Xh0 ddlmZddlmZddlmZddlmZddlm 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>Gdde Z?Gdde?Z@Gdde?ZAGdde?ZBGdde?ZCGd d!e?ZDGd"d#e?ZEd$ZFGd%d&e?ZGd'ZHd(ZIGd)d*eGZJGd+d,eGZKGd-d.eGZLGd/d0eLZMGd1d2eLZNdGd5ZOGd6d7e ZPGd8d9ePZQGd:d;ePZRGd<d=ePZSGd>d?ePZTGd@dAe ZUGdBdCe ZVGdDdEe ZWdFS)Hwraps)S)Add)cacheit)Expr)DefinedFunctionArgumentIndexError_mexpand)fuzzy_or fuzzy_not)RationalpiI)Pow)Dummyuniquely_named_symbolWild)sympify) factorialRisingFactorial)sincoscsccot)ceiling)explog)cbrtsqrtroot)Absreim polar_lift unpolarify)gammadigamma uppergamma)hyper)spherical_bessel_fn)mpworkpreccteZdZdZedZedZedZd dZ dZ dZ d Z d Z d S) BesselBasea Abstract base class for Bessel-type functions. This class is meant to reduce code duplication. All Bessel-type functions can 1) be differentiated, with the derivatives expressed in terms of similar functions, and 2) be rewritten in terms of other Bessel-type functions. Here, Bessel-type functions are assumed to have one complex parameter. To use this base class, define class attributes ``_a`` and ``_b`` such that ``2*F_n' = -_a*F_{n+1} + b*F_{n-1}``. c|jdS)z( The order of the Bessel-type function. rargsselfs p/var/www/tools.fuzzalab.pt/emblema-extractor/venv/lib/python3.11/site-packages/sympy/functions/special/bessel.pyorderzBesselBase.order4y|c|jdS)z+ The argument of the Bessel-type function. r1r3s r5argumentzBesselBase.argument9r7r8cdSNclsnuzs r5evalzBesselBase.eval>sr8c|dkrt|||jdz ||jdz |jz|jdz ||jdz|jzz SNrDr:)r _b __class__r6r;_ar4argindexs r5fdiffzBesselBase.fdiffBso q==$T844 4 DNN4:>4=III DNN4:>4=IIIJ Kr8c|j}|jdur?||j|SdSNF)r;is_extended_negativerHr6 conjugater4rBs r5_eval_conjugatezBesselBase._eval_conjugateHsI M !U * *>>$*"6"6"8"8!++--HH H + *r8c |j|j}}||rdS|||sdS|||}|jrOt |ttttttfs|j st|jStt!|j |jgSrN)r6r;has_eval_is_meromorphicsubs is_integer isinstancebesseljbesselihn1hn2jnynis_zeror is_infiniter )r4xarArBz0s r5rUzBesselBase._eval_is_meromorphicMs DMA 66!99 5%%a++ 4 VVAq\\ = 1$'3R DEE 1RZ 1 0002:r~">??@@@r8c |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|SNr:rD) r6r;rHis_real is_positiverIrG_eval_expand_func is_negative)r4hintsrArBfs r5rhzBesselBase._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 r8c $ddlm}||S)Nr) besselsimp)sympy.simplify.simplifyrm)r4kwargsrms r5_eval_simplifyzBesselBase._eval_simplifyes$666666z$r8NrD)__name__ __module__ __qualname____doc__propertyr6r; classmethodrCrLrRrUrhrpr>r8r5r/r/$s  XX[KKKK III A A A        r8r/cxeZdZdZejZejZedZ dZ dZ dZ fdZ dZd fd ZxZS) rYa4 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)NFTrrD)r_rOnerWr#rgZeroriComplexInfinity is_imaginaryNaNInfinityNegativeInfinitycould_extract_minus_signrY NegativeOneextract_multiplicativelyrrZr&extract_branch_factorrrr@rArBnewznnnus r5rCz besselj.evals 9 z u - BJ%$7$7BrFF>0II I " !r8c |td|ztz t|tjz |jzSr)r rr]rHalfr;rs r5_eval_rewrite_as_jnzbesselj._eval_rewrite_as_jns-AaCF||BrAF{DM::::r8c|j\}} ||}n#t$r|cYSwxYw||\}}|jr||zd|zt |dzzz S|jrf|dkrdn|}|||zz} | jsKtdt|td|zdzzdz z ztt|zz S|Stt| |||S)NrDr:rlogxcdir) r2as_leading_termNotImplementedErroras_coeff_exponentrgr'rir rrsuperrY_eval_as_leading_term r4rarrrArBargcesignrHs r5rzbesselj._eval_as_leading_terms, A ##A&&CC"   KKK $$Q''1 = 7ArE%Q--/0 0 ]  11tDT1W9D# CAwws1r1R4!8}Q#6777RT BBKWd##99!