gddlmZddlmZddlmZddlmZddlm Z ddl m Z m Z m Z ddlmZmZddlmZmZmZdd lmZdd lmZmZmZdd lmZdd lmZdd lm Z m!Z!m"Z"m#Z#ddl$m%Z%ddl&m'Z'm(Z(ddl)m*Z*m+Z+m,Z,ddl-m.Z.m/Z/m0Z0m1Z1m2Z2ddl3m4Z4m5Z5m6Z6ddl7m8Z8ddl9m:Z:ddl;mGdde>Z?Gdde>Z@Gdde>ZAGdde>ZBGd d!e>ZCGd"d#e>ZDd$ZEGd%d&e>ZFd'ZGd(ZHGd)d*eFZIGd+d,eFZJGd-d.eFZKGd/d0eKZLGd1d2eKZMdCd5ZNGd6d7e ZOGd8d9eOZPGd:d;eOZQGd<d=eOZRGd>d?eOZSGd@dAe ZTdBS)Dwraps)S)Add)cacheit)Expr)FunctionArgumentIndexError_mexpand)fuzzy_or fuzzy_not)RationalpiI)Pow)Dummyuniquely_named_symbolWild)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 Z/var/www/html/ai-engine/env/lib/python3.11/site-packages/sympy/functions/special/bessel.pyorderzBesselBase.order4y|c|jdS)z+ The argument of the Bessel-type function. r0r2s r4argumentzBesselBase.argument9r6r7cdSNclsnuzs r4evalzBesselBase.eval>sr7c|dkrt|||jdz ||jdz |jz|jdz ||jdz|jzz SNrCr9)r _b __class__r5r:_ar3argindexs r4fdiffzBesselBase.fdiffBso q==$T844 4 DNN4:>4=III DNN4:>4=IIIJ Kr7c|j}|jdur?||j|SdSNF)r:is_extended_negativerGr5 conjugater3rAs r4_eval_conjugatezBesselBase._eval_conjugateHsI M !U * *>>$*"6"6"8"8!++--HH H + *r7c |j|j}}||rdS|||sdS|||}|jrOt |ttttttfs|j st|jStt!|j |jgSrM)r5r:has_eval_is_meromorphicsubs is_integer isinstancebesseljbesselihn1hn2jnynis_zeror is_infiniter )r3xar@rAz0s r4rTzBesselBase._eval_is_meromorphicMs DMA 66!99 5%%a++ 4 VVAq\\ = 1$'3R DEE 1RZ 1 0002:r~">??@@@r7c |j|j|j}}}|jr|dz jrh|j |jz||dz |zd|jz|dz z||dz |z|z zS|dzjrgd|jz|dzz||dz|z|z |j|jz||dz|zz S|SNr9rC) r5r:rGis_real is_positiverHrF_eval_expand_func is_negative)r3hintsr@rAfs r4rgzBesselBase._eval_expand_funcZs :t}dnqA : JQ# J(261)G)G)I)II$' 26*11R!VQ<<+I+I+K+KKAMNOq&% J$' 26*11R!VQ<<+I+I+K+KKAM"q&! (F(F(H(HHIJ r7c $ddlm}||S)Nr) besselsimp)sympy.simplify.simplifyrl)r3kwargsrls r4_eval_simplifyzBesselBase._eval_simplifyes$666666z$r7NrC)__name__ __module__ __qualname____doc__propertyr5r: classmethodrBrKrQrTrgror=r7r4r.r.$s  XX[KKKK III A A A        r7r.czeZdZdZejZejZedZ dZ dZ dZ d fd Z d Zd fd ZxZS) rXa4 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/ 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||SdS)NFTrrC)r^rOnerVr"rfZerorhComplexInfinity is_imaginaryNaNInfinityNegativeInfinitycould_extract_minus_signrX NegativeOneextract_multiplicativelyrrYr%extract_branch_factorrrr?r@rAnewznnnus r4rBz besselj.evals 9 z u - BJ%$7$7BrFF>0II I " !r7c |td|ztz t|tjz |jzSr)rrr\rHalfr:rs r4_eval_rewrite_as_jnzbesselj._eval_rewrite_as_jns-AaCF||BrAF{DM::::r7Nrc|j\}} ||}n#t$r|cYSwxYw||\}}|jr||zd|zt |dzzz S|jrf|dkrdn|}|||zz} | jsKtdt|td|zdzzdz z ztt|zz S|Stt| |||S)NrCr9r) r1as_leading_termNotImplementedErroras_coeff_exponentrfr&rhrrrsuperrX_eval_as_leading_term r3r`logxcdirr@rAargcesignrGs r4rzbesselj._eval_as_leading_terms) A ##A&&CC"   KKK $$Q''1 = 7ArE%Q--/0 0 ]  11tDT1W9D# CAwws1r1R4!8}Q#6777RT BBKWd##99!T4HHH # 22c>|j\}}|jr |jrdSdSdSNTr1rVis_extended_realr3r@rAs r4_eval_is_extended_realzbesselj._eval_is_extended_real: A = Q/ 4    r7c ddlm}|j\}} ||\}} n#tt f$r|cYSwxYw| jr t|| z } |||z|} |dz |||| } | tj ur| St| dz| z } | |zt|dzz }|g}td| dzdzD]J}|| |||zzz z}t|| z }||Kt!|| zSt#t$|||||SNrOrderrCr9)sympy.series.orderrr1leadterm ValueErrorrrfr _eval_nseriesremoveOrrzr r&rangeappendrrrXr3r`rrrrr@rA_rnewnorttermskrGs r4rzbesselj._eval_nseriess -,,,,, A ZZ]]FAss/0   KKK  ? 1S5>>DadAA1##Aq$55==??AAF{{!Q$!#,,..Ab5rAv&DA1tax!m,,  ArAvJ' *33557Q; Wd##11!QdCCC,AANrr)rqrrrsrtrryrHrFrvrBrrrrrr __classcell__rGs@r4rXrXjsBBH B B!#!#[!#F<<<JJJ;;;IIIIII* DDDDDDDDDDr7rXczeZdZdZejZejZedZ dZ dZ dZ d fd Z d Zd fd ZxZS) ra` Bessel function of the second kind. Explanation =========== The Bessel $Y$ function of order $\nu$ is defined as .. math :: Y_\nu(z) = \lim_{\mu \to \nu} \frac{J_\mu(z) \cos(\pi \mu) - J_{-\mu}(z)}{\sin(\pi \mu)}, where $J_\mu(z)$ is the Bessel function of the first kind. It is a solution to Bessel's equation, and linearly independent from $J_\nu$. Examples ======== >>> from sympy import bessely, yn >>> from sympy.abc import z, n >>> b = bessely(n, z) >>> b.diff(z) bessely(n - 1, z)/2 - bessely(n + 1, z)/2 >>> b.rewrite(yn) sqrt(2)*sqrt(z)*yn(n - 1/2, z)/sqrt(pi) See Also ======== besselj, besseli, besselk References ========== .. [1] https://functions.wolfram.com/Bessel-TypeFunctions/BesselY/ c|jrU|jr tjSt|jdur tjSt|jr tjS|tjtjfvr tjS|ttjzkr2tttz|dzzdz tjzS|ttjzkr3tt tz|dzzdz tjzS|j r6| r$tj | zt| |zSdSdS)NFr9rC)r^rrr"r{r}r~rzrrrrVrrrr>s r4rBz bessely.evalFs/ 9 z ))B5((((B u Q/0 0 06M !*  qtR!V}Q''!*4 4 !$$ $ $r"ub1f~a'((1:5 5 = <**,, <}s+GRCOO;; < < < @=MY1 }YrAv%6%66r99STSYHQ3)R " %56Q!,8VWJ(J78HHQUHVVCJ ]  11tDT1W9D# oAwwRU1Wq[2a4%7!8!8 81SBq1rRStAS=T=T;TVWXYVY;Z Z[\`abcdad\e\eefjkmfnfnnnKWd##99!T4HHHrc>|j\}}|jr |jrdSdSdSrr1rVrfrs r4rzbessely._eval_is_extended_real9 A = Q] 4    r7cddlm}|j\}} ||\}} n#tt f$r|cYSwxYw| jr|jrt|| z } t||} dtz t|dz z| z ||||} gg}} |||z|}|dz |||| }|tjur|St!