o e@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. rargsselfr1ND:\Projects\ConvertPro\env\Lib\site-packages\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;Zz0r1r1r2rKMs    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@Zis_real is_positiverAr?_eval_expand_func is_negative)r0hintsr:r;fr1r1r2r[Zs & &zBesselBase._eval_expand_funccKsddlm}||S)Nr) besselsimp)Zsympy.simplify.simplifyr_)r0kwargsr_r1r1r2_eval_simplifyes zBesselBase._eval_simplifyNr=)__name__ __module__ __qualname____doc__propertyr3r6 classmethodr<rDrIrKr[rar1r1r1r2r+$s     r+cheZdZdZejZejZeddZ ddZ ddZ dd Z dfd d Z ddZdfdd ZZS)rOa4 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] https://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||SdS)NFTrr=)rUrOnerMr rZZeror\ComplexInfinity is_imaginaryNaNInfinityNegativeInfinitycould_extract_minus_signrO NegativeOneextract_multiplicativelyrrPr#extract_branch_factorrrr9r:r;ZnewznZnnur1r1r2r<s>    " z besselj.evalcKs(ttt|dt|tt |SNr=)rrrrPr"r0r:r;r`r1r1r2_eval_rewrite_as_besseli(z besselj._eval_rewrite_as_besselicKs<|jdurtt|t| |tt|t||SdSrE)rMrrbesselyrrxr1r1r2_eval_rewrite_as_bessely .z besselj._eval_rewrite_as_besselycK"td|tt|tj|jSrw)rrrSrHalfr6rxr1r1r2_eval_rewrite_as_jn"zbesselj._eval_rewrite_as_jnNrc s|j\}}z||}Wn ty|YSw||\}}|jr0||d|t|dS|jr^|dkr9dn|}|||} | js\tdt|t d|ddtt |S|St t | |||S)Nr=r5r) r.as_leading_termNotImplementedErroras_coeff_exponentrZr$r\rrrsuperrO_eval_as_leading_term r0rWlogxcdirr:r;argcesignr@r1r2rs   0zbesselj._eval_as_leading_termcC"|j\}}|jr |jrdSdSdSNTr.rMis_extended_realr0r:r;r1r1r2_eval_is_extended_real  zbesselj._eval_is_extended_realc s"ddlm}|j\}}z ||\}} Wn ttfy!|YSw| jrt|| } ||||} |d|||| } | t j urE| St | d|  } | |t |d}|g}td| ddD]}|| |||9}t ||  }||qet|| Stt|||||SNrOrderr=r5)sympy.series.orderrr.leadterm ValueErrorrrZr _eval_nseriesremoveOrrkr r$rangeappendrrrOr0rWrvrrrr:r;_rnewnorttermskrr1r2rs,      zbesselj._eval_nseriesNrr)rcrdrerfrrjrAr?rhr<ryr|rrrr __classcell__r1r1rr2rOjsD #rOcri)r{a` 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] https://functions.wolfram.com/Bessel-TypeFunctions/BesselY/ cCs|jr|jr tjSt|jdurtjSt|jrtjS|tjtjfvr&tjS|ttjkr>> 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] https://functions.wolfram.com/Bessel-TypeFunctions/BesselI/ cCsv|jr,|jr tjS|jr|jdust|jrtjSt|jr&|jdur&tjS|j r,tj St |tj tj fvr9tjS|tj urAtj S|tj urMd|tj S|ra||| | t|| S|jr|rnt| |S|t}|rt| t|| S|jrt|}||krt||Sn|\}}|dkrtd|t|tt||St|}||krt||SdS)NFTrr=)rUrrjrMr