a kh0 @sddlmZddlmZddlmZddlmZddlm Z ddl m Z m Z m Z ddlmZmZddlmZmZmZdd lmZdd lmZmZmZdd lmZdd lmZmZdd l m!Z!m"Z"m#Z#m$Z$ddl%m&Z&ddl'm(Z(m)Z)ddl*m+Z+m,Z,m-Z-ddl.m/Z/m0Z0m1Z1m2Z2m3Z3ddl4m5Z5m6Z6m7Z7ddl8m9Z9ddl:m;Z;ddlZ>Gddde Z?Gddde?Z@Gddde?ZAGddde?ZBGddde?ZCGd d!d!e?ZDGd"d#d#e?ZEd$d%ZFGd&d'd'e?ZGd(d)ZHd*d+ZIGd,d-d-eGZJGd.d/d/eGZKGd0d1d1eGZLGd2d3d3eLZMGd4d5d5eLZNdKd8d9ZOGd:d;d;e ZPGdd?d?ePZRGd@dAdAePZSGdBdCdCePZTGdDdEdEe ZUGdFdGdGe ZVGdHdIdIe ZWdJS)Lwraps)S)Add)cacheit)Expr)DefinedFunctionArgumentIndexError_mexpand)fuzzy_or fuzzy_not)RationalpiI)Pow)Dummyuniquely_named_symbolWild)sympify) factorialRisingFactorial)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}``. cCs |jdS)z( The order of the Bessel-type function. rargsselfr2L/var/www/auris/lib/python3.9/site-packages/sympy/functions/special/bessel.pyorder4szBesselBase.ordercCs |jdS)z+ The argument of the Bessel-type function. r.r0r2r2r3argument9szBesselBase.argumentcCsdSNr2clsnuzr2r2r3eval>szBesselBase.evalcCsN|dkrt|||jd||jd|j|jd||jd|jSNr=r5)r _b __class__r4r6_ar1argindexr2r2r3fdiffBs  zBesselBase.fdiffcCs*|j}|jdur&||j|SdSNF)r6is_extended_negativer@r4 conjugater1r;r2r2r3_eval_conjugateHs zBesselBase._eval_conjugatecCsx|j|j}}||rdS|||s,dS|||}|jrdt|ttt t t t fsZ|j sdt|jStt|j |jgSrE)r4r6Zhas_eval_is_meromorphicsubs is_integer isinstancebesseljbesselihn1hn2jnynis_zeror is_infiniter )r1xar:r;Zz0r2r2r3rJMs    zBesselBase._eval_is_meromorphiccKs|j|j|j}}}|jr|djrn|j |j||d|d|j|d||d||S|djrd|j|d||d|||j|j||d|S|SNr5r=) r4r6r@Zis_real is_positiverAr?_eval_expand_func is_negative)r1hintsr:r;fr2r2r3rZZs & &zBesselBase._eval_expand_funccKsddlm}||S)Nr) besselsimp)Zsympy.simplify.simplifyr^)r1kwargsr^r2r2r3_eval_simplifyes zBesselBase._eval_simplifyN)r=)__name__ __module__ __qualname____doc__propertyr4r6 classmethodr<rDrIrJrZr`r2r2r2r3r-$s      r-csfeZdZdZejZejZeddZ ddZ ddZ dd Z fd d Z d d Zdfdd ZZS)rNa4 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/ cCsZ|jrX|jrtjS|jr"|jdus,t|jr2tjSt|jrL|jdurLtjS|j rXtj S|tj tj fvrntjS| r||| | t|| S|jr| rtj| t| |S|t}|rt|t||S|jrt|}||kr:t||Sn8|\}}|dkr:td|t|tt||St|}||krVt||SdS)NFTrr=)rTrOnerLr"rYZeror[ComplexInfinity is_imaginaryNaNInfinityNegativeInfinitycould_extract_minus_signrN NegativeOneextract_multiplicativelyrrOr%extract_branch_factorrrr9r:r;ZnewznZnnur2r2r3r<s:      " z besselj.evalcKs(ttt|dt|tt |SNr=)rrrrOr$r1r:r;r_r2r2r3_eval_rewrite_as_besselisz besselj._eval_rewrite_as_besselicKs<|jdur8tt|t| |tt|t||SdSrE)rLrrbesselyrrur2r2r3_eval_rewrite_as_besselys z besselj._eval_rewrite_as_besselycKs"td|tt|tj|jSrt)rrrRrHalfr6rur2r2r3_eval_rewrite_as_jnszbesselj._eval_rewrite_as_jnc s|j\}}z||}Wnty.