2rIrzhankel2._eval_conjugateNrr1r1r1r2ros % rcstfdd}|S)Ncs|jr |||SdSr7)rMrfnr1r2gs zassume_integer_order..gr)rrr1rr2assume_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. expansionrr0r_r1r1r2_expandszSphericalBesselBase._expandcKs|jjr |jdi|S|SNr1)r3 is_Integerrrr1r1r2r]sz%SphericalBesselBase._eval_expand_funcr=cCs:|dkr t||||jd|j||jd|jSr>)r r@r3r6rBr1r1r2rDs  zSphericalBesselBase.fdiffNre)rfrgrhrirr]rDr1r1r1r2rs  rcCs8t||t|tj|dt| d|t|SNr5)r(rrrvrr{r;r1r1r2_jns$rcCs8tj|dt| d|t|t||t|Sr)rrvr(rrrr1r1r2_yns$rc@sDeZdZdZeddZddZddZdd Zd d Z d d Z dS)rSa 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 tjSdSr7) rUrrnrMr\rorprtrsr8r1r1r2r<szjn.evalcK ttd|t|tj|Sr})rrrOrrr~r1r1r2rs zjn._eval_rewrite_as_besseljcKs,tj|ttd|t| tj|Sr})rrvrrrrr~r1r1r2rs,zjn._eval_rewrite_as_besselycKstj|t| d|Sr)rrvrTr~r1r1r2rszjn._eval_rewrite_as_yncKt|j|jSr7)rr3r6rr1r1r2rz jn._expandcC|jjr |t|SdSr7r3rrrO _eval_evalfr0precr1r1r2r zjn._eval_evalfN) rfrgrhrirkr<rrrrrr1r1r1r2rSs9   rSc@s@eZdZdZeddZeddZddZdd Zd d Z d S) rTa 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 cKs0tj|dttd|t| tj|SrZ)rrvrrrOrr~r1r1r2rTs0zyn._eval_rewrite_as_besseljcKrr})rrrrrr~r1r1r2rXs zyn._eval_rewrite_as_besselycKstj|dt| d|Sr)rrvrSr~r1r1r2r\szyn._eval_rewrite_as_jncKrr7)rr3r6rr1r1r2r_rz yn._expandcCrr7)r3rrrrrr1r1r2rbrzyn._eval_evalfN) rfrgrhrirrrrrrr1r1r1r2rT%s.   rTc@sLeZdZeddZeddZddZddZd d Zd d Z d dZ dS)SphericalHankelBasecKsN|j}ttd|t|tj||ttj|dt| tj|Sr>)_hankel_kind_signrrrOrrrrvr0r:r;rchksr1r1r2ris&z,SphericalHankelBase._eval_rewrite_as_besseljcKsJ|j}ttd|tj|t| tj||tt|tj|Sr})rrrrrvrrrrr1r1r2rrs(z,SphericalHankelBase._eval_rewrite_as_besselycKs(|j}t||t|tt||Sr7)rrSrrTrrr1r1r2r{"z'SphericalHankelBase._eval_rewrite_as_yncKs(|j}t|||tt||tSr7)rrSrrTrrr1r1r2rrz'SphericalHankelBase._eval_rewrite_as_jncKsF|jjr |jdi|S|j}|j}|j}t|||tt||Sr)r3rrr6rrSrrT)r0r_r:r;rr1r1r2r]s z%SphericalHankelBase._eval_expand_funccKs2|j}|j}|j}t|||tt||Sr7)r3r6rrrrexpand)r0r_r{r;rr1r1r2rs zSphericalHankelBase._expandcCrr7rrr1r1r2rrzSphericalHankelBase._eval_evalfN) rfrgrhrrrrrr]rrr1r1r1r2rgs   rc@s"eZdZdZejZeddZdS)rQa 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||Sr})rrrr~r1r1r2_eval_rewrite_as_hankel1zhn1._eval_rewrite_as_hankel1N) rfrgrhrirrnrrrr1r1r1r2rQs 0rQc@s$eZdZdZej ZeddZdS)rRa 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 cKrr})rrrr~r1r1r2_eval_rewrite_as_hankel2 rzhn2._eval_rewrite_as_hankel2N) rfrgrhrirrnrrrr1r1r1r2rRs 0rRsympyc sddlm}dkr*ddlmddlm}||fddtd|dDSd krZdd lmzdd l m fd d }Wnt yYddl m fdd }Ynwt dfdd}|}|||}|g} t|dD]} ||||}| |qw| 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)rr) besseljzero) dps_to_preccs0g|]}ttdt|qS)g?)r _from_mpmathr _to_mpmathint).0l)rr{rr1r2 Cszjn_zeros..r5scipy)newton) spherical_jncs |Sr7r1rW)r{rr1r2Js zjn_zeros..)sph_jncs|ddS)Nrr1r)r{rr1r2rMsUnknown method.csdkr ||}|Std)Nrrr)r`rWr)methodrr1r2solverQs zjn_zeros..solver)mathrmpmathrmpmath.libmp.libmpfrrscipy.optimizer scipy.specialr