$T9RRR # 22c>|j\}}|jr |jrdSdSdSNTr2rWis_extended_realr4rArBs r5_eval_is_extended_realzbesselj._eval_is_extended_real: A = Q/ 4    r8rc ddlm}|j\}} ||\}} n#tt f$r|cYSwxYw| jr t|| z } |||z|} |dz |||| } | tj ur| St| dz| z } | |zt|dzz }|g}td| dzdzD]J}|| |||zzz z}t|| z }||Kt!|| zSt#t$|||||SNrOrderrDr:)sympy.series.orderrr2leadterm ValueErrorrrgr _eval_nseriesremoveOrr{r r'rangeappendrrrYr4rarrrrrArB_rnewnorttermskrHs r5rzbesselj._eval_nseriess -,,,,, A ZZ]]FAss/0   KKK  ? 1S5>>DadAA1##Aq$55==??AAF{{!Q$!#,,..Ab5rAv&DA1tax!m,,  ArAvJ' *33557Q; Wd##11!QdCCC,AAr)rrrsrtrurrzrIrGrwrCrrrrrr __classcell__rHs@r5rYrYjsBBH B B!#!#[!#F<<<JJJ;;;SSSSS* DDDDDDDDDDr8rYcxeZdZdZejZejZedZ dZ dZ dZ fdZ dZd fd ZxZS) ra` Bessel function of the second kind. Explanation =========== The Bessel $Y$ function of order $\nu$ is defined as .. math :: Y_\nu(z) = \lim_{\mu \to \nu} \frac{J_\mu(z) \cos(\pi \mu) - J_{-\mu}(z)}{\sin(\pi \mu)}, where $J_\mu(z)$ is the Bessel function of the first kind. It is a solution to Bessel's equation, and linearly independent from $J_\nu$. Examples ======== >>> from sympy import bessely, yn >>> from sympy.abc import z, n >>> b = bessely(n, z) >>> b.diff(z) bessely(n - 1, z)/2 - bessely(n + 1, z)/2 >>> b.rewrite(yn) sqrt(2)*sqrt(z)*yn(n - 1/2, z)/sqrt(pi) See Also ======== besselj, besseli, besselk References ========== .. [1] https://functions.wolfram.com/Bessel-TypeFunctions/BesselY/ c|jrU|jr tjSt|jdur tjSt|jr tjS|tjtjfvr tjS|ttjzkr2tttz|dzzdz tjzS|ttjzkr3tt tz|dzzdz tjzS|j r6| r$tj | zt| |zSdSdS)NFr:rD)r_rrr#r|r~rr{rrrrWrrrr?s r5rCz bessely.evalFs/ 9 z ))B5((((B u Q/0 0 06M !*  qtR!V}Q''!*4 4 !$$ $ $r"ub1f~a'((1:5 5 = <**,, <}s+GRCOO;; < < < @=MY1 }YrAv%6%66r99STSYHQ3)R " %56Q!,8VWJ(J78HHQUHVVCJ ]  11tDT1W9D# oAwwRU1Wq[2a4%7!8!8 81SBq1rRStAS=T=T;TVWXYVY;Z Z[\`abcdad\e\eefjkmfnfnnnKWd##99!$T9RRRrc>|j\}}|jr |jrdSdSdSrr2rWrgrs r5rzbessely._eval_is_extended_real9 A = Q] 4    r8rcddlm}|j\}} ||\}} n#tt f$r|cYSwxYw| jr|jrt|| z } t||} dtz t|dz z| z ||||} gg}} |||z|}|dz |||| }|tjur|St!|dz|z }|tjkr|| zt#|dz ztz }| |t'd|D]d}||z |z}|tjkr |||z z}n|||z z}t!||z }| |e||ztt#|zz }|t)|dztjz z}||t'd| dzdzD]u}|| |||zzz z}t!||z }|t)||zdzt)|dzzz}||v| t-| z t-|z St/t0| ||||Sr)rrr2rrrrgrWrrYrrrrrr{r rrrr(rrrr)r4rarrrrrArBrrrbnrbbrrrrrrdenomprHs r5rzbessely._eval_nseriess -,,,,, A ZZ]]FAss/0   KKK  ? )r} )1S5>>DQBB$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!QdCCCrr)rrrsrtrurrzrIrGrwrCrrrrrrrrs@r5rrs&&P B B<<[<&LLL''' ===SSSSS2 /D/D/D/D/D/D/D/D/D/Dr8rceZdZdZej ZejZedZ d dZ dZ dZ dZ dZfd Zdfd Zfd ZxZS)rZa 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)NFTrrD)r_rrzrWr#rgr{rir|r}r~r$rrrrZrrrYr&rrrrs r5rCz besseli.evals 9 z u - BJ%$7$7BrFF|j\}}|jr |jrdSdSdSrrrs r5rzbesseli._eval_is_extended_realrr8c|j\}} ||}n#t$r|cYSwxYw||\}}|jr||zd|zt |dzzz S|jrE|dkrdn|}|||zz} | js*t|tdtz|zz S|Stt| |||S)NrDr:rr) r2rrrrgr'rirr rrrZrrs r5rzbesseli._eval_as_leading_terms  A ##A&&CC"   KKK $$Q''1 = 7ArE%Q--/0 0 ]  11tDT1W9D# +1vvd1R46ll**KWd##99!$T9RRRrrcddlm}|j\}} ||\}} n#tt f$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)rrr2rrrrgrrrrr{r r'rrrrrZrs r5rzbesseli._eval_nseries2s -,,,,, A ZZ]]FAss/0   KKK  ? 1S5>>DadAA1##Aq$55==??AAF{{!Q$!