|YS0||\}}|jr`||d|t|dS|jr|dkrrdn|}|||} | jstdt|t d|ddtt |S|St t |j |||dS)Nr=r5rlogxcdir) r/as_leading_termNotImplementedErroras_coeff_exponentrYr&r[rrrsuperrN_eval_as_leading_term r1rVr}r~r:r;argcesignr@r2r3rs    0zbesselj._eval_as_leading_termcCs|j\}}|jr|jrdSdSNTr/rLis_extended_realr1r:r;r2r2r3_eval_is_extended_reals  zbesselj._eval_is_extended_realrc s$ddlm}|j\}}z||\}} WnttfyB|YS0| jrt|| } ||||} |d|||| } | t j ur| St | d|  } | |t |d}|g}td| ddD]4}|| |||9}t ||  }||qt|| Stt|||||SNrOrderr=r5)sympy.series.orderrr/leadterm ValueErrorrrYr _eval_nseriesremoveOrrhr r&rangeappendrrrNr1rVrsr}r~rr:r;_rnewnorttermskrr2r3rs*       zbesselj._eval_nseries)r)rarbrcrdrrgrAr?rfr<rvrxrzrrr __classcell__r2r2rr3rNjsD # rNcsfeZdZdZejZejZeddZ ddZ ddZ dd Z fd d Z d d Zdfdd ZZS)rwa` 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|jr6|jrtjSt|jdur&tjSt|jr6tjS|tjtjfvrLtjS|ttjkrxt tt |ddtjS|ttjkrt t t |ddtjS|j r| rtj | t| |SdS)NFr5r=)rTrrmr"rirkrlrhrrrrLrnrorwr8r2r2r3r<Fs   z bessely.evalcKs<|jdur8tt|tt|t||t| |SdSrE)rLrrrrNrur2r2r3_eval_rewrite_as_besseljZs z bessely._eval_rewrite_as_besseljcKs|j|j}|r|tSdSr7)rr/rewriterOr1r:r;r_Zajr2r2r3rv^s z bessely._eval_rewrite_as_besselicKs"td|tt|tj|jSrt)rrrSrryr6rur2r2r3_eval_rewrite_as_yncszbessely._eval_rewrite_as_ync s~|j\}}z||}Wnty.|YS0||\}}|jrdtt|dt||} |jr|d|  t|dtnt j } |d| tt|t |dt j } t | | | gj||d}|S|jrh|dkrdn|}|||} | jsdtdtt|d|td dtt|d|tdd|td|ttS|Stt|j|||dS) Nr=r5r}rr{r|)r/rrrrYrrrNrrrhr' EulerGammarr[rrrrrwr) r1rVr}r~r:r;rrrZterm_oneZterm_twoZ term_threerrr2r3rfs&   ,, bzbessely._eval_as_leading_termcCs|j\}}|jr|jrdSdSrr/rLrYrr2r2r3rs  zbessely._eval_is_extended_realrc s6ddlm}|j\}}z||\}} WnttfyB|YS0| jr |jr t|| } t ||} dt t |d|  ||||} gg} }||||}|d |||| }|tjur|St|d| }|tjkrn|| t|dt }| |td|D]R}|||}|tjkrD|||9}n |||9}t|| }| |q||t t|}|t|dtj}||td| ddD]V}|| |||9}t|| }|t||dt|d}||q| t| t|Stt| ||||Sr)rrr/rrrrYrLrrNrrrrrrhr rrrr'rrrrw)r1rVrsr}r~rr:r;rrrbnrWbrrrrrrZdenomprr2r3rsH     $         zbessely._eval_nseries)r)rarbrcrdrrgrAr?rfr<rrvrrrrrr2r2rr3rws(  rwcs~eZdZdZej ZejZeddZ dddZ ddZ d d Z d d Z d dZfddZdfdd ZfddZZS)rOa 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] https://functions.wolfram.com/Bessel-TypeFunctions/BesselI/ cCs|jrX|jrtjS|jr"|jdus,t|jr2tjSt|jrL|jdurLtjS|j rXtj St |tj tj fvrrtjS|tj urtj S|tj urd|tj S|r||| | t|| S|jr|rt| |S|t}|rt| t|| S|jr*t|}||krbt||Sn8|\}}|dkrbtd|t|tt||St|}||kr~t||SdS)NFTrr=)rTrrgrLr"rYrhr[rirjrkr#rlrmrnrOrprrNr%rqrrrrr2r2r3r<sB         " z besseli.evalNcKs|jrt|t||SdSr7)rr_besselir1r:r;Zlimitvarr_r2r2r3_eval_rewrite_as_tractable sz"besseli._eval_rewrite_as_tractablecKs(tt t|dt|tt|Srt)rrrrNr$rur2r2r3r sz besseli._eval_rewrite_as_besseljcKs|j|j}|r|tSdSr7rr/rrwrr2r2r3rxs z besseli._eval_rewrite_as_besselycKs|j|jtSr7)rr/rrRrur2r2r3rzszbesseli._eval_rewrite_as_jncCs|j\}}|jr|jrdSdSrrrr2r2r3rs  zbesseli._eval_is_extended_realc s|j\}}z||}Wnty.