|dz|z }|tjkr|| zt#|dz ztz }| |t'd|D]d}||z |z}|tjkr |||z z}n|||z z}t!||z }| |e||ztt#|zz }|t)|dztjz z}||t'd| dzdzD]u}|| |||zzz z}t!||z }|t)||zdzt)|dzzz}||v| t-| z t-|z St/t0| ||||Sr)rrr1rrrrfrVrrXrrrrrrzr rrrr'rrrrr3r`rrrrr@rArrrbnrabrrrrrrdenomprGs r4rzbessely._eval_nseriess -,,,,, A ZZ]]FAss/0   KKK  ? )r} )1S5>>DQBB$AaC#221atDDArqAadAA1##Aq$55==??AAF{{!Q$!#,,..AAF{{B3x "q& 1 11"4q"##A!VQJE! %$TNNQ.7799DHHTNNNN2r)B--'(Agb1foo 45D HHTNNN1tax!m,,  aRAF_$a[[1_--//'!b&1*--A>?sAw;a( (Wd##11!QdCCCrrr)rqrrrsrtrryrHrFrvrBrrrrrrrrs@r4rrs&&P B B<<[<&LLL''' ===IIIIII2 /D/D/D/D/D/D/D/D/D/Dr7rc|eZdZdZej ZejZedZ dZ dZ dZ dZ d fd Zd fd ZxZS) rYa 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/ cf|jr|jr tjS|jr |jdust |jr tjSt |jr|jdur tjS|j r tj St|tj tj fvr tjS|tj ur tj S|tj urd|ztj zS|r||z| | zzt|| zS|jr^|rt| |S|t"}|rt"| zt%|| zS|jr&t'|}||krt||SnS|\}}|dkr6t+d|zt,z|zt"zt||zSt'|}||krt||SdS)NFTrrC)r^rryrVr"rfrzrhr{r|r}r#r~rrrYrrrXr%rrrrs r4rBz besseli.evals 9 z u - BJ%$7$7BrFF|j\}}|jr |jrdSdSdSrrrs r4rzbesseli._eval_is_extended_realrr7Nrc|j\}} ||}n#t$r|cYSwxYw||\}}|jr||zd|zt |dzzz S|jrE|dkrdn|}|||zz} | js*t|tdtz|zz S|Stt| |||SNrCr9r) r1rrrrfr&rhrrrrrYrrs r4rzbesseli._eval_as_leading_terms  A ##A&&CC"   KKK $$Q''1 = 7ArE%Q--/0 0 ]  11tDT1W9D# +1vvd1R46ll**KWd##99!T4HHHrcddlm}|j\}} ||\}} n#tt f$r|cYSwxYw| jr t|| z } |||z|} |dz |||| } | tj ur| St| dz| z } | |zt|dzz }|g}td| dzdzD]I}|| |||zzz z}t|| z }||Jt!|| zSt#t$|||||Sr)rrr1rrrrfrrrrrzr r&rrrrrYrs r4rzbesseli._eval_nseries.s -,,,,, A ZZ]]FAss/0   KKK  ? 1S5>>DadAA1##Aq$55==??AAF{{!Q$!#,,..Ab5rAv&DA1tax!m,,  1b1f:& *33557Q; Wd##11!QdCCCrrr)rqrrrsrtrryrHrFrvrBrrrrrrrrs@r4rYrYs%%N %B B%#%#[%#N<<<''' EEE IIIIII*DDDDDDDDDDr7rYceZdZdZejZej ZedZ dZ dZ dZ dZ dZd fd Zd fd ZxZS)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/ c|jrU|jr tjSt|jdur tjSt|jr tjS|tjt tjzt tjzfvr tjS|j r%| rt| |SdSdSrM) r^rr~r"r{r}rrrzrVrrr>s r4rBz besselk.evalws 9 z z!B5((((B u Qqz\1Q-?+?