#,,..Ab5rAv&DA1tax!m,,  1b1f:& *33557Q; Wd##11!QdCCCrc  ddlmddlm}|d}|tjtjfvr|j\  fdt|D|d td|zdzdzz |gz}t tdtzz t|zSt||||S)Nrrrr:c g|]s}tdzdz d|tdzdzd|zd|ztd|zdzdzzt|zz tSrDr:rr.0rrrArBs r5 z)besseli._eval_aseries..XsQQQBC"/(1R4!8Q"7"7;;OOHUVWYUY\]U]_`LaLacdQQQr8rD(sympy.functions.combinatorial.factorialsrrrrrrr2rrrr rrr _eval_aseries r4rargs0rarrpointrrrArBrHs @@@r5rzbesseli._eval_aseriesQsLLLLLL,,,,,,a QZ!34 4 4IEBQQQQQQGLQxxQQQTYTYZ[\]`hijklilopiprs`t`t\uZuwxTyTySz{Aq66$qt**$Q0 0ww$$Qq$777r8r=r)rrrsrtrurrzrIrGrwrCrrrrrrrrrrs@r5rZrZs%%N %B B%#%#[%#N****<<<''' EEE SSSSS*DDDDDD> 8 8 8 8 8 8 8 8 8r8rZceZdZdZejZej ZedZ dZ dZ dZ dZ dZdd Zd Zdfd Zfd ZxZS)besselka Modified Bessel function of the second kind. Explanation =========== The Bessel $K$ function of order $\nu$ is defined as .. math :: K_\nu(z) = \lim_{\mu \to \nu} \frac{\pi}{2} \frac{I_{-\mu}(z) -I_\mu(z)}{\sin(\pi \mu)}, where $I_\mu(z)$ is the modified Bessel function of the first kind. It is a solution of the modified Bessel equation, and linearly independent from $Y_\nu$. Examples ======== >>> from sympy import besselk >>> from sympy.abc import z, n >>> besselk(n, z).diff(z) -besselk(n - 1, z)/2 - besselk(n + 1, z)/2 See Also ======== besselj, besseli, bessely References ========== .. [1] https://functions.wolfram.com/Bessel-TypeFunctions/BesselK/ c|jrU|jr tjSt|jdur tjSt|jr tjS|tjt tjzt tjzfvr tjS|j r%| rt| |SdSdSrN) r_rrr#r|r~rrr{rWrrr?s r5rCz besselk.evals 9 z z!B5((((B u Qqz\1Q-?+?@ @ @6M = '**,, 'sA& ' ' ' 'r8c |jdurEttt|zzt| |t||z zdz SdS)NFrD)rWrrrZrs r5rz besselk._eval_rewrite_as_besselisL =E ! !c"R%jj='2#q//GBNN"BCAE E " !r8c \|j|j}|r|tSdSr=)rr2rrY)r4rArBroais r5rz besselk._eval_rewrite_as_besseljrr8c \|j|j}|r|tSdSr=rrs r5rz besselk._eval_rewrite_as_besselyrr8c \|j|j}|r|tSdSr=)rr2rr^)r4rArBroays r5rzbesselk._eval_rewrite_as_yns5 *T *DI 6  "::b>> ! " "r8c>|j\}}|jr |jrdSdSdSrrrs r5rzbesselk._eval_is_extended_realrr8Nc V|jr!t| t||zSdSr=)rr_besselkrs r5rz"besselk._eval_rewrite_as_tractables2  +r778B??* * + +r8c|j\}} ||}n#t$r|cYSwxYw||\}}|jr|jr.t | tjz t dz} nQ|j r7tt||dz t| zzdz } ntd|d| ||S|j r8ttt| ztd|zz S|||S)NrDz"Cannot proceed without knowing if z is zero or not.r)r2rrrrgr_rrr is_nonzeror'r"rir rrfunc) r4rarrrArBrrrrs r5rzbesselk._eval_as_leading_termsJ A ##A&&CC"   KKK $$Q''1 = &z eAw-A6 eSWW~~qss2wwh&779)*cr*c*c*cddd'''55 5 ] &88CII%d1S5kk1 199R%% %s " 11rchddlm}|j\}} ||\}} n#tt f$r|cYSwxYw| jr|dz ||||} | tj ur||| z||zz|S|||z|} |j rt|| z } t||} d|dz zt|dz z| z||||}gg}}t| dz}|tj kr| | zt!|dz zdz }||t%d|D]I}||||z |zz z}t|| z}||J| |zd|zzdt!|zz }|t'|dztjz z}||t%d| dzdzD]t}|||||zzz z}t|| z}|t'||zdzt'|dzzz}||u|t+|zt+|z| zS|jr=t||z| z }t||z | z }gg}}t%|dzdzD]f}t/|| d|z|z zzdt1d|z |zt!|zz }|t|gt%|dzdzD]g}t/| | d|z|zzzdt1|dz|zt!|zz }|t|ht+|t+|z| zSt dt3t4|||||S)NrrrDrr:z4besselk expansion is only implemented for real order)rrr2rrrrgrrrr{rWrrZrr rrrr(rr is_nonintegerr'rrr)r4rarrrrrArBrrrrrrrbrrrrrrnewn_anewn_brHs r5rzbesselk._eval_nseriessV,,,,,, A ZZ]]FAss/0   KKK  ?6 b1##Aq$55==??AAF{{uQ"X2-q111adAA}0 bqu~~R^^BF^C!HH,R/>>q!T4PP21QTNN;;s8Ib1f$5$55a7DHHTNNN"1b\\''AFA:. ( 2;;==rE2(NAimmO4'"q&//AL89q4!8a-00##AAq2vJ'A!!q1133Aga"fqj11GAENNBCDHHTNNNN37{S!W,q00! b!!B$,, !B$,,21q1}----A 99Q1R[0!OAbD!4L4L2LYWX\\2YZDHHXd^^,,,,q1}----A "::a!A#b&k11_RT15M5M3MiXYll3ZHHXd^^,,,,Awa(1,,)*`aaaWd##11!QdCCCrc  ddlmddlm}|d}|tjtjfvr|j\  fdt|D|d td|zdzdzz |gz}t ttdz zt|zSt||||S)Nrrrr:c g|]s}tdzdz d|tdzdzd|zd|ztd|zdzdzzt|zz tSrDr:rrs r5rz)besselk._eval_aseries..sRRRCD"/(1R4!8Q"7"7;;OOHUVWYUY\]U]_`LaLacd>> 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|SdSrN)r;rOhankel2r6rPrQs