|YS0||\}}|jr`||d|t|dS|jr|dkrrdn|}|||} | jst|tdt |S|St t |j |||dS)Nr=r5rr|) r/rrrrYr&r[rrrrrOrrrr2r3rs    zbesseli._eval_as_leading_termrc s"ddlm}|j\}}z||\}} WnttfyB|YS0| jr t|| } ||||} |d|||| } | t j ur| St | d|  } | |t |d}|g}td| ddD]2}|| |||9}t ||  }||qt|| Stt|||||Sr)rrr/rrrrYrrrrrhr r&rrrrrOrrr2r3r2s*       zbesseli._eval_nseriescsddlmddlm}|d}|tjtjfvr|j\fddt|D|dt d|dd|g}t t dt t |St||||S)Nrrrr5csbg|]Z}tddd|tddd|d|td|ddt|qSr=r5r r.0rrr:r;r2r3 Xs.$z)besseli._eval_aseries..r=(sympy.functions.combinatorial.factorialsrrrrrlrmr/rr rrrrr _eval_aseriesr1rsargs0rVr}rpointrrrr3rQs    zbesseli._eval_aseries)N)r)rarbrcrdrrgrAr?rfr<rrrxrzrrrrrr2r2rr3rOs' '  rOcseZdZdZejZej ZeddZ ddZ ddZ dd Z d d Z d d ZdddZddZdfdd ZfddZZS)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/ cCsv|jr6|jrtjSt|jdur&tjSt|jr6tjS|tjttjttjfvrXtjS|j rr| rrt | |SdSrE) rTrrlr"rirkrrmrhrLrnrr8r2r2r3r<s z besselk.evalcKs8|jdur4ttt|t| |t||dSdS)NFr=)rLrrrOrur2r2r3rvs z besselk._eval_rewrite_as_besselicKs|j|j}|r|tSdSr7)rvr/rrN)r1r:r;r_Zair2r2r3rs z besselk._eval_rewrite_as_besseljcKs|j|j}|r|tSdSr7rrr2r2r3rxs z besselk._eval_rewrite_as_besselycKs|j|j}|r|tSdSr7)rxr/rrS)r1r:r;r_Zayr2r2r3rs zbesselk._eval_rewrite_as_yncCs|j\}}|jr|jrdSdSrrrr2r2r3rs  zbesselk._eval_is_extended_realNcKs|jrt| t||SdSr7)rr_besselkrr2r2r3rsz"besselk._eval_rewrite_as_tractablec Cs|j\}}z||}Wnty.|YS0||\}}|jr|jrdt| tjtd} n:|j rt t ||dt | d} ntd|d| j||dS|j rt tt| t d|S|||SdS)Nr=z"Cannot proceed without knowing if z is zero or not.r)r/rrrrYrTrrrZ is_nonzeror&r!r[rrrfunc) r1rVr}r~r:r;rrrrr2r2r3rs    $zbesselk._eval_as_leading_termrc s2ddlm}|j\}}z||\}} WnttfyB|YS0| jr|d||||} | t j ur||| |||S||||} |j rt || } t ||} d|dt|d| ||||}gg}}t| d}|t j kr`| | t|dd}||td|D]4}|||||9}t|| }||q*| |d|dt|}|t|dt j}||td| ddD]T}|||||9}t|| }|t||dt|d}||q|t|t|| S|jrt ||| }t ||| }gg}}t|ddD]F}t|| d||dtd||t|}|t|q^t|ddD]H}t| | d||dt|d|t|}|t|qt|t|| Stdtt|||||S)Nrrr=rr5z4besselk expansion is only implemented for real order)rrr/rrrrYrrrrhrLrrOrr rrrr'rrZ is_nonintegerr&rrr)r1rVrsr}r~rr:r;rrrrrrrWrrrrrrZnewn_aZnewn_brr2r3rs\      (        24zbesselk._eval_nseriescsddlmddlm}|d}|tjtjfvr|j\fddt|D|dt d|dd|g}t  t t dt |St||||S)Nrrrr5csbg|]Z}tddd|tddd|d|td|ddt|qSr=r5rrrr2r3rs.