@ @ @6M = '**,, 'sA& ' ' ' 'r7c |jdurEttt|zzt| |t||z zdz SdS)NFrC)rVrrrYrs r4rz besselk._eval_rewrite_as_besselisL =E ! !c"R%jj='2#q//GBNN"BCAE E " !r7c \|j|j}|r|tSdSr<)rr1rrX)r3r@rArnais r4rz besselk._eval_rewrite_as_besseljrr7c \|j|j}|r|tSdSr<rrs r4rz besselk._eval_rewrite_as_besselyrr7c \|j|j}|r|tSdSr<)rr1rr])r3r@rArnays r4rzbesselk._eval_rewrite_as_yns5 *T *DI 6  "::b>> ! " "r7c>|j\}}|jr |jrdSdSdSrrrs r4rzbesselk._eval_is_extended_realrr7Nrc|j\}} ||}n#t$r|cYSwxYw||\}}|jrd|dz zt |dz zt ||z} |jr|dz | zt|dz zdz n tj } d|z|dz |zzdt|zz t|dztj z z} t| | | g||}|S|j r8ttt!| ztd|zz St#t$||||S)Nrr9rCr)r1rrrrfrrYrrrzr'rrrhrrrrrr) r3r`rrr@rArrrrrrrGs r4rzbesselk._eval_as_leading_terms A ##A&&CC"   KKK $$Q''1 = .r1u c!A#hh.wr1~~=H<>;KW!s|Ib1f$5$55a77QRQWHr1Q3)+Qy}}_=wrAvQRQ]?]^J(J78HHQUHVVCJ ] .88CGG#D1II- -Wd##99!T4HHHrcddlm}|j\}} ||\}} n#tt f$r|cYSwxYw| jr|jryt|| z } t||} d|dz zt|dz z| z ||||} gg}} |||z|}|dz |||| }|tjur|St|dz|z }|tjkr|| zt!|dz zdz }| |t%d|D]d}||z |z}|tjkr |||z z}n|||z z}t||z }| |e||zd|zzdt!|zz }|t'|dztjz z}||t%d| dzdzD]t}|||||zzz z}t||z }|t'||zdzt'|dzzz}||u| t+| zt+|zSt-t.| ||||S)Nrrrr9rC)rrr1rrrrfrVrrYrrrrrzr rrrr'rrrrrs r4rzbesselk._eval_nseriess -,,,,, A ZZ]]FAss/0   KKK  ? )r} )1S5>>DQBQAaC(+::1atLLArqAadAA1##Aq$55==??AAF{{!Q$!#,,..AAF{{B3x "q& 1 11!3q"##AVQJE! %$TNNQ.7799DHHTNNNN2rBh)B--0Agb1foo 45D HHTNNN1tax!m,,  Q1r6 ^#a[[1_--//'!b&1*--A>?sAw;a( (Wd##11!QdCCCrrr)rqrrrsrtrryrHrFrvrBrrrrrrrrrs@r4rrNs##J B %B ' '[ 'FFF''' ''' """  IIIIII*/D/D/D/D/D/D/D/D/D/Dr7rc4eZdZdZejZejZdZdS)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/ c|j}|jdur9t|j|SdSrM)r:rNhankel2r5rOrPs r4rQzhankel1._eval_conjugateE M !U * *4://111;;==AA A + *r7N rqrrrsrtrryrHrFrQr=r7r4rrsC""H B BBBBBBr7rc4eZdZdZejZejZdZdS)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/ c|j}|jdur9t|j|SdSrM)r:rNrr5rOrPs r4rQzhankel2._eval_conjugate=rr7Nrr=r7r4rrsC##J B BBBBBBr7rc<tfd}|S)Nc0|jr |||SdSr<)rV)r3r@rAfns r4gzassume_integer_order..gDs) = #2dB?? " # #r7r)rrs` r4assume_integer_orderrCs3 2YY####Y# Hr7c&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. expansionrr3ris r4_expandzSphericalBesselBase._expandWs!+...r7c 8|jjr |jdi|S|SNr=)r5 is_Integerrrs r4rgz%SphericalBesselBase._eval_expand_func[s, :  )4<((%(( ( r7rCc|dkrt||||jdz |j||jdzz|jz z SrE)r rGr5r:rIs r4rKzSphericalBesselBase.fdiff`sS q==$T844 4~~dj1ndm<< DJN #DM 12 2r7Nrp)rqrrrsrtrrgrKr=r7r4rrKsP  /// 222222r7rct||t|ztj|dzzt| dz |zt |zzSNr9)r*rrrrrrAs r4_jnrgsU 1 % %c!ff , MAE "#6rAvq#A#A A#a&& H IJr7ctj|dzzt| dz |zt|zt||t |zz Sr )rrr*rrrs r4_ynrlsS MAE "%8!a%C%C CCFF J 1 % %c!ff , -.r7cFeZdZdZedZdZdZdZdZ 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] https://dlmf.nist.gov/10.47 c|jr9|jr tjS|jr|jr tjStjS|tjtjfvr tjSdSr<) r^rryrVrfrzr{rr~r>s r4rBzjn.evalse 9 -z -u  ->-6M,, #QZ0 0 06M 1 0r7c rttd|zz t|tjz|zSr)rrrXrrrs r4rzjn._eval_rewrite_as_besseljs+B!H~~QV Q 7 777r7c tj|zttd|zz zt | tjz |zSr)rrrrrrrs r4rzjn._eval_rewrite_as_besselys9}b 4AaC>>1GRC!&L!4L4LLLr7c Jtj|zt| dz |zSr )rrr]rs r4rzjn._eval_rewrite_as_yns"}r"Ra^^33r7c 6t|j|jSr<)rr5r:rs r4rz jn._expand4:t}---r7cx|jjr-|t|SdSr<r5r rrX _eval_evalfr3precs r4rzjn._eval_evalf9 :  ;<<((44T:: : ; ;r7N) rqrrrsrtrvrBrrrrrr=r7r4r\r\rs88r  [ 888MMM444...;;;;;r7r\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] https://dlmf.nist.gov/10.47 c tj|dzzttd|zz zt | tjz |zSrd)rrrrrXrrs r4rzyn._eval_rewrite_as_besseljs=}r!t$tB!H~~5af a8P8PPPr7c rttd|zz t|tjz|zSr)rrrrrrs r4rzyn._eval_rewrite_as_besselys+B!H~~QV Q 7 777r7c Ptj|dzzt| dz |zSr )rrr\rs r4rzyn._eval_rewrite_as_jns&}rAv&RC!GQ77r7c 6t|j|jSr<)rr5r:rs r4rz yn._expandrr7cx|jjr-|t|SdSr<)r5r rrrrs r4rzyn._eval_evalfrr7N) rqrrrsrtrrrrrrr=r7r4r]r]s--\QQQ888888...;;;;;r7r]cXeZdZedZedZdZdZdZdZ dZ dS) SphericalHankelBasec |j}ttd|zz t|tjz||t ztj|dzzzt| tjz |zzzSrE)_hankel_kind_signrrrXrrrrr3r@rArnhkss r4rz,SphericalHankelBase._eval_rewrite_as_besseljsp $B!H~~wrAF{A66"1uQ]RT%::7B3 >r7c |j}t||t|tzt||zzSr<)r(r\rr]rr)s r4rz'SphericalHankelBase._eval_rewrite_as_yn s;$"ayy  $$s1uRAYY66r7c |j}t|||tzt||tzzSr<)r(r\rr]rr)s r4rz'SphericalHankelBase._eval_rewrite_as_jn$s<$"ayy3q5B!2!22!6!6666r7c |jjr |jdi|S|j}|j}|j}t |||t zt||zzSr )r5r rr:r(r\rr])r3rir@rAr*s r4rgz%SphericalHankelBase._eval_expand_func(s` :  /4<((%(( (B A(Cb!99s1uRAYY. .r7c |j}|j}|j}t|||tzt ||zzSr<)r5r:r(rrrexpand)r3rirrAr*s r4rzSphericalHankelBase._expand1sI J M$Aq CE#a))O+33555r7cx|jjr-|t|SdSr<rrs r4rzSphericalHankelBase._eval_evalf@rr7N) rqrrrsrrrrrrgrrr=r7r4r&r& sUUU>>>777777/// 6 6 6;;;;;r7r&c6eZdZdZejZedZdS)rZa 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 c Xttd|zz t||zSr)rrrrs r4_eval_rewrite_as_hankel1zhn1._eval_rewrite_as_hankel1x#B!H~~gb!nn,,r7N) rqrrrsrtrryr(rr4r=r7r4rZrZEsC..`-----r7rZc8eZdZdZej ZedZdS)r[a 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 c Xttd|zz t||zSr)rrrrs r4_eval_rewrite_as_hankel2zhn2._eval_rewrite_as_hankel2r5r7N) rqrrrsrtrryr(rr8r=r7r4r[r[}sE..