EdB|ddlmZddlmZddlmZddlmZddlm 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;mGdde=Z?Gdde=Z@Gdde=ZAGd d!e=ZBGd"d#e=ZCd$ZDGd%d&e=ZEd'ZFd(ZGGd)d*eEZHGd+d,eEZIGd-d.eEZJGd/d0eJZKGd1d2eJZLdCd5ZMGd6d7e ZNGd8d9eNZOGd:d;eNZPGd<d=eNZQGd>d?eNZRGd@dAe ZSdBS)Dwraps)S)Add)cacheit)Expr)FunctionArgumentIndexError_mexpand)fuzzy_or fuzzy_not)RationalpiI)Pow)DummyWild)sympify) factorial)sincoscsccot)ceiling)explog)cbrtsqrtroot)Absreim polar_lift unpolarify)gammadigamma uppergamma)hyper)spherical_bessel_fn)mpworkpreccteZdZdZedZedZedZd dZ dZ dZ d Z d Z d S) BesselBasea Abstract base class for Bessel-type functions. This class is meant to reduce code duplication. All Bessel-type functions can 1) be differentiated, with the derivatives expressed in terms of similar functions, and 2) be rewritten in terms of other Bessel-type functions. Here, Bessel-type functions are assumed to have one complex parameter. To use this base class, define class attributes ``_a`` and ``_b`` such that ``2*F_n' = -_a*F_{n+1} + b*F_{n-1}``. c|jdS)z( The order of the Bessel-type function. rargsselfs >lib/python3.11/site-packages/sympy/functions/special/bessel.pyorderzBesselBase.order4y|c|jdS)z+ The argument of the Bessel-type function. r/r1s r3argumentzBesselBase.argument9r5r6cdSNclsnuzs r3evalzBesselBase.eval>sr6c|dkrt|||jdz ||jdz |jz|jdz ||jdz|jzz SNrBr8)r _b __class__r4r9_ar2argindexs r3fdiffzBesselBase.fdiffBsr q= 5$T844 4 DNN4:>4=III DNN4:>4=IIIJ Kr6c|j}|jdur?||j|SdSNF)r9is_extended_negativerFr4 conjugater2r@s r3_eval_conjugatezBesselBase._eval_conjugateHsP M !U * I>>$*"6"6"8"8!++--HH H I Ir6c |j|j}}||rdS|||sdS|||}|jrOt |ttttttfs|j st|jStt!|j |jgSrL)r4r9has_eval_is_meromorphicsubs is_integer isinstancebesseljbesselihn1hn2jnynis_zeror is_infiniter )r2xar?r@z0s r3rSzBesselBase._eval_is_meromorphicMs DMA 66!99 5%%a++ 4 VVAq\\ = 1$'3R DEE 1RZ 1 0002:r~">??@@@r6c |j|j|j}}}|jr|dz jrh|j |jz||dz |zd|jz|dz z||dz |z|z zS|dzjrgd|jz|dzz||dz|z|z |j|jz||dz|zz S|SNr8rB) r4r9rFis_real is_positiverGrE_eval_expand_func is_negative)r2hintsr?r@fs r3rfzBesselBase._eval_expand_funcZs :t}dnqA : JQ# J(261)G)G)I)II$' 26*11R!VQ<<+I+I+K+KKAMNOq&% J$' 26*11R!VQ<<+I+I+K+KKAM"q&! (F(F(H(HHIJ r6c $ddlm}||S)Nr) besselsimp)sympy.simplify.simplifyrk)r2kwargsrks r3_eval_simplifyzBesselBase._eval_simplifyes$666666z$r6NrB)__name__ __module__ __qualname____doc__propertyr4r9 