rZrkr\rlrmrnr!rorprqrPrsrrOr#rtrrrur1r1r2r<sF       " z besseli.evalcKs(tt t|dt|tt|Srw)rrrrOr"rxr1r1r2r rzz besseli._eval_rewrite_as_besseljcKrr7rr.rr{rr1r1r2r| rz besseli._eval_rewrite_as_besselycKs|j|jtSr7)rr.rrSrxr1r1r2rszbesseli._eval_rewrite_as_jncCrrrrr1r1r2rrzbesseli._eval_is_extended_realNrc s|j\}}z||}Wn ty|YSw||\}}|jr0||d|t|dS|jrR|dkr9dn|}|||} | jsPt|tdt |S|St t | |||SNr=r5r) r.rrrrZr$r\rrrrrPrrrr1r2rs   zbesseli._eval_as_leading_termc s ddlm}|j\}}z ||\}} Wn ttfy!|YSw| jrt|| } ||||} |d|||| } | t j urE| St | d|  } | |t |d}|g}td| ddD]}|| |||9}t ||  }||qet|| Stt|||||Sr)rrr.rrrrZrrrrrkr r$rrrrrPrrr1r2r.s,      zbesseli._eval_nseriesrr)rcrdrerfrrjrAr?rhr<rr|rrrrrr1r1rr2rPs' 'rPcsreZdZdZejZej ZeddZ ddZ ddZ dd Z d d Z d d Zdfdd Zdfdd ZZS)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] https://functions.wolfram.com/Bessel-TypeFunctions/BesselK/ cCsz|jr|jr tjSt|jdurtjSt|jrtjS|tjttjttjfvr,tjS|j r9| r;t | |SdSdSrE) rUrror rlrnrrprkrMrqrr8r1r1r2r<ws  z besselk.evalcKs8|jdurttt|t| |t||dSdS)NFr=)rMrrrPrxr1r1r2rys *z besselk._eval_rewrite_as_besselicKrr7)ryr.rrO)r0r:r;r`Zair1r1r2rrz besselk._eval_rewrite_as_besseljcKrr7rrr1r1r2r|rz besselk._eval_rewrite_as_besselycKrr7)r|r.rrT)r0r:r;r`Zayr1r1r2rrzbesselk._eval_rewrite_as_yncCrrrrr1r1r2rrzbesselk._eval_is_extended_realNrc s|j\}}z||}Wn ty|YSw||\}}|jrod|dt|dt||} |jrE|d| t|ddntj } d||d|dt|t |dtj } t | | | gj||d}|S|j rttt| td|Stt||||S)Nrr5r=r)r.rrrrZrrPrrrkr%rrr\rrrrrr) r0rWrrr:r;rrrrrrrr1r2rs  "*2zbesselk._eval_as_leading_termc s8ddlm}|j\}}z ||\}} Wn ttfy!|YSw| jr|jrt|| } t ||} d|dt |d|  ||||} gg} }||||}|d |||| }|t jurh|St|d| }|t jkr|| t|dd}| |td|D]'}|||}|t jkr|||9}n|||9}t|| }| |q||d|dt|}|t|dt j}||td| ddD])}|||||9}t|| }|t||dt|d}||q| t| t|Stt| ||||S)Nrrrr5r=)rrr.rrrrZrMrrPrrrrrkr rrrr%rrrrrrr1r2rsJ    (           zbesselk._eval_nseriesrr)rcrdrerfrrjrAr?rhr<ryrr|rrrrrr1r1rr2rNs% 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] https://functions.wolfram.com/Bessel-TypeFunctions/HankelH1/ cC(|j}|jdurt|j|SdSrE)r6rFhankel2r3rGrHr1r1r2rI zhankel1._eval_conjugateN rcrdrerfrrjrAr?rIr1r1r1r2rs $ 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] https://functions.wolfram.com/Bessel-TypeFunctions/HankelH2/ cCrrE)r6rFrr3rGrHr1r1r2rI=rzhankel2._eval_conjugateNrr1r1r1r2rs % rcstfdd}|S)Ncs|jr |||SdSr7)rMrfnr1r2gDs zassume_integer_order..gr)rrr1rr2assume_integer_orderCsrc@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 