`-----r7r[sympyc 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)rr9) besseljzero) dps_to_precc g|]Q}tjtdzt |RS)g?)r _from_mpmathr _to_mpmathint).0lr<rrs r4 zjn_zeros..sj***!++aCjj.C.CD.I.I.1!ff#6#67;==***r7r9scipy)newton) spherical_jnc|Sr<r=)r`rrGs r4zjn_zeros..s,,q!,,r7)sph_jnc4|ddS)Nrrr=)r`rrJs r4rIzjn_zeros..s&&A,,q/"-r7Unknown method.cLdkr ||}ntd|S)NrErLr)rjr`r methodrFs r4solverzjn_zeros..solvers3 W  6!Q<A?c.eZdZdZdZdZddZddZdS) AiryBasezg Abstract base class for Airy functions. This class is meant to reduce code duplication. cf||jdSr)funcr1rOr2s r4rQzAiryBase._eval_conjugates&yy1//11222r7c&|jdjSr)r1rr2s r4rzAiryBase._eval_is_extended_realsy|,,r7Tc |jd}|}|j}||||zdz }t||||z zdz }||fS)NrrC)r1rOr^r)r3deeprirAzcrjuvs r4 as_real_imagzAiryBase.as_real_imagsi IaL [[]] I QqTT!!B%%ZN qquuQQqTTzN1 !t r7c @|jdd|i|\}}||tzzS)Nrar=)rer)r3rarire_partim_parts r4_eval_expand_complexzAiryBase._eval_expand_complexs3,4,@@$@%@@""r7N)T)rqrrrsrtrQrrerir=r7r4r\r\ sd333---######r7r\cveZdZdZdZdZedZd 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] https://dlmf.nist.gov/9 .. [3] https://encyclopediaofmath.org/wiki/Airy_functions .. [4] https://mathworld.wolfram.com/AiryFunctions.html r9Tc|jr|tjur tjS|tjur tjS|tjur tjS|jr>tjdtddzttddzz S|jr>tjdtddzttddzz SdS)NrrC) is_Numberrr}r~rzrr^ryrr&r?rs r4rBz airyai.evals = Kae||u  ""v ***v  Ku8Aq>> 1E(1a..4I4I IJJ ; G5Ax1~~-hq!nn0E0EEF F G Gr7cb|dkrt|jdSt||Nr9r) airyaiprimer1r rIs r4rKz airyai.fdiff/ q==ty|,, ,$T844 4r7c |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)Nrr9rrrCr) rrzrlenrrrrrr&ryrr`previous_termsrs r4 taylor_termzairyai.taylor_terms q556M A>""Q&&"2&aqb)47719A*>>s2qRSUVGWZbcdfgZhZhGhCi?j?jjktuvkwkwwacHQNN233458Qx1~~=MPXYZ\]P^P^=^9_5`5`ajklopkpaqaq5qrwxyz{x{GHIKLMMyMsNsN6NORSSTq(1a..034uagqtt^7L7LLsS[\]_`SaSabdSdfghihmfmSnOoOoo!! %(,Q A~67r7c $tdd}tdd}t| tdd}t|jr<|t | zt | ||zt |||zzzSdSNr9rrCrrr"rhrrXr3rArnotttras r4rzairyai._eval_rewrite_as_besseljs a^^ a^^ HQNN # # a55  JdA2hh;'2#r!t"4"4wr2a47H7H"HI I J Jr7c tdd}tdd}t|tdd}t|jr;|t |zt | ||zt |||zz zS|t||t | ||zz|t|| zt |||zzz zSryrrr"rfrrYr{s r4rzairyai._eval_rewrite_as_besselis a^^ a^^ 8Aq>> " " a55  Xd1gg:"bd!3!3gb"Q$6G6G!GH Hs1bzz'2#r!t"4"44qQ}WRQSTUQUEVEV7VVW Wr7c tjdtddzttddzz }|t ddttddzz }|t gtddg|dzdz z|t gtddg|dzdz zz S)NrrCr9 r)rryrr&r r)r3rArnpf1pf2s r4_eval_rewrite_as_hyperzairyai._eval_rewrite_as_hyperseq(1a..