ImportErrorrrr) r{rrdpsmath_pirr`rrrootsir1)rrr{rrrrr2jn_zeross4 -         r c@s4eZdZdZddZddZd ddZd d d Zd S) AiryBasezg Abstract base class for Airy functions. This class is meant to reduce code duplication. cCs||jdSr)rr.rGr/r1r1r2rIlszAiryBase._eval_conjugatecCs |jdjSr)r.rr/r1r1r2ros zAiryBase._eval_is_extended_realTcKsL|jd}|}|j}||||d}t||||d}||fS)Nrr=)r.rGrr)r0deepr_r;zcr`uvr1r1r2 as_real_imagrs zAiryBase.as_real_imagcKs&|jdd|i|\}}||tjS)Nr r1)rr ImaginaryUnit)r0r r_re_partim_partr1r1r2_eval_expand_complexzszAiryBase._eval_expand_complexN)T)rfrgrhrirIrrrr1r1r1r2r ds  r c@^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 r5TcCs|jr/|tjur tjS|tjurtjS|tjurtjS|jr/tjdtddt tddS|jrCtjdtddt tddSdS)Nr=) is_NumberrrrrsrortrUrnr r$r9rr1r1r2r<s   ""z airyai.evalcC |dkr t|jdSt||Nr5r) airyaiprimer.r rBr1r1r2rD z airyai.fdiffcGs6|dkrtjSt|}t|dkrj|d}td|| td||dtt|tddtddt|t |dtddtt|tddtddt|dt |dtdd|Stj dtddtt |tj tdttddt|tj t|td||S)Nrr5rrr=) rrorlenrrrr rr$rnr{rWprevious_termsrr1r1r2 taylor_terms" L@Hzairyai.taylor_termcKs`tdd}tdd}t| tdd}t|jr.|t| t| ||t|||SdSNr5rr=r rr r^rrOr0r;rcotttrXr1r1r2r   ,zairyai._eval_rewrite_as_besseljcKstdd}tdd}t|tdd}t|jr,|t|t| ||t|||S|t||t| |||t|| t|||Sr"r rr r\rrPr$r1r1r2rs   *<zairyai._eval_rewrite_as_besselicKs~tjdtddttdd}|tddttdd}|tgtddg|dd|tgtddg|ddS)Nrr=r5 r)rrnr r$rr'r0r;rcpf1pf2r1r1r2_eval_rewrite_as_hypers"@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 Nrr5r)excludedmr{r) r.rrpoprmatchrMrrrnrrairybi r0r_rsymbsr;rr0r1r{Mpfnewargr1r1r2r]*   $2zairyai._eval_expand_funcNr5rfrgrhrinargs unbranchedrkr<rD staticmethodrr!rrr-r]r1r1r1r2rsX     rc@r)r4a 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 r5TcCs|jr/|tjur tjS|tjurtjS|tjurtjS|jr/tjdtddt tddS|jrCtjdtddt tddSdS)Nrr5r=) rrrrrsrtrorUrnr r$rr1r1r2r<s   ""z airybi.evalcCrr) airybiprimer.r rBr1r1r2rDrz airybi.fdiffcGs|dkrtjSt|}t|dkrW|d}td|tttddt|tj t |tj td|tj tt tddt|tj t |dtd|Stj t ddtt|tj tdtttddt|tj t |td||S)Nrr5rrr=r@)rrorrrrrr rrnrrrrr$rr1r1r2r!s @<Hzairybi.taylor_termcKs`tdd}tdd}t| tdd}t|jr.t| dt| ||t|||SdSr"r#r$r1r1r2rr'zairybi._eval_rewrite_as_besseljcKstdd}tdd}t|tdd}t|jr.t|tdt| ||t|||St||}t|| }t||t| ||||t|||Sr"r(r0r;rcr%r&rXrrr1r1r2rs   .  2zairybi._eval_rewrite_as_besselicKsztjtddttdd}|tddttdd}|tgtddg|dd|tgtddg|ddS)Nrr@r=r5r)r)rrnrr$r r'r*r1r1r2r-s@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.rrr2rr3rMrrrrnrr4r5r1r1r2r]r:zairybi._eval_expand_funcNr;r<r1r1r1r2r4+sZ     r4c@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 r5TcCsR|jr|tjur tjS|tjurtjS|jr'tjdtddttddSdS)Nrr5) rrrrrsrorUrvr r$rr1r1r2r<1s  "zairyaiprime.evalcC*|dkr|jdt|jdSt||r)r.rr rBr1r1r2rD< zairyaiprime.fdiffcCR|jd|}t|tj|dd}Wdn1swYt||SNrr5) derivative)r.rr*r)rrrr0rr;resr1r1r2rB   zairyaiprime._eval_evalfcKsPtdd}t| tdd}t|jr&|dt| ||t|||SdSNr=r)r rr r^rOr0r;rcr&rXr1r1r2rHs  &z$airyaiprime._eval_rewrite_as_besseljcKstdd}tdd}|t|tdd}t|jr(|dt||t| |St|tdd}t||}t|| }||d|t||||t| ||Sr")r rr r\rPrBr1r1r2rNs     