classmethodrArJrPrSrfrnr<r6r3r-r-$s  XX[KKKK III A A A        r6r-cveZdZdZejZejZedZ dZ dZ dZ d dZ d Zd fd ZxZS) rWa3 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/ c|jr|jr tjS|jr |jdust |jr tjSt |jr|jdur tjS|j r tj S|tj tj fvr tjS| r||z| | zzt|| zS|jrm| r"tj| zt| |zS|t"}|rt"|zt%||zS|jr&t'|}||krt||SnS|\}}|dkr6t+d|zt,z|zt"zt||zSt'|}||krt||SdSNFTrrB)r]rOnerUr!reZerorgComplexInfinity is_imaginaryNaNInfinityNegativeInfinitycould_extract_minus_signrW NegativeOneextract_multiplicativelyrrXr$extract_branch_factorrrr>r?r@newznnnus r3rAz besselj.evals 9 z u - BJ%$7 BrFF>0II I J Jr6c |td|ztz t|tjz |jzSr)rrr[rHalfr9rs r3_eval_rewrite_as_jnzbesselj._eval_rewrite_as_jns-AaCF||BrAF{DM::::r6Nrc|j\}}||}||jvr||zd|zt|dzzz S||||dSNrBr8r)r0as_leading_term free_symbolsr%funcrT)r2r_logxcdirr?r@args r3_eval_as_leading_termzbesselj._eval_as_leading_termsr A""   /7ArE%Q--/0 099R1.. .r6c>|j\}}|jr |jrdSdSdSNTr0rUis_extended_realr2r?r@s r3_eval_is_extended_realzbesselj._eval_is_extended_real: A = Q/ 4    r6c ddlm}|j\}} ||\}} n#tt f$r|cYSwxYw| jr t|| z } |||z|} |dz |||| } | tj ur| St| dz| z } | |zt|dzz }|g}td| dzdzD]J}|| |||zzz z}t|| z }||Kt!|| zSt#t$|||||SNr)OrderrBr8)sympy.series.orderrr0leadterm ValueErrorNotImplementedErrorrer _eval_nseriesremoveOrrzr r%rangeappendrsuperrW)r2r_rrrrr?r@_rnewnorttermskrFs r3rzbesselj._eval_nseriess,,,,,, A ZZ]]FAss/0   KKK  ? 1S5>>DadAA1##Aq$55==??AAF{ !Q$!#,,..Ab5rAv&DA1tax!m,,  ArAvJ' *33557Q; Wd##11!QdCCC,AANrr)rprqrrrsrryrGrErurArrrrrr __classcell__rFs@r3rWrWjsBBH B B!#!#[!#F<<<JJJ;;;//// DDDDDDDDDDr6rWcveZdZdZejZejZedZ dZ dZ dZ d dZ d Zd fd ZxZS) 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/ c|jrU|jr tjSt|jdur tjSt|jr tjS|tjtjfvr tjS|jr6| r$tj | zt| |zSdSdSrL) r]rrr!r{r}r~rzrUrrrr=s r3rAz bessely.eval7s 9 z ))B5( ((B u Q/0 0 6M = <**,, <}s+GRCOO;; < < < |j\}}|jr |jrdSdSdSrr0rUrers r3rzbessely._eval_is_extended_real^9 A = Q] 4    r6cddlm}|j\}} ||\}} n#tt f$r|cYSwxYw| jrs|jrkt|| z } t||} dtz t|dz z| z ||||} gg}} |||z|}|dz |||| }|tjur|St!|dz|z }|tjkr|| zt%|dz ztz }| |t)d|dz D]L}||||z dz z|z z}t!||z }| |M||ztt%|zz }|t+|dztjz z}||t)d| dzdzD]u}|| |||zzz z}t!||z }|t+||zdzt+|dzzz}||v| t/| z t/|z St1t2| ||||Sr)rrr0rrrrerUrrWrrrrrrzr ryrrrr&rrrr)r2r_rrrrr?r@rrrbnr`bcrrrrrprFs r3rzbessely._eval_nseriescs,,,,,, A ZZ]]FAss/0   KKK  ? )r} )1S5>>DQBB$AaC#221atDDArqAadAA1##Aq$55==??AAF{ !Q$!