expansionrr0r]r1r1r2_expandWszSphericalBesselBase._expandcKs|jjr |jdi|S|SNr1)r3 is_Integerrrr1r1r2r[[sz%SphericalBesselBase._eval_expand_funcr=cCs:|dkr t||||jd|j||jd|jSr>)r r@r3r6rBr1r1r2rD`s  zSphericalBesselBase.fdiffNrb)rcrdrerfrr[rDr1r1r1r2rKs  rcCs8t||t|tj|dt| d|t|SNr5)r(rrrrrrvr;r1r1r2_jngs$rcCs8tj|dt| d|t|t||t|Sr)rrrr(rrrr1r1r2_ynls$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] https://dlmf.nist.gov/10.47 cCsD|jr|jr tjS|jr|jrtjStjS|tjtjfvr tjSdSr7) rUrrjrMrZrkrlrpror8r1r1r2r<szjn.evalcK ttd|t|tj|Srw)rrrOrrrxr1r1r2rs zjn._eval_rewrite_as_besseljcKs,tj|ttd|t| tj|Srw)rrrrrr{rrxr1r1r2r|s,zjn._eval_rewrite_as_besselycKstj|t| d|Sr)rrrrTrxr1r1r2rszjn._eval_rewrite_as_yncKt|j|jSr7)rr3r6rr1r1r2rz jn._expandcC|jjr |t|SdSr7r3rrrO _eval_evalfr0precr1r1r2rzjn._eval_evalfN) rcrdrerfrhr<rr|rrrr1r1r1r2rSrs9   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] https://dlmf.nist.gov/10.47 cKs0tj|dttd|t| tj|SrY)rrrrrrOrrxr1r1r2rs0zyn._eval_rewrite_as_besseljcKrrw)rrr{rrrxr1r1r2r|s zyn._eval_rewrite_as_besselycKstj|dt| d|Sr)rrrrSrxr1r1r2rszyn._eval_rewrite_as_jncKrr7)rr3r6rr1r1r2rrz yn._expandcCrr7)r3rrr{rrr1r1r2rrzyn._eval_evalfN) rcrdrerfrrr|rrrr1r1r1r2rTs.   rTc@sLeZdZeddZeddZddZddZd d Zd d Z d dZ dS)SphericalHankelBasecKsN|j}ttd|t|tj||ttj|dt| tj|Sr>)_hankel_kind_signrrrOrrrrrr0r:r;r`hksr1r1r2rs&z,SphericalHankelBase._eval_rewrite_as_besseljcKsJ|j}ttd|tj|t| tj||tt|tj|Srw)rrrrrrr{rrrr1r1r2r|s(z,SphericalHankelBase._eval_rewrite_as_besselycKs(|j}t||t|tt||Sr7)rrSrrTrrr1r1r2r "z'SphericalHankelBase._eval_rewrite_as_yncKs(|j}t|||tt||tSr7)rrSrrTrrr1r1r2r$rz'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]rvr;rr1r1r2r1s zSphericalHankelBase._expandcCrr7rrr1r1r2r@rzSphericalHankelBase._eval_evalfN) rcrdrerrr|rrr[rrr1r1r1r2r s   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] https://dlmf.nist.gov/10.47 cKttd|t||Srw)rrrrxr1r1r2_eval_rewrite_as_hankel1xzhn1._eval_rewrite_as_hankel1N) rcrdrerfrrjrrrr1r1r1r2rQEs 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] https://dlmf.nist.gov/10.47 cKrrw)rrrrxr1r1r2_eval_rewrite_as_hankel2rzhn2._eval_rewrite_as_hankel2N) rcrdrerfrrjrrrr1r1r1r2rR}s 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)rrvrr1r2 szjn_zeros..r5scipy)newton) spherical_jncs |Sr7r1rW)rvrr1r2s zjn_zeros..)sph_jncs|ddS)Nrrr1r)rvrr1r2rsUnknown method.csdkr ||}|Std)Nrrr)r^rWr)methodrr1r2solvers