(x1~~)>)>>?41::eHQNN3334U2A/Aa8883rHUVXYNNK[]^`a]abc]cAdAd;dddr7c |jd}|j}t|dkr0|}t d|g}t d|g}t d|g}t d|g}|||||zz|zz} | | |}d|zjr| |}| |}| |}|||zz|z||z|||zzzz } |||zz|||zzz} tj| tj zt| z| tj z tdz t| zz zSdSdSdS Nrr9r)excludedmrr) r1 free_symbolsrtpoprmatchrVrrryrkrairybi r3rirsymbsrArrrrMpfnewargs r4rgzairyai._eval_expand_funcs~il  u::?? AS1#&&&AS1#&&&AS1#&&&AS1#&&&A !Qq!tVaK-((A}aDaC#h!A!A!Aad(Q!Q$QqS/:BAXAaC0F6b15j&..%@BJPTUVPWPWCWX^_eXfXfCf%fgg# ?}hhr7Nr9rqrrrsrtnargs unbranchedrvrBrK staticmethodrrwrrrrgr=r7r4rkrk$sVVp EJ G G[ G5555   7 7 W\ 7JJJXXXeee hhhhhr7rkcveZdZdZdZdZedZd 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] https://dlmf.nist.gov/9 .. [3] https://encyclopediaofmath.org/wiki/Airy_functions .. [4] https://mathworld.wolfram.com/AiryFunctions.html r9Tc|jr|tjur tjS|tjur tjS|tjur tjS|jr>tjdtddzttddzz S|jr>tjdtddzttddzz SdS)Nrr9rC) rmrr}r~rrzr^ryrr&rns r4rBz airybi.eval.s = Kae||u  ""z!***v  Ku8Aq>> 1E(1a..4I4I IJJ ; G5Ax1~~-hq!nn0E0EEF F G Gr7cb|dkrt|jdSt||rp) airybiprimer1r rIs r4rKz airybi.fdiff=rrr7c |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)Nrr9rrrCr)rrzrrtrr!rrrryrrrr r&rus r4rwzairybi.taylor_termCs q556M A>""Q&&"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~67r7c $tdd}tdd}t| tdd}t|jr> " " a55  K77477?grc2a4&8&872r!t;L;L&LM MAr AAs A88QwsBqD111AaCBqD8I8I4IIJ Jr7c tjtddtt ddzz }|tddztt ddz }|t gt ddg|dzdz z|t gt ddg|dzdz zzS)NrrrCr9rr)rryr r&rr)rs r4rzairybi._eval_rewrite_as_hyperdsetAqzz%A"7"778Q lU8Aq>>222U2A/Aa8883rHUVXYNNK[]^`a]abc]cAdAd;dddr7c |jd}|j}t|dkr0|}t d|g}t d|g}t d|g}t d|g}|||||zz|zz} | | |}d|zjr| |}| |}| |}|||zz|z||z|||zzzz } |||zz|||zzz} tjtdtj | z zt| ztj | zt| zzzSdSdSdSr) r1rrtrrrrVrrrryrkrrs r4rgzairybi._eval_expand_funcis{il  u::?? AS1#&&&AS1#&&&AS1#&&&AS1#&&&A !Qq!tVaK-((A}aDaC#h!A!A!Aad(Q!Q$QqS/:BAXAaC0F6T!WWaebj%9&..%HAETVJX^_eXfXfKf%fgg# ?}hhr7Nrrr=r7r4rrsXXt EJ G G[ G5555   7 7 W\ 7III K K Keee hhhhhr7rcVeZdZdZdZdZedZd dZdZ dZ dZ d Z d Z d S) rqa% 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 .. 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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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