#,,..AAEz #B3x "q& 1 11"4q"q&))##AArAvzN1,,D$TNNQ.7799DHHTNNNN2r)B--'(Agb1foo 45D HHTNNN1tax!m,,  aRAF_$a[[1_--//'!b&1*--A>?sAw;a( (Wd##11!QdCCCrrr)rprqrrrsrryrGrErurArrrrrrrrs@r3rr s&&P B B < <[ <LLL''' === 8 8 8 8 )D)D)D)D)D)D)D)D)D)Dr6rc^eZdZdZej ZejZedZ dZ dZ dZ dZ dS)rXa 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/ c|jr|jr tjS|jr |jdust |jr tjSt |jr|jdur tjS|j r tj St|tj tj fvr tjS|r||z| | zzt|| zS|jr^|rt| |S|t"}|rt"| zt%|| zS|jr&t'|}||krt||SnS|\}}|dkr6t+d|zt,z|zt"zt||zSt'|}||krt||SdSrx)r]rryrUr!rerzrgr{r|r}r"r~rrrXrrrWr$rrrrs r3rAz besseli.evals 9 z u - BJ%$7 BrFF|j\}}|jr |jrdSdSdSrrrs r3rzbesseli._eval_is_extended_realrr6N)rprqrrrsrryrGrErurArrrrr<r6r3rXrXs%%N %B B!#!#[!#F<<<''' EEEr6rXcdeZdZdZejZej ZedZ dZ dZ dZ dZ dZdS) 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/ c|jrU|jr tjSt|jdur tjSt|jr tjS|tjt tjzt tjzfvr tjS|j r%| rt| |SdSdSrL) r]rr~r!r{r}rrrzrUrrr=s r3rAz besselk.evals 9 z z!B5( ((B u Qqz\1Q-?+?@ @ 6M = '**,, 'sA& ' ' ' 'r6c |jdurEttt|zzt| |t||z zdz SdS)NFrB)rUrrrXrs r3rz besselk._eval_rewrite_as_besseli(sS =E ! Fc"R%jj='2#q//GBNN"BCAE E F Fr6c \|j|j}|r|tSdSr;)rr0rrW)r2r?r@rmais r3rz besselk._eval_rewrite_as_besselj,rr6c \|j|j}|r|tSdSr;rrs r3rz besselk._eval_rewrite_as_bessely1rr6c \|j|j}|r|tSdSr;)rr0rr\)r2r?r@rmays r3rzbesselk._eval_rewrite_as_yn6s5 *T *DI 6  "::b>> ! " "r6c>|j\}}|jr |jrdSdSdSrrrs r3rzbesselk._eval_is_extended_real;rr6N)rprqrrrsrryrGrErurArrrrrr<r6r3rrs##J B %B ' '[ 'FFF''' ''' """ r6rc4eZdZdZejZejZdZdS)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/ c|j}|jdur9t|j|SdSrL)r9rMhankel2r4rNrOs r3rPzhankel1._eval_conjugateiL M !U * B4://111;;==AA A B Br6N rprqrrrsrryrGrErPr<r6r3rrAsC""H B BBBBBBr6rc4eZdZdZejZejZdZdS)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/ c|j}|jdur9t|j|SdSrL)r9rMrr4rNrOs r3rPzhankel2._eval_conjugaterr6Nrr<r6r3rrosC##J B BBBBBBr6rc<tfd}|S)Nc0|jr |||SdSr;)rU)r2r?r@fns r3gzassume_integer_order..gs) = #2dB?? " # #r6r)rrs` r3assume_integer_orderrs3 2YY####Y# Hr6c&eZdZdZdZdZddZdS)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. c td)z@ Expand self into a polynomial. Nu is guaranteed to be Integer. expansionrr2rhs r3_expandzSphericalBesselBase._expands!