$z)besselk._eval_aseries..r=rrrrr3rs    zbesselk._eval_aseries)N)r)rarbrcrdrrgrAr?rfr<rvrrxrrrrrrrr2r2rr3r_s%  Erc@s$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/ cCs(|j}|jdur$t|j|SdSrE)r6rFhankel2r4rGrHr2r2r3rIHs zhankel1._eval_conjugateN rarbrcrdrrgrAr?rIr2r2r2r3r s$rc@s$eZdZdZejZejZddZdS)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/ cCs(|j}|jdur$t|j|SdSrE)r6rFrr4rGrHr2r2r3rIws zhankel2._eval_conjugateNrr2r2r2r3rNs%rcstfdd}|S)Ncs|jr|||SdSr7)rLrfnr2r3g~szassume_integer_order..gr)rrr2rr3assume_integer_order}src@s*eZdZdZddZddZd ddZd S) SphericalBesselBasea- Base class for spherical Bessel functions. These are thin wrappers around ordinary Bessel functions, since spherical Bessel functions differ from the ordinary ones just by a slight change in order. To use this class, define the ``_eval_evalf()`` and ``_expand()`` methods. cKs tddS)z@ Expand self into a polynomial. Nu is guaranteed to be Integer. Z expansionNrr1r\r2r2r3_expandszSphericalBesselBase._expandcKs|jjr|jfi|S|Sr7)r4 is_Integerrrr2r2r3rZsz%SphericalBesselBase._eval_expand_funcr=cCs:|dkrt||||jd|j||jd|jSr>)r r@r4r6rBr2r2r3rDs  zSphericalBesselBase.fdiffN)r=)rarbrcrdrrZrDr2r2r2r3rs rcCs8t||t|tj|dt| d|t|SNr5)r*rrrorrsr;r2r2r3_jns$rcCs8tj|dt| d|t|t||t|Sr)rror*rrrr2r2r3_yns$rc@sDeZdZdZeddZddZddZdd Zd d Z d d Z dS)rRa 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*|jrtjS|jr*|jr$tjStjS|tjtjfvr@tjSdSr7) rTrrgrLrYrhrirmrlr8r2r2r3r<szjn.evalcKs ttd|t|tj|Srt)rrrNrryrur2r2r3rszjn._eval_rewrite_as_besseljcKs,tj|ttd|t| tj|Srt)rrorrrwryrur2r2r3rxszjn._eval_rewrite_as_besselycKstj|t| d|Sr)rrorSrur2r2r3rszjn._eval_rewrite_as_yncKst|j|jSr7)rr4r6rr2r2r3rsz jn._expandcCs|jjr|t|SdSr7r4rrrN _eval_evalfr1precr2r2r3rszjn._eval_evalfN) rarbrcrdrfr<rrxrrrr2r2r2r3rRs9 rRc@s@eZdZdZeddZeddZddZdd Zd d Z d S) rSa 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|SrX)rrorrrNryrur2r2r3r3szyn._eval_rewrite_as_besseljcKs ttd|t|tj|Srt)rrrwrryrur2r2r3rx7szyn._eval_rewrite_as_besselycKstj|dt| d|Sr)rrorRrur2r2r3rz;szyn._eval_rewrite_as_jncKst|j|jSr7)rr4r6rr2r2r3r>sz yn._expandcCs|jjr|t|SdSr7)r4rrrwrrr2r2r3rAszyn._eval_evalfN) rarbrcrdrrrxrzrrr2r2r2r3rSs.  rSc@sLeZdZeddZeddZddZddZd d Zd d Z d dZ dS)SphericalHankelBasecKsN|j}ttd|t|tj||ttj|dt| tj|Sr>)_hankel_kind_signrrrNrryrror1r:r;r_hksr2r2r3rHs&z,SphericalHankelBase._eval_rewrite_as_besseljcKsJ|j}ttd|tj|t| tj||tt|tj|Srt)rrrrrorwryrrr2r2r3rxQs(z,SphericalHankelBase._eval_rewrite_as_besselycKs(|j}t||t|tt||Sr7)rrRrrSrrr2r2r3rZsz'SphericalHankelBase._eval_rewrite_as_yncKs(|j}t|||tt||tSr7)rrRrrSrrr2r2r3rz^sz'SphericalHankelBase._eval_rewrite_as_jncKsJ|jjr|jfi|S|j}|j}|j}t|||tt||SdSr7)r4rrr6rrRrrS)r1r\r:r;rr2r2r3rZbs