2z$airyaiprime._eval_rewrite_as_besselicKs|dddtddttdd}dtddttdd}|tgtddg|dd|tgtddg|ddS)Nr=rr5r))r r$rr'r*r1r1r2r-Zs(@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.rrr2rr3rMrrrnrrrAr5r1r1r2r]_*   $2zairyaiprime._eval_expand_funcNr;rfrgrhrir=r>rkr<rDrrrr-r]r1r1r1r2rsQ   rc@rC)rAa< 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 .. [4] http://mathworld.wolfram.com/AiryFunctions.html r5TcCs~|jr,|tjur tjS|tjurtjS|tjurtjS|jr,dtddttddS|jr=dtddttddSdS)Nrr5r@) rrrrrsrtrorUr r$rr1r1r2r<s   zairybiprime.evalcCrDr)r.r4r rBr1r1r2rDrEzairybiprime.fdiffcCrFrG)r.rr*r)r4rrrIr1r1r2rrKzairybiprime._eval_evalfcKsRtdd}|t| tdd}t|jr'| tdt| |t||SdSrLr#rMr1r1r2rs  $z$airybiprime._eval_rewrite_as_besseljcKstdd}tdd}|t|tdd}t|jr*|tdt| |t||St|tdd}t||}t|| }t||t| |||d|t|||Sr"r(rBr1r1r2rs   "  6z$airybiprime._eval_rewrite_as_besselicKs||ddtddttdd}tddttdd}|tgtddg|dd|tgtddg|ddS)Nr=rr@r5rNr))rr$r r'r*r1r1r2r-s$@z"airybiprime._eval_rewrite_as_hyperc Ks"|jd}|j}t|dkr|}td|gd}td|gd}td|gd}td|gd}||||||} | dur| |}d|jr| |}| |}| |}|||||||||} ||||||} tjt d| tj t | | tj t | SdSdSdSr.) r.rrr2rr3rMrrrrnrrAr5r1r1r2r]rOzairybiprime._eval_expand_funcNr;rPr1r1r1r2rAysS   rAc@sFeZdZdZeddZdddZddZd d Zd d Z d dZ dS)marcumqa The Marcum Q-function. 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 .. [2] http://mathworld.wolfram.com/MarcumQ-Function.html cCsT|tjur |tjur|tjurtjSt||dtjt|S|tjur7|tjur7ddt|dtjS||kry|tjurSdt|d td|dtjS|dkrytjtjt|d td|dt|d td|dS|jr|jr|jrtjSt||dtjt|S|jr|jrddt|dtjSdSdSr) rror&rr$rrnrPrU)r9r1rXrr1r1r2r<Ns$  &D  z marcumq.evalr=cCs|j\}}}|dkr|t||| td|||S|dkr@|| ||dt|d|d dt|d||St||)Nr=r5r)r.rQrrPr )r0rCr1rXrr1r1r2rDfs "B z marcumq.fdiffcKsjddlm}|dtd}|d||||t|d|d dt|d||||tjgS)Nr)IntegralrWr5r=)sympy.integrals.integralsrRgetrrrPrrs)r0r1rXrrcrRrWr1r1r2_eval_rewrite_as_Integralos  @z!marcumq._eval_rewrite_as_IntegralcKsbddlm}|dtd}t|d|d d||||t||||d|tjgS)Nr)Sumrr=r5)sympy.concrete.summationsrVrTrrrPrrs)r0r1rXrrcrVrr1r1r2_eval_rewrite_as_Sumus Fzmarcumq._eval_rewrite_as_Sumc s|krK|dkrdtd tdddS|jrM|dkrOtfddtd|D}tjtd tdddtd |SdSdSdS)Nr5r=rcsg|] }t|dqSre)rP)rrrXr1r2rsz4marcumq._eval_rewrite_as_besseli..)rrPrsumrrr)r0r1rXrrcrr1rYr2rzs$8z marcumq._eval_rewrite_as_besselicCstdd|jDr dSdS)Ncss|]}|jVqdSr7)rU)rrr1r1r2 sz(marcumq._eval_is_zero..T)allr.r/r1r1r2 _eval_is_zeroszmarcumq._eval_is_zeroNre) rfrgrhrirkr<rDrUrXrr]r1r1r1r2rQs0    rQN)rr)T functoolsr sympy.corersympy.core.addrsympy.core.cachersympy.core.exprrsympy.core.functionrr r sympy.core.logicr r sympy.core.numbersr rrsympy.core.powerrsympy.core.symbolrrsympy.core.sympifyr(sympy.functions.combinatorial.factorialsr(sympy.functions.elementary.trigonometricrrrr#sympy.functions.elementary.integersr&sympy.functions.elementary.exponentialrr(sympy.functions.elementary.miscellaneousrrr$sympy.functions.elementary.complexesrr r!r"r#'sympy.functions.special.gamma_functionsr$r%r&sympy.functions.special.hyperr'sympy.polys.orthopolysr(rr)r*r+rOrrPrrrrrrrrSrTrrQrRr r rr4rrArQr1r1r1r2sd           F"`R./XB98 8T-2%