+...r6c 8|jjr |jdi|S|SNr<)r4 is_Integerrrs r3rfz%SphericalBesselBase._eval_expand_funcs, :  )4<((%(( ( r6rBc|dkrt||||jdz |j||jdzz|jz z SrD)r rFr4r9rHs r3rJzSphericalBesselBase.fdiffsV q= 5$T844 4~~dj1ndm<< DJN #DM 12 2r6Nro)rprqrrrsrrfrJr<r6r3rrsP  /// 222222r6rct||t|ztj|dzzt| dz |zt |zzSNr8)r)rrrrrr@s r3_jnrsU 1 % %c!ff , MAE "#6rAvq#A#A A#a&& H IJr6ctj|dzzt| dz |zt|zt||t |zz Sr)rrr)rrrs r3_ynrsS MAE "%8!a%C%C CCFF J 1 % %c!ff , -.r6cFeZdZdZedZdZdZdZdZ dZ dS) r[a 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 c|jr9|jr tjS|jr|jr tjStjS|tjtjfvr tjSdSr;) r]rryrUrerzr{rr~r=s r3rAzjn.evalsi 9 -z -u  ->-6M,, #QZ0 0 6M  r6c rttd|zz t|tjz|zSr)rrrWrrrs r3rzjn._eval_rewrite_as_besseljs+B!H~~QV Q 7 777r6c tj|zttd|zz zt | tjz |zSr)rrrrrrrs r3rzjn._eval_rewrite_as_besselys9}b 4AaC>>1GRC!&L!4L4LLLr6c Jtj|zt| dz |zSr)rrr\rs r3rzjn._eval_rewrite_as_yns"}r"Ra^^33r6c 6t|j|jSr;)rr4r9rs r3rz jn._expand4:t}---r6cx|jjr-|t|SdSr;r4rrrW _eval_evalfr2precs r3r zjn._eval_evalf 9 :  ;<<((44T:: : ; ;r6N) rprqrrrsrurArrrrr r<r6r3r[r[s88r  [ 888MMM444...;;;;;r6r[cPeZdZdZedZedZdZdZdZ dS)r\a 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 c tj|dzzttd|zz zt | tjz |zSrc)rrrrrWrrs r3rzyn._eval_rewrite_as_besseljTs=}r!t$tB!H~~5af a8P8PPPr6c rttd|zz t|tjz|zSr)rrrrrrs r3rzyn._eval_rewrite_as_besselyXs+B!H~~QV Q 7 777r6c Ptj|dzzt| dz |zSr)rrr[rs r3rzyn._eval_rewrite_as_jn\s&}rAv&RC!GQ77r6c 6t|j|jSr;)rr4r9rs r3rz yn._expand_r r6cx|jjr-|t|SdSr;)r4rrrr r s r3r zyn._eval_evalfbrr6N) rprqrrrsrrrrrr r<r6r3r\r\%s--\QQQ888888...;;;;;r6r\cXeZdZedZedZdZdZdZdZ dZ dS) SphericalHankelBasec |j}ttd|zz t|tjz||t ztj|dzzzt| tjz |zzzSrD)_hankel_kind_signrrrWrrrrr2r?r@rmhkss r3rz,SphericalHankelBase._eval_rewrite_as_besseljisp $B!H~~wrAF{A66"1uQ]RT%::7B3 >r6c |j}t||t|tzt||zzSr;)rr[rr\rrs r3rz'SphericalHankelBase._eval_rewrite_as_yn{s;$"ayy  $$s1uRAYY66r6c |j}t|||tzt||tzzSr;)rr[rr\rrs r3rz'SphericalHankelBase._eval_rewrite_as_jns<$"ayy3q5B!2!22!6!6666r6c |jjr |jdi|S|j}|j}|j}t |||t zt||zzSr)r4rrr9rr[rr\)r2rhr?r@rs r3rfz%SphericalHankelBase._eval_expand_funcs` :  /4<((%(( (B A(Cb!99s1uRAYY. .r6c |j}|j}|j}t|||tzt ||zzSr;)r4r9rrrrexpand)r2rhrr@rs r3rzSphericalHankelBase._expandsI J M$Aq CE#a))O+33555r6cx|jjr-|t|SdSr;r r s r3r zSphericalHankelBase._eval_evalfrr6N) rprqrrrrrrrrfrr r<r6r3rrgsUUU>>>777777/// 6 6 6;;;;;r6rc6eZdZdZejZedZdS)rYa 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 c Xttd|zz t||zSr)rrrrs r3_eval_rewrite_as_hankel1zhn1._eval_rewrite_as_hankel1#B!H~~gb!nn,,r6N) rprqrrrsrryrrr%r<r6r3rYrYsC..