zjn_zeros..solver)mathrmpmathrZmpmath.libmp.libmpfrrZscipy.optimizerZ scipy.specialr ImportErrorrrr) rvrrZdpsZmath_pirr^rrrootsir1)rrrvrrrrr2jn_zeross4 -         rc@s4eZdZdZddZddZd ddZd d d Zd S) AiryBasezg Abstract base class for Airy functions. This class is meant to reduce code duplication. cCs||jdSr)funcr.rGr/r1r1r2rIszAiryBase._eval_conjugatecCs |jdjSr)r.rr/r1r1r2rs zAiryBase._eval_is_extended_realTcKsL|jd}|}|j}||||d}t||||d}||fS)Nrr=)r.rGrr)r0deepr]r;Zzcr^uvr1r1r2 as_real_imags zAiryBase.as_real_imagcKs$|jdd|i|\}}||tS)Nrr1)rr)r0rr]Zre_partZim_partr1r1r2_eval_expand_complexs zAiryBase._eval_expand_complexN)T)rcrdrerfrIrrr r1r1r1r2r 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] https://dlmf.nist.gov/9 .. [3] https://encyclopediaofmath.org/wiki/Airy_functions .. [4] https://mathworld.wolfram.com/AiryFunctions.html r5TcCs|jr/|tjur tjS|tjurtjS|tjurtjS|jr/tjdtddt tddS|jrCtjdtddt tddSdS)Nrr=) is_NumberrrnrorkrprUrjr r$r9rr1r1r2r<s   ""z airyai.evalcC |dkr t|jdSt||Nr5r) airyaiprimer.r rBr1r1r2rD z airyai.fdiffcGs6|dkrtjSt|}t|dkrj|d}td|| td||dtt|tddtddt|t |dtddtt|tddtddt|dt |dtdd|Stj dtddtt |tj tdttddt|tj t|td||S)Nrr5rrr=r) rrkrlenrrrr rr$rjrvrWZprevious_termsrr1r1r2 taylor_terms" L@Hzairyai.taylor_termcKs`tdd}tdd}t| tdd}t|jr.|t| t| ||t|||SdSNr5rr=r rr r\rrOr0r;r`otttrXr1r1r2r   ,zairyai._eval_rewrite_as_besseljcKstdd}tdd}t|tdd}t|jr,|t|t| ||t|||S|t||t| |||t|| t|||Srr rr rZrrPrr1r1r2rys   *<zairyai._eval_rewrite_as_besselicKs~tjdtddttdd}|tddttdd}|tgtddg|dd|tgtddg|ddS)Nrr=r5 r)rrjr r$rr'r0r;r`Zpf1Zpf2r1r1r2_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)excludedmrvr) r. free_symbolsrpoprmatchrMrrrjr rairybi r0r]rZsymbsr;rr!r"rvMpfZnewargr1r1r2r[*   $2zairyai._eval_expand_funcNr5rcrdrerfnargs unbranchedrhr<rD staticmethodrrrryrr[r1r1r1r2r $sX     r c@r )r&a 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] https://dlmf.nist.gov/9 .. [3] https://encyclopediaofmath.org/wiki/Airy_functions .. [4] https://mathworld.wolfram.com/AiryFunctions.html r5TcCs|jr/|tjur tjS|tjurtjS|tjurtjS|jr/tjdtddt tddS|jrCtjdtddt tddSdS)Nrr5r=) r rrnrorprkrUrjr r$r r1r1r2r<.s   ""z airybi.evalcCrr) airybiprimer.r rBr1r1r2rD=rz airybi.fdiffcGs|dkrtjSt|}t|dkrW|d}td|tttddt|tj t |tj td|tj tt tddt|tj t |dtd|Stj t ddtt|tj tdtttddt|tj t |td||S)Nrr5rrr=r0)rrkrrrrrr rrjrrrrr$rr1r1r2rCs @<Hzairybi.taylor_termcKs`tdd}tdd}t| tdd}t|jr.t| dt| ||t|||SdSrrrr1r1r2rRrzairybi._eval_rewrite_as_besseljcKstdd}tdd}t|tdd}t|jr.t|tdt| ||t|||St||}t|| }t||t| ||||t|||Srrr0r;r`rrrXrrr1r1r2ryYs   .  