z%SphericalHankelBase._eval_expand_funccKs2|j}|j}|j}t|||tt||Sr7)r4r6rrrrexpand)r1r\rsr;rr2r2r3rks zSphericalHankelBase._expandcCs|jjr|t|SdSr7rrr2r2r3rzszSphericalHankelBase._eval_evalfN) rarbrcrrrxrrzrZrrr2r2r2r3rFs   rc@s"eZdZdZejZeddZdS)rPa 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 cKsttd|t||Srt)rrrrur2r2r3_eval_rewrite_as_hankel1szhn1._eval_rewrite_as_hankel1N) rarbrcrdrrgrrrr2r2r2r3rPs0rPc@s$eZdZdZej ZeddZdS)rQa 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 cKsttd|t||Srt)rrrrur2r2r3_eval_rewrite_as_hankel2szhn2._eval_rewrite_as_hankel2N) rarbrcrdrrgrrrr2r2r2r3rQs0rQsympyc sddlm}dkrTddlmddlm}||fddtd|dDSd krdd lmzdd l m fd d }Wqt yddl m fdd }Yq0nt dfdd}|}|||}|g} t|dD]} ||||}| |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)rr) besseljzero) dps_to_preccs0g|](}ttdt|qS)g?)r _from_mpmathr _to_mpmathint)rl)rrsrr2r3r"s zjn_zeros..r5scipy)newton) spherical_jncs |Sr7r2rV)rsrr2r3)zjn_zeros..)sph_jncs|ddS)Nrrr2r)rsrr2r3r,rUnknown method.cs dkr||}ntd|S)Nrrr)r]rVr )methodrr2r3solver0s zjn_zeros..solver)mathrmpmathrZmpmath.libmp.libmpfrrZscipy.optimizerZ scipy.specialr ImportErrorrrr) rsrrZdpsZmath_pirr]rr rootsir2)rrrsrrrrr3jn_zeross2-          rc@s4eZdZdZddZddZd ddZd d d Zd S)AiryBasezg Abstract base class for Airy functions. This class is meant to reduce code duplication. cCs||jdSNr)rr/rGr0r2r2r3rIKszAiryBase._eval_conjugatecCs |jdjSr)r/rr0r2r2r3rNszAiryBase._eval_is_extended_realTcKsL|jd}|}|j}||||d}t||||d}||fS)Nrr=)r/rGrr)r1deepr\r;Zzcr]uvr2r2r3 as_real_imagQs  zAiryBase.as_real_imagcKs$|jfd|i|\}}||tS)Nr)rr)r1rr\Zre_partZim_partr2r2r3_eval_expand_complexYszAiryBase._eval_expand_complexN)T)T)rarbrcrdrIrrrr2r2r2r3rCs  rc@s^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^|tjurtjS|tjur&tjS|tjur6tjS|jr^tjdtddt tddS|jrtjdtddt tddSdS)Nrr=) is_NumberrrkrlrhrmrTrgr r&r9rr2r2r3r<s   "z airyai.evalcCs$|dkrt|jdSt||dSNr5r) airyaiprimer/r rBr2r2r3rDsz airyai.fdiffcGs:|dkrtjSt|}t|dkr|d}td|| td||dtt|tddtddt|t |dtddtt|tddtddt|dt |dtdd|Stj dtddtt |tj tdttddt|tj t|td||SdS)Nrr5rrr=r{) rrhrlenrrrr rr&rgrsrVZprevious_termsrr2r2r3 taylor_terms" L@Hzairyai.taylor_termcKs`tdd}tdd}t| tdd}t|jr\|t| t| ||t|||SdSNr5rr=r rr"r[rrNr1r;r_otttrWr2r2r3rs    zairyai._eval_rewrite_as_besseljcKstdd}tdd}t|tdd}t|jrX|t|t| ||t|||S|t||t| |||t|| t|||SdSrr rr"rYrrOr r2r2r3rvs    *zairyai._eval_rewrite_as_besselicKs~tjdtddttdd}|tddttdd}|tgtddg|dd|tgtddg|ddS)Nrr=r5 r{)rrgr r&r r)r1r;r_Zpf1Zpf2r2r2r3_eval_rewrite_as_hypers"zairyai._eval_rewrite_as_hyperc