r5rRzhankel1._eval_conjugateHE M !U * *4://111;;==AA A + *r8N rrrsrtrurrzrIrGrRr>r8r5rr sC""H B BBBBBBr8rc4eZdZdZejZejZdZdS)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|SdSrN)r;rOrr6rPrQs r5rRzhankel2._eval_conjugatewrr8Nrr>r8r5rrNsC##J B BBBBBBr8rc<tfd}|S)Nc0|jr |||SdSr=)rW)r4rArBfns r5gzassume_integer_order..g~s) = #2dB?? " # #r8r)rrs` r5assume_integer_orderr}s3 2YY####Y# Hr8c&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. expansionrr4rjs r5_expandzSphericalBesselBase._expands!+...r8c 8|jjr |jdi|S|SNr>)r6 is_Integerr$r#s r5rhz%SphericalBesselBase._eval_expand_funcs, :  )4<((%(( ( r8rDc|dkrt||||jdz |j||jdzz|jz z SrF)r rHr6r;rJs r5rLzSphericalBesselBase.fdiffsS q==$T844 4~~dj1ndm<< DJN #DM 12 2r8Nrq)rrrsrtrur$rhrLr>r8r5rrsP  /// 222222r8rct||t|ztj|dzzt| dz |zt |zzSNr:)r+rrrrrrBs r5_jnr,sU 1 % %c!ff , MAE "#6rAvq#A#A A#a&& H IJr8ctj|dzzt| dz |zt|zt||t |zz Sr*)rrr+rrr+s r5_ynr.sS MAE "%8!a%C%C CCFF J 1 % %c!ff , -.r8cFeZdZdZedZdZdZdZdZ 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_rrzrWrgr{r|rrr?s r5rCzjn.evalse 9 -z -u  ->-6M,, #QZ0 0 06M 1 0r8c rttd|zz t|tjz|zSr)r rrYrrrs r5rzjn._eval_rewrite_as_besseljs+B!H~~QV Q 7 777r8c tj|zttd|zz zt | tjz |zSr)rrr rrrrs r5rzjn._eval_rewrite_as_besselys9}b 4AaC>>1GRC!&L!4L4LLLr8c Jtj|zt| dz |zSr*)rrr^rs r5rzjn._eval_rewrite_as_yns"}r"Ra^^33r8c 6t|j|jSr=)r,r6r;r#s r5r$z jn._expand4:t}---r8cx|jjr-|t|SdSr=r6r'rrY _eval_evalfr4precs r5r8zjn._eval_evalf9 :  ;<<((44T:: : ; ;r8N) rrrsrtrurwrCrrrr$r8r>r8r5r]r]s88r  [ 888MMM444...;;;;;r8r]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 |zSre)rrr rrYrrs r5rzyn._eval_rewrite_as_besselj3s=}r!t$tB!H~~5af a8P8PPPr8c rttd|zz t|tjz|zSr)r rrrrrs r5rzyn._eval_rewrite_as_bessely7s+B!H~~QV Q 7 777r8c Ptj|dzzt| dz |zSr*)rrr]rs r5rzyn._eval_rewrite_as_jn;s&}rAv&RC!GQ77r8c 6t|j|jSr=)r.r6r;r#s r5r$z yn._expand>r5r8cx|jjr-|t|SdSr=)r6r'rrr8r9s r5r8zyn._eval_evalfAr;r8N) rrrsrtrurrrrr$r8r>r8r5r^r^s--\QQQ888888...;;;;;r8r^cXeZdZedZedZdZdZdZdZ dZ dS) SphericalHankelBasec |j}ttd|zz t|tjz||t ztj|dzzzt| tjz |zzzSrF)_hankel_kind_signr rrYrrrrr4rArBrohkss r5rz,SphericalHankelBase._eval_rewrite_as_besseljHsp $B!H~~wrAF{A66"1uQ]RT%::7B3 >r8c |j}t||t|tzt||zzSr=)rEr]rr^rrFs r5rz'SphericalHankelBase._eval_rewrite_as_ynZs;$"ayy  $$s1uRAYY66r8c |j}t|||tzt||tzzSr=)rEr]rr^rrFs r5rz'SphericalHankelBase._eval_rewrite_as_jn^s<$"ayy3q5B!2!22!6!6666r8c |jjr |jdi|S|j}|j}|j}t |||t zt||zzSr&)r6r'r$r;rEr]rr^)r4rjrArBrGs r5rhz%SphericalHankelBase._eval_expand_funcbs` :  /4<((%(( (B A(Cb!99s1uRAYY. .r8c |j}|j}|j}t|||tzt ||zzSr=)r6r;rEr,rr.expand)r4rjrrBrGs r5r$zSphericalHankelBase._expandksI J M$Aq CE#a))O+33555r8cx|jjr-|t|SdSr=r7r9s r5r8zSphericalHankelBase._eval_evalfzr;r8N) rrrsrtrrrrrrhr$r8r>r8r5rCrCFsUUU>>>777777/// 6 6 6;;;;;r8rCc6eZdZdZejZedZdS)r[a 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)r rrrs r5_eval_rewrite_as_hankel1zhn1._eval_rewrite_as_hankel1#B!H~~gb!nn,,r8N) rrrsrtrurrzrErrQr>r8r5r[r[sC..`-----r8r[c8eZdZdZej 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)r rrrs r5_eval_rewrite_as_hankel2zhn2._eval_rewrite_as_hankel2rRr8N) rrrsrtrurrzrErrUr>r8r5r\r\sE..`-----r8r\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)rrV) besseljzero) dps_to_precc g|]Q}tjtdzt |RS)g?)r _from_mpmathr _to_mpmathint)rlrYrr:s r5rzjn_zeros.."sj***!