`-----r6rYc8eZdZdZej ZedZdS)rZa 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 c Xttd|zz t||zSr)rrrrs r3_eval_rewrite_as_hankel2zhn2._eval_rewrite_as_hankel2 r&r6N) rprqrrrsrryrrr)r<r6r3rZrZsE..`-----r6rZsympyc ddlm}dkr8ddlm ddlm}|| fdt d|dzDSdkr0dd lm  dd l m fd }n+#t$rdd l m fd }YnwxYwtd fd}|z}|||}|g} t |dz D]&} ||||z}| |'| 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_precc g|]Q}tjtdzt |RS)g?)r _from_mpmathr _to_mpmathint).0lr-rrs r3 zjn_zeros..Csj***!++aCjj.C.CD.I.I.1!ff#6#67;==***r6r8scipy)newton) spherical_jnc|Sr;r<)r_rr8s r3zjn_zeros..Js,,q!,,r6)sph_jnc4|ddS)Nrr<)r_rr;s r3r:zjn_zeros..Ms&&A,,q/"-r6Unknown method.cLdkr ||}ntd|S)Nr6r>r)rir_rmethodr7s r3solverzjn_zeros..solverQs5 W  96!Q<A?c.eZdZdZdZdZddZddZdS) AiryBasezg Abstract base class for Airy functions. This class is meant to reduce code duplication. cf||jdSr)rr0rNr1s r3rPzAiryBase._eval_conjugatels&yy1//11222r6c&|jdjSr)r0rr1s r3rzAiryBase._eval_is_extended_realosy|,,r6Tc |jd}|}|j}||||zdz }t||||z zdz }||fS)NrrB)r0rNrr)r2deeprhr@zcriuvs r3 as_real_imagzAiryBase.as_real_imagrsi IaL [[]] I QqTT!!B%%ZN qquuQQqTTzN1 !t r6c J|jdd|i|\}}||tjzzS)NrRr<)rVr ImaginaryUnit)r2rRrhre_partim_parts r3_eval_expand_complexzAiryBase._eval_expand_complexzs6,4,@@$@%@@000r6N)T)rprqrrrsrPrrVr[r<r6r3rNrNdsd333---111111r6rNcveZdZdZdZdZedZd dZe e dZ dZ dZ d Zd Zd S) 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 r8Tc|jr|tjur tjS|tjur tjS|tjur tjS|jr>tjdtddzttddzz S|jr>tjdtddzttddzz SdS)NrB) is_Numberrr}r~rzrr]ryrr%r>rs r3rAz airyai.evals = Kae| Ku  " Kv ** Kv  Ku8Aq>> 1E(1a..4I4I IJJ ; G5Ax1~~-hq!nn0E0EEF F G Gr6cb|dkrt|jdSt||Nr8r) airyaiprimer0r rHs r3rJz airyai.fdiff2 q= 5ty|,, ,$T844 4r6c |dkr tjSt|}t|dkr|d}t d|z| zt d|z|dzzzt t |tddztddzzzt|zt|dz tddzzt t |tddztddzzt|dzzt|dz tddzzz |zStj dtddzt zz t|tj ztdz zt tddt z|tj zzzt|z t d|z|zzS)Nrr8r=r_rB) rrzrlenrrrrrr%ryrr_previous_termsrs r3 taylor_termzairyai.taylor_terms  q5 76M A>""Q& 7"2&aqb)47719A*>>s2qRSUVGWZbcdfgZhZhGhCi?j?jjktuvkwkwwacHQNN233458Qx1~~=MPXYZ\]P^P^=^9_5`5`ajklopkpaqaq5qrwxyz{x{GHIKLMMyMsNsN6NORSSTq(1a..034uagqtt^7L7LLsS[\]_`SaSabdSdfghihmfmSnOoOoo!! %(,Q A~67r6c $tdd}tdd}t| tdd}t|jr<|t | zt | ||zt |||zzzSdSNr8r_rBrrr!rgrrWr2r@rmotttr`s r3rzairyai._eval_rewrite_as_besseljs a^^ a^^ HQNN # # a55  JdA2hh;'2#r!t"4"4wr2a47H7H"HI I J Jr6c tdd}tdd}t|tdd}t|jr;|t |zt | ||zt |||zz zS|t||t | ||zz|t|| zt |||zzz zSrmrrr!rerrXros r3rzairyai._eval_rewrite_as_besselis a^^ a^^ 8Aq>> " " a55  Xd1gg:"bd!3!3gb"Q$6G6G!GH Hs1bzz'2#r!t"4"44qQ}WRQSTUQUEVEV7VVW Wr6c tjdtddzttddzz }|t ddttddzz }|t gtddg|dzdz z|t gtddg|dzdz zz S)Nr_rBr8 rg)rryrr%rr(r2r@rmpf1pf2s r3_eval_rewrite_as_hyperzairyai._eval_rewrite_as_hyperseq(1a..