2zairybi._eval_rewrite_as_besselicKsztjtddttdd}|tddttdd}|tgtddg|dd|tgtddg|ddS)Nrr0r=r5rr)rrjrr$r r'rr1r1r2rds@zairybi._eval_rewrite_as_hyperc Ks"|jd}|j}t|dkr|}td|gd}td|gd}td|gd}td|gd}||||||} | dur| |}d|jr| |}| |}| |}|||||||||} ||||||} tjt dtj | t | tj | t | SdSdSdSr) r.r#rr$rr%rMrrrrjr r&r'r1r1r2r[ir*zairybi._eval_expand_funcNr+r,r1r1r1r2r&sZ     r&c@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] https://dlmf.nist.gov/9 .. [3] https://encyclopediaofmath.org/wiki/Airy_functions .. [4] https://mathworld.wolfram.com/AiryFunctions.html r5TcCsR|jr|tjur tjS|tjurtjS|jr'tjdtddttddSdS)Nrr5) r rrnrorkrUrrr r$r r1r1r2r<s  "zairyaiprime.evalcC*|dkr|jdt|jdSt||r)r.r r rBr1r1r2rD zairyaiprime.fdiffcCR|jd|}t|tj|dd}Wdn1swYt||SNrr5)Z derivative)r.rr*r)r rrr0rr;resr1r1r2r   zairyaiprime._eval_evalfcKsPtdd}t| tdd}t|jr&|dt| ||t|||SdSNr=r)r rr r\rOr0r;r`rrXr1r1r2rs  &z$airyaiprime._eval_rewrite_as_besseljcKstdd}tdd}|t|tdd}t|jr(|dt||t| |St|tdd}t||}t|| }||d|t||||t| ||Sr)r rr rZrPr2r1r1r2rys     2z$airyaiprime._eval_rewrite_as_besselicKs|dddtddttdd}dtddttdd}|tgtddg|dd|tgtddg|ddS)Nr=rr5r)r r$rr'rr1r1r2rs(@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.r#rr$rr%rMrrrjrrr1r'r1r1r2r[*   $2zairyaiprime._eval_expand_funcNr+rcrdrerfr-r.rhr<rDrrryrr[r1r1r1r2rsQ   rc@r3)r1a6 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] https://dlmf.nist.gov/9 .. [3] https://encyclopediaofmath.org/wiki/Airy_functions .. [4] https://mathworld.wolfram.com/AiryFunctions.html r5TcCs~|jr,|tjur tjS|tjurtjS|tjurtjS|jr,dtddttddS|jr=dtddttddSdS)Nrr5r0) r rrnrorprkrUr r$r r1r1r2r<us   zairybiprime.evalcCr4r)r.r&r rBr1r1r2rDr5zairybiprime.fdiffcCr6r7)r.rr*r)r&rrr8r1r1r2rr:zairybiprime._eval_evalfcKsRtdd}|t| tdd}t|jr'| tdt| |t||SdSr;rr<r1r1r2rs  $z$airybiprime._eval_rewrite_as_besseljcKstdd}tdd}|t|tdd}t|jr*|tdt| |t||St|tdd}t||}t|| }t||t| |||d|t|||Srrr2r1r1r2rys   "  6z$airybiprime._eval_rewrite_as_besselicKs||ddtddttdd}tddttdd}|tgtddg|dd|tgtddg|ddS)Nr=rr0r5r=r)rr$r r'rr1r1r2rs$@z"airybiprime._eval_rewrite_as_hyperc Ks"|jd}|j}t|dkr|}td|gd}td|gd}td|gd}td|gd}||||||} | dur| |}d|jr| |}| |}| |}|||||||||} ||||||} tjt d| tj t | | tj t | SdSdSdSr) r.r#rr$rr%rMrrrrjrr1r'r1r1r2r[r>zairybiprime._eval_expand_funcNr+r?r1r1r1r2r1sS   r1c@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 .. 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