++aCjj.C.CD.I.I.1!ff#6#67;==***r8r:scipy)newton) spherical_jnc|Sr=r>)rarrbs r5zjn_zeros..)s,,q!,,r8)sph_jnc4|ddS)Nrrr>)rarres r5rdzjn_zeros..,s&&A,,q/"-r8Unknown method.cLdkr ||}ntd|S)Nr`rgr")rkrar!methodras r5solverzjn_zeros..solver0s3 W  6!Q<A?c.eZdZdZdZdZddZddZdS) AiryBasezg Abstract base class for Airy functions. This class is meant to reduce code duplication. cf||jdSNr)rr2rPr3s r5rRzAiryBase._eval_conjugateKs&yy1//11222r8c&|jdjSry)r2rr3s r5rzAiryBase._eval_is_extended_realNsy|,,r8Tc |jd}|}|j}||||zdz }t||||z zdz }||fS)NrrD)r2rPrr)r4deeprjrBzcrkuvs r5 as_real_imagzAiryBase.as_real_imagQsi IaL [[]] I QqTT!!B%%ZN qquuQQqTTzN1 !t r8c @|jdd|i|\}}||tzzS)Nr|r>)rr)r4r|rjre_partim_parts r5_eval_expand_complexzAiryBase._eval_expand_complexYs3,4,@@$@%@@""r8N)T)rrrsrtrurRrrrr>r8r5rwrwCsd333---######r8rwcveZdZdZdZdZedZd 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 r:Tc|jr|tjur tjS|tjur tjS|tjur tjS|jr>tjdtddzttddzz S|jr>tjdtddzttddzz SdS)NrrD) is_Numberrr~rr{rr_rzrr'r@rs r5rCz airyai.evals = Kae||u  ""v ***v  Ku8Aq>> 1E(1a..4I4I IJJ ; G5Ax1~~-hq!nn0E0EEF F G Gr8cb|dkrt|jdSt||Nr:r) airyaiprimer2r rJs r5rLz airyai.fdiff/ q==ty|,, ,$T844 4r8c |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)Nrr:rrrDr) rr{rlenrrrrrr'rzrraprevious_termsrs r5 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~67r8c $tdd}tdd}t| tdd}t|jr<|t | zt | ||zt |||zzzSdSNr:rrDrrr#rir rYr4rBrootttrbs r5rzairyai._eval_rewrite_as_besseljs a^^ a^^ HQNN # # a55  JdA2hh;'2#r!t"4"4wr2a47H7H"HI I J Jr8c tdd}tdd}t|tdd}t|jr;|t |zt | ||zt |||zz zS|t||t | ||zz|t|| zt |||zzz zSrrrr#rgr rZrs r5rzairyai._eval_rewrite_as_besselis a^^ a^^ 8Aq>> " " a55  Xd1gg:"bd!3!3gb"Q$6G6G!GH Hs1bzz'2#r!t"4"44qQ}WRQSTUQUEVEV7VVW Wr8c tjdtddzttddzz }|t ddttddzz }|t gtddg|dzdz z|t gtddg|dzdz zz S)NrrDr: r)rrzrr'r!r*r4rBropf1pf2s r5_eval_rewrite_as_hyperzairyai._eval_rewrite_as_hyperseq(1a..(x1~~)>)>>?41::eHQNN3334U2A/Aa8883rHUVXYNNK[]^`a]abc]cAdAd;dddr8c |jd}|j}t|dkr0|}t d|g}t d|g}t d|g}t d|g}|||||zz|zz} | | |}d|zjr| |}| |}| |}|||zz|z||z|||zzzz } |||zz|||zzz} tj| tj zt| z| tj z tdz t| zz zSdSdSdS Nrr:r)excludedmrr) r2 free_symbolsrpoprmatchrWrrrzrr airybi r4rjrsymbsrBrrrrMpfnewargs r5rhzairyai._eval_expand_funcs~il  u::?? AS1#&&&AS1#&&&AS1#&&&AS1#&&&A !Qq!tVaK-((A}aDaC#h!A!A!Aad(Q!Q$QqS/:BAXAaC0F6b15j&..%@BJPTUVPWPWCWX^_eXfXfCf%fgg# ?}hhr8Nr:rrrsrtrunargs unbranchedrwrCrL staticmethodrrrrrrhr>r8r5rr^sVVp EJ G G[ G5555   7 7 W\ 7JJJXXXeee hhhhhr8rcveZdZdZdZdZedZd 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 r:Tc|jr|tjur tjS|tjur tjS|tjur tjS|jr>tjdtddzttddzz S|jr>tjdtddzttddzz SdS)Nrr:rD) rrr~rrr{r_rzrr'rs r5rCz airybi.evalhs = Kae||u  ""z!***v  Ku8Aq>> 1E(1a..4I4I IJJ ; G5Ax1~~-hq!nn0E0EEF F G Gr8cb|dkrt|jdSt||r) airybiprimer2r rJs r5rLz airybi.fdiffwrr8c |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)Nrr:rrrDr)rr{rrrr"rrrrzrrrr!r'rs r5rzairybi.taylor_term}s 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~67r8c $tdd}tdd}t| tdd}t|jr> " " a55  K77477?grc2a4&8&872r!t;L;L&LM MAr AAs A88QwsBqD111AaCBqD8I8I4IIJ Jr8c tjtddtt ddzz }|tddztt ddz }|t gt ddg|dzdz z|t gt ddg|dzdz zzS)NrrrDr:rr)rrzr!r'rr*rs r5rzairybi._eval_rewrite_as_hypersetAqzz%A"7"778Q lU8Aq>>222U2A/Aa8883rHUVXYNNK[]^`a]abc]cAdAd;dddr8c |jd}|j}t|dkr0|}t d|g}t d|g}t d|g}t d|g}|||||zz|zz} | | |}d|zjr| |}| |}| |}|||zz|z||z|||zzzz } |||zz|||zzz} tjtdtj | z zt| ztj | zt| zzzSdSdSdSr) r2rrrrrrWrrr rzrrrs r5rhzairybi._eval_expand_funcs{il  u::?? AS1#&&&AS1#&&&AS1#&&&AS1#&&&A !Qq!tVaK-((A}aDaC#h!A!A!Aad(Q!Q$QqS/:BAXAaC0F6T!WWaebj%9&..