(x1~~)>)>>?41::eHQNN3334U2A/Aa8883rHUVXYNNK[]^`a]abc]cAdAd;dddr6c |jd}|j}t|dkr0|}t d|g}t d|g}t d|g}t d|g}|||||zz|zz} | | |}d|zjr| |}| |}| |}|||zz|z||z|||zzzz } |||zz|||zzz} tj| tj zt| z| tj z tdz t| zz zSdSdSdS Nrr8r)excludedmrr_) r0rrhpoprmatchrUrrryr]rairybi r2rhrsymbsr@rr}r~rMpfnewargs r3rfzairyai._eval_expand_funcsil  u::? h AS1#&&&AS1#&&&AS1#&&&AS1#&&&A !Qq!tVaK-((A haDaC#h!A!A!Aad(Q!Q$QqS/:BAXAaC0F6b15j&..%@BJPTUVPWPWCWX^_eXfXfCf%fgg# h h h hhhr6Nr8rprqrrrsnargs unbranchedrurArJ staticmethodrrkrrryrfr<r6r3r]r]sVVp EJ G G[ G5555   7 7 W\ 7JJJXXXeee hhhhhr6r]cveZdZdZdZdZedZd dZe e dZ dZ dZ d Zd Zd S) ra The Airy function $\operatorname{Bi}$ of the second kind. Explanation =========== The Airy function $\operatorname{Bi}(z)$ is defined to be the function satisfying Airy's differential equation .. math:: \frac{\mathrm{d}^2 w(z)}{\mathrm{d}z^2} - z w(z) = 0. Equivalently, for real $z$ .. math:: \operatorname{Bi}(z) := \frac{1}{\pi} \int_0^\infty \exp\left(-\frac{t^3}{3} + z t\right) + \sin\left(\frac{t^3}{3} + z t\right) \mathrm{d}t. Examples ======== Create an Airy function object: >>> from sympy import airybi >>> from sympy.abc import z >>> airybi(z) airybi(z) Several special values are known: >>> airybi(0) 3**(5/6)/(3*gamma(2/3)) >>> from sympy import oo >>> airybi(oo) oo >>> airybi(-oo) 0 The Airy function obeys the mirror symmetry: >>> from sympy import conjugate >>> conjugate(airybi(z)) airybi(conjugate(z)) Differentiation with respect to $z$ is supported: >>> from sympy import diff >>> diff(airybi(z), z) airybiprime(z) >>> diff(airybi(z), z, 2) z*airybi(z) Series expansion is also supported: >>> from sympy import series >>> series(airybi(z), z, 0, 3) 3**(1/3)*gamma(1/3)/(2*pi) + 3**(2/3)*z*gamma(2/3)/(2*pi) + O(z**3) We can numerically evaluate the Airy function to arbitrary precision on the whole complex plane: >>> airybi(-2).evalf(50) -0.41230258795639848808323405461146104203453483447240 Rewrite $\operatorname{Bi}(z)$ in terms of hypergeometric functions: >>> from sympy import hyper >>> airybi(z).rewrite(hyper) 3**(1/6)*z*hyper((), (4/3,), z**3/9)/gamma(1/3) + 