Ks |jd}|j}t|dkr|}td|gd}td|gd}td|gd}td|gd}||||||} | dur| |}d|jr| |}| |}| |}|||||||||} ||||||} tj| tj t | | tj t dt | SdS Nrr5r)excludedmrsr) r/ free_symbolsrpoprmatchrLrryrgrrairybi r1r\rZsymbsr;rrrrsMpfZnewargr2r2r3rZs$   $zairyai._eval_expand_funcN)r5rarbrcrdnargs unbranchedrfr<rD staticmethodrrrrvrrZr2r2r2r3r^sX   rc@s^eZdZdZdZdZeddZdddZe e dd Z d d Z d d Z ddZddZdS)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] https://dlmf.nist.gov/9 .. [3] https://encyclopediaofmath.org/wiki/Airy_functions .. [4] https://mathworld.wolfram.com/AiryFunctions.html r5TcCs|jr^|tjurtjS|tjur&tjS|tjur6tjS|jr^tjdtddt tddS|jrtjdtddt tddSdS)Nrr5r=) rrrkrlrmrhrTrgr r&rr2r2r3r<hs   "z airybi.evalcCs$|dkrt|jdSt||dSr) airybiprimer/r rBr2r2r3rDwsz airybi.fdiffcGs|dkrtjSt|}t|dkr|d}td|tttddt|tj t |tj td|tj tt tddt|tj t |dtd|Stj t ddtt|tj tdtttddt|tj t |td||SdS)Nrr5rrr=r )rrhrrrr!rr rrgrrryr r&rr2r2r3r}s @<Hzairybi.taylor_termcKs`tdd}tdd}t| tdd}t|jr\t| dt| ||t|||SdSrr r r2r2r3rs    zairybi._eval_rewrite_as_besseljcKstdd}tdd}t|tdd}t|jr\t|tdt| ||t|||St||}t|| }t||t| ||||t|||SdSrr r1r;r_r r rWrrr2r2r3rvs   .  zairybi._eval_rewrite_as_besselicKsztjtddttdd}|tddttdd}|tgtddg|dd|tgtddg|ddS)Nrr r=r5rr{)rrgr r&r r)rr2r2r3rszairybi._eval_rewrite_as_hyperc Ks |jd}|j}t|dkr|}td|gd}td|gd}td|gd}td|gd}||||||} | dur| |}d|jr| |}| |}| |}|||||||||} ||||||} tjt dtj | t | tj | t | SdSr) r/rrrrrrLrryrrgrrrr2r2r3rZs$   $zairybi._eval_expand_funcN)r5rr2r2r2r3r sZ    rc@sVeZdZdZdZdZeddZdddZdd 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&|tjurtjS|tjur&tjS|jrNtjdtddttddSdS)Nrr5) rrrkrlrhrTror r&rr2r2r3r<s  zairyaiprime.evalcCs.|dkr |jdt|jdSt||dSr)r/rr rBr2r2r3rDszairyaiprime.fdiffcCsR|jd|}t|tj|dd}Wdn1s<0Yt||SNrr5)Z derivative)r/rr,r+rrrr1rr;resr2r2r3r!s ,zairyaiprime._eval_evalfcKsPtdd}t| tdd}t|jrL|dt| ||t|||SdSNr=r)r rr"r[rNr1r;r_r rWr2r2r3r's  z$airyaiprime._eval_rewrite_as_besseljcKstdd}tdd}|t|tdd}t|jrP|dt||t| |St|tdd}t||}t|| }||d|t||||t| ||SdSr)r rr"rYrOr"r2r2r3rv-s     z$airyaiprime._eval_rewrite_as_besselicKs|dddtddttdd}dtddttdd}|tgtddg|dd|tgtddg|ddS)Nr=rr5r)r r&r r)rr2r2r3r9s(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 | SdSr) r/rrrrrrLrryrgrrr!rr2r2r3rZ>s$   $zairyaiprime._eval_expand_funcN)r5rarbrcrdrrrfr<rDrrrvrrZr2r2r2r3rsQ  rc@sVeZdZdZdZdZeddZdddZdd Z d d Z d d Z ddZ ddZ dS)r!a6 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 .. 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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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