%HAETVJX^_eXfXfKf%fgg# ?}hhr8Nrrr>r8r5rr sXXt EJ G G[ G5555   7 7 W\ 7III K K Keee hhhhhr8rcVeZdZdZdZdZedZd dZdZ dZ dZ d Z d Z d S) 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 r:Tc|jr4|tjur tjS|tjur tjS|jr>tjdtddzttddzz SdS)Nrr:) rrr~rr{r_rrr'rs r5rCzairyaiprime.evalsx = ae||u  ""v ; O=Ax1~~$5hq!nn8M8M$MN N O Or8c~|dkr(|jdt|jdzSt||r)r2rr rJs r5rLzairyaiprime.fdiff: q==9Q<ty| 4 44 4$T844 4r8c|jd|}t|5tj|d}dddn #1swxYwYt j||SNrr:) derivative)r2r]r-r,rrr\r4r:rBress r5r8zairyaiprime._eval_evalf! IaL # #D ) ) d^^ - -)A!,,,C - - - - - - - - - - - - - - - d+++AAAc tdd}t| tdd}t|jr.|dz t | ||zt |||zz zSdSNrDr)rrr#rirYr4rBrorrbs r5rz$airyaiprime._eval_rewrite_as_besselj'sx a^^ HQNN # # a55  BQ3'2#r!t,,wr2a4/@/@@A A B Br8c tdd}tdd}|t|tddz}t|jr(|dz t ||t | |z zSt|tdd}t||}t|| }||dz|zt |||zz|t | ||zzz zSr)rrr#rgrZrs r5rz$airyaiprime._eval_rewrite_as_besseli-s a^^ a^^ QA'' ' a55  JQ3'"a..7B3??:; ;Ax1~~&&AAr AAs AAaBqD 1 11Agrc2a46H6H4HHI Ir8c ~|dzddtddzzttddzz }dtddttddzz }|tgtddg|dzdz z|tgtddg|dzdz zz S)NrDrr:r)rr'r!r*rs r5rz"airyaiprime._eval_rewrite_as_hyper9sda8Aq>>))%A*?*??@41::eHQNN3334U2A/Aa8883rHUVXYNNK[]^`a]abc]cAdAd;dddr8c |jd}|j}t|dkr0|}t d|g}t d|g}t d|g}t d|g}|||||zz|zz} | | |}d|zjr| |}| |}| |}||z|||zzz|||zz|zz } |||zz|||zzz} tj| tj zt| z| tj z tdz t| zzzSdSdSdSr) r2rrrrrrWrrrzrr rrs r5rhzairyaiprime._eval_expand_func>sil  u::?? AS1#&&&AS1#&&&AS1#&&&AS1#&&&A !Qq!tVaK-((A}aD aC#r!A!A!AQ$QqS/a!Q$h]:BAXAaC0F6b15j+f2E2E%EaeUYZ[U\U\H\]hio]p]pHp%pqq' ?} rrr8NrrrrsrtrurrrwrCrLr8rrrrhr>r8r5rrsOOb EJOO[O5555 ,,, BBB J J Jeee rrrrrr8rcVeZdZdZdZdZedZd dZdZ dZ dZ d Z d Z d S) ra6 The derivative $\operatorname{Bi}^\prime$ of the Airy function of the first kind. Explanation =========== The Airy function $\operatorname{Bi}^\prime(z)$ is defined to be the function .. math:: \operatorname{Bi}^\prime(z) := \frac{\mathrm{d} \operatorname{Bi}(z)}{\mathrm{d} z}. Examples ======== Create an Airy function object: >>> from sympy import airybiprime >>> from sympy.abc import z >>> airybiprime(z) airybiprime(z) Several special values are known: >>> airybiprime(0) 3**(1/6)/gamma(1/3) >>> from sympy import oo >>> airybiprime(oo) oo >>> airybiprime(-oo) 0 The Airy function obeys the mirror symmetry: >>> from sympy import conjugate >>> conjugate(airybiprime(z)) airybiprime(conjugate(z)) Differentiation with respect to $z$ is supported: >>> from sympy import diff >>> diff(airybiprime(z), z) z*airybi(z) >>> diff(airybiprime(z), z, 2) z*airybiprime(z) + airybi(z) Series expansion is also supported: >>> from sympy import series >>> series(airybiprime(z), z, 0, 3) 3**(1/6)/gamma(1/3) + 3**(5/6)*z**2/(6*gamma(2/3)) + O(z**3) We can numerically evaluate the Airy function to arbitrary precision on the whole complex plane: >>> airybiprime(-2).evalf(50) 0.27879516692116952268509756941098324140300059345163 Rewrite $\operatorname{Bi}^\prime(z)$ in terms of hypergeometric functions: >>> from sympy import hyper >>> airybiprime(z).rewrite(hyper) 3**(5/6)*z**2*hyper((), (5/3,), z**3/9)/(6*gamma(2/3)) + 3**(1/6)*hyper((), (1/3,), z**3/9)/gamma(1/3) See Also ======== airyai: Airy function of the first kind. airybi: Airy function of the second kind. airyaiprime: Derivative of the Airy function of the first kind. References ========== .. [1] https://en.wikipedia.org/wiki/Airy_function .. [2] https://dlmf.nist.gov/9 .. [3] https://encyclopediaofmath.org/wiki/Airy_functions .. [4] https://mathworld.wolfram.com/AiryFunctions.html r:Tc|jr|tjur tjS|tjur tjS|tjur tjS|jr1dtddzttddz S|jr1dtddzttddz SdS)Nrr:r) rrr~rrr{r_rr'rs r5rCzairybiprime.evals = Aae||u  ""z!