3**(5/6)*hyper((), (2/3,), z**3/9)/(3*gamma(2/3)) See Also ======== airyai: Airy function of the first kind. airyaiprime: Derivative of the Airy function of the first kind. airybiprime: Derivative of the Airy function of the second kind. References ========== .. [1] https://en.wikipedia.org/wiki/Airy_function .. [2] http://dlmf.nist.gov/9 .. [3] http://www.encyclopediaofmath.org/index.php/Airy_functions .. [4] http://mathworld.wolfram.com/AiryFunctions.html r8Tc|jr|tjur tjS|tjur tjS|tjur tjS|jr>tjdtddzttddzz S|jr>tjdtddzttddzz SdS)Nr_r8rB) r`rr}r~rrzr]ryrr%ras r3rAz airybi.evals = Kae| Ku  " Kz!** Kv  Ku8Aq>> 1E(1a..4I4I IJJ ; G5Ax1~~-hq!nn0E0EEF F G Gr6cb|dkrt|jdSt||rc) airybiprimer0r rHs r3rJz airybi.fdiffrer6c |dkr tjSt|}t|dkr|d}t d|zt t tddtz|tj zzzt|tj z tdz z|tj zt ttddtz|tj zzzt|dz tdz zz |zStj tddtzz t|tj ztdz zt t tddtz|tj zzzt|z t d|z|zzS)Nrr8r=r_rBr)rrzrrhrr rrrryrrrrr%ris r3rkzairybi.taylor_terms q5 76M A>""Q& 7"2&Q CHQNN2,=q15y,I(J(J$K$KKiYZ]^]bYbdefgdhdhXhNiNiiae)s3x1~~b/@!af*/M+N+N'O'OOR[]^ab]bdefgdhdh\hRiRiikmnoptAqzz"}-q15y!A$$6F0G0GG#cRZ[\^_R`R`acRcefijinenRoNpNpJqJqq!! %(,Q A~67r6c $tdd}tdd}t| tdd}t|jr> " " a55  K77477?grc2a4&8&872r!t;L;L&LM MAr AAs A88QwsBqD111AaCBqD8I8I4IIJ Jr6c tjtddtt ddzz }|tddztt ddz }|t gt ddg|dzdz z|t gt ddg|dzdz zzS)Nr_rrBr8rurg)rryrr%rr(rvs r3ryzairybi._eval_rewrite_as_hypersetAqzz%A"7"778Q lU8Aq>>222U2A/Aa8883rHUVXYNNK[]^`a]abc]cAdAd;dddr6c |jd}|j}t|dkr0|}t d|g}t d|g}t d|g}t d|g}|||||zz|zz} | | |}d|zjr| |}| |}| |}|||zz|z||z|||zzzz } |||zz|||zzz} tjtdtj | z zt| ztj | zt| zzzSdSdSdSr{) r0rrhrrrrUrrrryr]rrs r3rfzairybi._eval_expand_funcsil  u::? h AS1#&&&AS1#&&&AS1#&&&AS1#&&&A !Qq!tVaK-((A haDaC#h!A!A!Aad(Q!Q$QqS/:BAXAaC0F6T!WWaebj%9&..%HAETVJX^_eXfXfKf%fgg# h h h hhhr6Nrrr<r6r3rr+sXXt EJ G G[ G5555   7 7 W\ 7III K K Keee hhhhhr6rcVeZdZdZdZdZedZd dZdZ dZ dZ d Z d Z d S) rda+ 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 .. 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Explanation =========== The Airy function $\operatorname{Bi}^\prime(z)$ is defined to be the function .. math:: \operatorname{Bi}^\prime(z) := \frac{\mathrm{d} \operatorname{Bi}(z)}{\mathrm{d} z}. Examples ======== Create an Airy function object: >>> from sympy import airybiprime >>> from sympy.abc import z >>> airybiprime(z) airybiprime(z) Several special values are known: >>> airybiprime(0) 