***v  A(1a..(5!Q+@+@@@ ; =hq!nn$uXa^^'<'<< < = =r8c~|dkr(|jdt|jdzSt||r)r2rr rJs r5rLzairybiprime.fdiffrr8c|jd|}t|5tj|d}dddn #1swxYwYt j||Sr)r2r]r-r,rrr\rs r5r8zairybiprime._eval_evalfrrc tdd}|t| tddz}t|jr6| t dz t | |t ||zzSdSrrrs r5rz$airybiprime._eval_rewrite_as_besseljsy a^^ aR!Q(( ( a55  C2d1gg:"a72q>>!AB B C Cr8c tdd}tdd}|t|tddz}t|jr5|t dz t | |t ||zzSt|tdd}t||}t|| }t ||t | ||zz|dz|zt |||zzzzSrrrs r5rz$airybiprime._eval_rewrite_as_besselis a^^ a^^ QA'' ' a55  PT!WW9Q'"a.. @A AAx1~~&&AAr AAs A88q"bd!3!33ad1fWRA=N=N6NNO Or8c r|dzdtddzttddzz }tddttddz }|tgtddg|dzdz z|tgtddg|dzdz zzS)NrDrrr:rr)r!r'rr*rs r5rz"airybiprime._eval_rewrite_as_hypersdaQ l5!Q#8#8891ajj5!Q000U2A/Aa8883rHUVXYNNK[]^`a]abc]cAdAd;dddr8c |jd}|j}t|dkr0|}t d|g}t d|g}t d|g}t d|g}|||||zz|zz} | | |}d|zjr| |}| |}| |}||z|||zzz|||zz|zz } |||zz|||zzz} tjtd| tj z zt| z| tj zt| zzzSdSdSdSr) r2rrrrrrWrrr rzrrrs r5rhzairybiprime._eval_expand_funcsil  u::?? AS1#&&&AS1#&&&AS1#&&&AS1#&&&A !Qq!tVaK-((A}aD aC#r!A!A!AQ$QqS/a!Q$h]:BAXAaC0F6T!WWb15j%9+f:M:M%MQSVWV[Q[]hio]p]pPp%pqq' ?} rrr8Nrrr>r8r5rrXsQQf EJ = =[ =5555 ,,, CCC P P Peee rrrrrr8rcHeZdZdZedZd dZdZdZdZ dZ d S) marcumqa The Marcum Q-function. Explanation =========== The Marcum Q-function is defined by the meromorphic continuation of .. math:: Q_m(a, b) = a^{- m + 1} \int_{b}^{\infty} x^{m} e^{- \frac{a^{2}}{2} - \frac{x^{2}}{2}} I_{m - 1}\left(a x\right)\, dx Examples ======== >>> from sympy import marcumq >>> from sympy.abc import m, a, b >>> marcumq(m, a, b) marcumq(m, a, b) Special values: >>> marcumq(m, 0, b) uppergamma(m, b**2/2)/gamma(m) >>> marcumq(0, 0, 0) 0 >>> marcumq(0, a, 0) 1 - exp(-a**2/2) >>> marcumq(1, a, a) 1/2 + exp(-a**2)*besseli(0, a**2)/2 >>> marcumq(2, a, a) 1/2 + exp(-a**2)*besseli(0, a**2)/2 + exp(-a**2)*besseli(1, a**2) Differentiation with respect to $a$ and $b$ is supported: >>> from sympy import diff >>> diff(marcumq(m, a, b), a) a*(-marcumq(m, a, b) + marcumq(m + 1, a, b)) >>> diff(marcumq(m, a, b), b) -a**(1 - m)*b**m*exp(-a**2/2 - b**2/2)*besseli(m - 1, a*b) References ========== .. [1] https://en.wikipedia.org/wiki/Marcum_Q-function .. [2] https://mathworld.wolfram.com/MarcumQ-Function.html c|tjurX|tjur|tjur tjSt||dztjzt |z S|tjur3|tjur%ddt |dztjzz z S||kr|tjur7dt |dz td|dzzztjzS|dkritjtjt |dz ztd|dzzzt |dz td|dzzzS|jrJ|jr|jr tjSt||dztjzt |z S|jr,|jr'ddt |dztjzz z SdSdS)NrDr:r) rr{r)rr'rrzrZr_)r@rrbrs r5rCz marcumq.eval-s ;;AF{{qAF{{v aA//%((: : ;;1;;q3q!taf}---- - 66AEzzCAJJAqD)9)99916AAAvvvadU 3gaA6F6F FFaQRdUV]^_abdeaeVfVfIfff 9 9y QY v aAaf--a8 8 9 , ,q3q!tAF{++++ + , , , ,r8rDc*|j\}}}|dkr*|t||| td|z||zzS|dkrC||z ||dz zz t|dz|dzz dz zt|dz ||zzSt ||)NrDr:r)r2rrrZr )r4rKrrbrs r5rLz marcumq.fdiffEs)1a q==Aq)))GAaCA,>,>>? ? ]]TEA!H$adQTkN1,<(=(==!QqS@Q@QQ Q$T844 4r8c *ddlm}|dtt dj}|d|z z|||zt |dz|dzz dz zt|dz ||zz||tj gzS)Nr)Integralrar:rD) sympy.integrals.integralsrgetrrnamerrZrr)r4rrbrrorras r5_eval_rewrite_as_Integralz!marcumq._eval_rewrite_as_IntegralNs666666 JJsE"7"<"<"ABB C CQU|x1sQTAqD[>!#3444wqsAaC7H7HH1aQRQ[J\]]^ ^r8c ddlm}|dtd}t |dz|dzz dz |||z |zt |||zz|d|z t jgzS)Nr)SumrrDr:)sympy.concrete.summationsrrrrrZrr)r4rrbrrorrs r5_eval_rewrite_as_Sumzmarcumq._eval_rewrite_as_SumTs111111 JJsE#JJ ' 'QTAqD[>A%&&acAX1Q3-G!QqSRSR\I])^)^^^r8c |kr|dkr-dtdz tddzzzdz S|jr|dkr{tfdt d|D}t jtdz tddzzdz ztdz |zzSdSdSdS)Nr:rDrc3>K|]}t|dzVdS)rDN)rZ)rrtrbs r5 z3marcumq._eval_rewrite_as_besseli..^s1>>Q1a4((>>>>>>r8)rrZr'sumrrr)r4rrbrrors ` r5rz marcumq._eval_rewrite_as_besseliYs 66AvvCAJJAqD)9)999Q>>| SQ>>>>%1++>>>>>vQTE WQ1-=-= = AACAJJQRNRR 6 S Sr8cFtd|jDrdSdS)Nc3$K|] }|jV dSr=)r_)rrs r5rz(marcumq._eval_is_zero..bs$00ss{000000r8T)allr2r3s r5 _eval_is_zerozmarcumq._eval_is_zeroas2 00di000 0 0 4  r8Nrq) rrrsrtrurwrCrLrrrrr>r8r5rrs..`,,[,.5555^^^ ___ SSSr8rc4eZdZdZfdZdZdfd ZxZS)rzq Helper function to make the $\mathrm{besseli}(nu, z)$ function tractable for the Gruntz algorithm. c  ddlmddlm}|d}|tjtjfvro|j\  fdt|D}ttdz t|z|d td|zdzdzz |zSt||||S)Nrrrr:c g|]s}tdzdz d|tdzdzd|zd|ztd|zdzdzzt|zz tSrrrs r5rz*_besseli._eval_aseries..ssuuufg#?8AbD1Ha#8#8!<<__QrTAXq))1>.>..12aXacAgq=Q=Q9R0RS\]^S_S_0_auuur8rDrrrrrrrr2rr rrrrr r4rrrarrrr_rrArBrHs @@@r5rz_besseli._eval_aserieslsLLLLLL,,,,,,a QZ!34 4 4IEBuuuuuukpqrksksuuuAA<<a)EE!A1q!8L8L4M2Mq,Q,QQ Qww$$Qq$777r8c Dt| t||zSr=)rrZrs r5_eval_rewrite_as_intractablez%_besseli._eval_rewrite_as_intractableysA2wwwr1~~%%r8rc|jd|d}|jr&|j|j}||||St |||Sryr2limitr_rrrr4rarrrx0rkrHs r5rz_besseli._eval_nseries|k Yq\  1 % % : /1149=A??1a.. .ww$$Q4000r8rrrrsrtrurrrrrs@r5rreso 8 8 8 8 8&&&1111111111r8rc4eZdZdZfdZdZdfd ZxZS)rzq Helper function to make the $\mathrm{besselk}(nu, z)$ function tractable for the Gruntz algorithm. c  ddlmddlm}|d}|tjtjfvro|j\  fdt|D}ttdz t|z|d td|zdzdzz |zSt||||S)Nrrrr:c g|]s}tdzdz d|tdzdzd|zd|ztd|zdzdzzt|zz tSrrrs r5rz*_besselk._eval_aseries..svvvgh#?8AbD1Ha#8#8!<<__QrTAXq))1>.>..13q !hqsQwPQ>R>R:S0ST]^_T`T`0`bvvvr8rDrrs @@@r5rz_besselk._eval_aseriessLLLLLL,,,,,,a QZ!34 4 4IEBvvvvvvlqrsltltvvvAA<<a)EE!A1q!8L8L4M2Mq,Q,QQ Qww$$Qq$777r8c Bt|t||zSr=)rrrs r5rz%_besselk._eval_rewrite_as_intractables1vvgb!nn$$r8rc|jd|d}|jr&|j|j}||||St |||Sryrrs r5rz_besselk._eval_nseriesrr8rrrs@r5rrso 8 8 8 8 8%%%1111111111r8rN)rVrW)X functoolsr sympy.corersympy.core.addrsympy.core.cachersympy.core.exprrsympy.core.functionr r r sympy.core.logicr r sympy.core.numbersrrrsympy.core.powerrsympy.core.symbolrrrsympy.core.sympifyrrrr(sympy.functions.elementary.trigonometricrrrr#sympy.functions.elementary.integersr&sympy.functions.elementary.exponentialrr(sympy.functions.elementary.miscellaneousrr r!$sympy.functions.elementary.complexesr"r#r$r%r&'sympy.functions.special.gamma_functionsr'r(r)sympy.functions.special.hyperr*sympy.polys.orthopolysr+rlr,r-r/rYrrZrrrrrr,r.r]r^rCr[r\rurwrrrrrrrr>r8r5rsp$$$$$$ MMMMMMMMMM00000000.......... @@@@@@@@@@&&&&&&OOOOOOOOGGGGGGGGGGGG777777;;;;;;;;EEEEEEEEEEVVVVVVVVVVVVVVNNNNNNNNNN//////666666C C C C C C C C LmDmDmDmDmDjmDmDmD`YDYDYDYDYDjYDYDYDxf8f8f8f8f8jf8f8f8R~8~8~8~8~8j~8~8~8B+B+B+B+B+Bj+B+B+B\,B,B,B,B,Bj,B,B,B^   22222*2228JJJ ... U;U;U;U;U; U;U;U;p?;?;?;?;?; ?;?;?;D6;6;6;6;6;-6;6;6;r5-5-5-5-5- 5-5-5-p5-5-5-5-5- 5-5-5-pQQQQh########6ihihihihihXihihihXnhnhnhnhnhXnhnhnhbZrZrZrZrZr(ZrZrZrzararararar(arararHgggggogggR11111111>1111111111r8