3**(1/6)/gamma(1/3) >>> from sympy import oo >>> airybiprime(oo) oo >>> airybiprime(-oo) 0 The Airy function obeys the mirror symmetry: >>> from sympy import conjugate >>> conjugate(airybiprime(z)) airybiprime(conjugate(z)) Differentiation with respect to $z$ is supported: >>> from sympy import diff >>> diff(airybiprime(z), z) z*airybi(z) >>> diff(airybiprime(z), z, 2) z*airybiprime(z) + airybi(z) Series expansion is also supported: >>> from sympy import series >>> series(airybiprime(z), z, 0, 3) 3**(1/6)/gamma(1/3) + 3**(5/6)*z**2/(6*gamma(2/3)) + O(z**3) We can numerically evaluate the Airy function to arbitrary precision on the whole complex plane: >>> airybiprime(-2).evalf(50) 0.27879516692116952268509756941098324140300059345163 Rewrite $\operatorname{Bi}^\prime(z)$ in terms of hypergeometric functions: >>> from sympy import hyper >>> airybiprime(z).rewrite(hyper) 3**(5/6)*z**2*hyper((), (5/3,), z**3/9)/(6*gamma(2/3)) + 3**(1/6)*hyper((), (1/3,), z**3/9)/gamma(1/3) See Also ======== airyai: Airy function of the first kind. airybi: Airy function of the second kind. airyaiprime: Derivative of the Airy function of the first kind. References ========== .. [1] https://en.wikipedia.org/wiki/Airy_function .. [2] http://dlmf.nist.gov/9 .. [3] http://www.encyclopediaofmath.org/index.php/Airy_functions .. 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Explanation =========== The Marcum Q-function is defined by the meromorphic continuation of .. math:: Q_m(a, b) = a^{- m + 1} \int_{b}^{\infty} x^{m} e^{- \frac{a^{2}}{2} - \frac{x^{2}}{2}} I_{m - 1}\left(a x\right)\, dx Examples ======== >>> from sympy import marcumq >>> from sympy.abc import m, a, b >>> marcumq(m, a, b) marcumq(m, a, b) Special values: >>> marcumq(m, 0, b) uppergamma(m, b**2/2)/gamma(m) >>> marcumq(0, 0, 0) 0 >>> marcumq(0, a, 0) 1 - exp(-a**2/2) >>> marcumq(1, a, a) 1/2 + exp(-a**2)*besseli(0, a**2)/2 >>> marcumq(2, a, a) 1/2 + exp(-a**2)*besseli(0, a**2)/2 + exp(-a**2)*besseli(1, a**2) Differentiation with respect to $a$ and $b$ is supported: >>> from sympy import diff >>> diff(marcumq(m, a, b), a) a*(-marcumq(m, a, b) + marcumq(m + 1, a, b)) >>> diff(marcumq(m, a, b), b) -a**(1 - m)*b**m*exp(-a**2/2 - b**2/2)*besseli(m - 1, a*b) References ========== .. [1] https://en.wikipedia.org/wiki/Marcum_Q-function .. 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