\h0 >SSKJr SSKJr SSKJr SSKJr SSKJ r SSK J r J r J r SSKJrJr SSKJrJrJr SS KJr SS KJrJrJr SS KJr SS KJrJr SS K J!r!J"r"J#r#J$r$ SSK%J&r& SSK'J(r(J)r) SSK*J+r+J,r,J-r- SSK.J/r/J0r0J1r1J2r2J3r3 SSK4J5r5J6r6J7r7 SSK8J9r9 SSK:J;r; SSKr> "SS\ 5r?"SS\?5r@"SS\?5rA"SS\?5rB"SS\?5rC"S S!\?5rD"S"S#\?5rES$rF"S%S&\?5rGS'rHS(rI"S)S*\G5rJ"S+S,\G5rK"S-S.\G5rL"S/S0\L5rM"S1S2\L5rNSES3jrO"S4S5\ 5rP"S6S7\P5rQ"S8S9\P5rR"S:S;\P5rS"S<S=\P5rT"S>S?\ 5rU"S@SA\ 5rV"SBSC\ 5rWgD)Fwraps)S)Add)cacheit)Expr)DefinedFunctionArgumentIndexError_mexpand)fuzzy_or fuzzy_not)RationalpiI)Pow)Dummyuniquely_named_symbolWild)sympify) factorialRisingFactorial)sincoscsccot)ceiling)explog)cbrtsqrtroot)Absreim polar_lift unpolarify)gammadigamma uppergamma)hyper)spherical_bessel_fn)mpworkpreccj\rSrSrSr\S5r\S5r\S5r S Sjr Sr Sr S r S rS rg ) BesselBase$a 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 URS$)z'The order of the Bessel-type function. rargsselfs V/var/www/auris/envauris/lib/python3.13/site-packages/sympy/functions/special/bessel.pyorderBesselBase.order4yy|c URS$)z*The argument of the Bessel-type function. r2r4s r6argumentBesselBase.argument9r9r:cgNclsnuzs r6evalBesselBase.eval>sr:c US:wa [X5eURS- URURS- UR5-UR S- URURS-UR5-- $Nr<)r _b __class__r7r=_ar5argindexs r6fdiffBesselBase.fdiffBsn q=$T4 4 DNN4::>4==II DNN4::>4==IIJ Kr:cURnURSLa8URURR 5UR 55$gNF)r=is_extended_negativerLr7 conjugater5rEs r6_eval_conjugateBesselBase._eval_conjugateHsB MM ! !U *>>$**"6"6"8!++-H H +r:c URURpCURU5(agURX5(dgUR X5nUR (aU[ U[[[[[[45(dUR(d[UR5$[[!URUR/55$rS)r7r=has_eval_is_meromorphicsubs is_integer isinstancebesseljbesselihn1hn2jnynis_zeror is_infiniter )r5xarDrEz0s r6r[BesselBase._eval_is_meromorphicMs DMMA 66!99%%a++ VVA\ ==$'3R DEERZZ 002::r~~">?@@r:c `URURURpCnUR(aUS- R(aiUR *UR -U"US- U5R5-SUR -US- -U"US- U5R5-U- -$US-R(ahSUR -US--U"US-U5R5-U- UR UR -U"US-U5R5-- $U$Nr<rJ) r7r=rLis_real is_positiverMrK_eval_expand_func is_negative)r5hintsrDrEfs r6roBesselBase._eval_expand_funcZs ::t}}dnnq ::Q##(261)G)G)II$'' 26*1R!VQ<+I+I+KKAMNOq&%%$'' 26*1R!VQ<+I+I+KKAM"q&! (F(F(HHIJ r:c SSKJn U"U5$)Nr) besselsimp)sympy.simplify.simplifyru)r5kwargsrus r6_eval_simplifyBesselBase._eval_simplifyes6$r:rANrJ)__name__ __module__ __qualname____firstlineno____doc__propertyr7r= classmethodrFrPrWr[rorx__static_attributes__rAr:r6r/r/$s_ K I A  r:r/c^\rSrSrSr\R r\R r\ S5r Sr Sr Sr U4SjrSrS U4S jjrS rU=r$) r_ja 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|UR(aUR(a[R$UR(aURSLd[ U5R (a[R $[ U5R(aURSLa[R$UR(a[R$U[R[R4;a[R $UR5(aX!-U*U*--[X*5-$UR(ahUR5(a"[RU*-[U*U5-$UR!["5nU(a["U-[%X5-$UR(a['U5nX2:wa [X5$OCUR)5up4US:wa+[+SU-[,-U-["-5[X5-$['U5nX:wa [XR5$g)NFTrrJ)rerOner]r#rnZerorpComplexInfinity is_imaginaryNaNInfinityNegativeInfinitycould_extract_minus_signr_ NegativeOneextract_multiplicativelyrr`r&extract_branch_factorrrrCrDrEnewznnnus r6rF besselj.evals 99zzuu --BJJ%$7BrF0II I "r:c |[SU-[- 5[U[R- UR 5-$r)r rrcrHalfr=rs r6_eval_rewrite_as_jnbesselj._eval_rewrite_as_jns,AaCF|BrAFF{DMM:::r:c>URupEURU5nURU5upxUR(aXd-SU-[ US-5-- $UR (aaUS:XaSOUnXsU--n U R (d=[S5[U[SU-S--S- - 5-[[U-5- $U$[[U]3XUS9$![a Us$f=f)NrJr<rlogxcdir) r3as_leading_termNotImplementedErroras_coeff_exponentrnr'rpr rrsuperr__eval_as_leading_term r5rgrrrDrEargcesignrLs r6rbesselj._eval_as_leading_terms  ##A&C$$Q' ==7ArE%Q-/0 0 ]] 1tD1W9D##Aws1r1R4!8}Q#677RT BBKWd9!T9RR# K sC## C21C2chURupUR(aUR(agggNTr3r]is_extended_realr5rDrEs r6_eval_is_extended_realbesselj._eval_is_extended_real&  ==Q//0=r:c>SSKJn URupgURU5upU R (a[X)- 5n U"X-U5n US- RXX45R5n U [RLaU $[U S-5U -R5n X-[US-5- nU/n[SU S-S-5H>nX*UUU--- -n[U5U -R5nURU5 M@ [!U6U -$["[$U]#XX45$![[ 4a Us$f=fNrOrderrJr<)sympy.series.orderrr3leadterm ValueErrorrrnr _eval_nseriesremoveOrrr r'rangeappendrrr_r5rgrrrrrDrE_rnewnorttermskrLs r6rbesselj._eval_nseriess> -  ZZ]FA ??15>DadAA1##A$5==?AAFF{!Q$!#,,.A5rAv&DA1tax!m,ArAvJ' *335-7Q; Wd1!CC'/0 K sD33EErAr)r{r|r}r~rrrrMrKrrFrrrrrrr __classcell__rLs@r6r_r_jsXBH B B!#!#F<J;S* DDr:r_c^\rSrSrSr\R r\R r\ S5r Sr Sr Sr U4SjrSrS U4S jjrS rU=r$) ria Bessel function of the second kind. Explanation =========== The Bessel $Y$ function of order $\nu$ is defined as .. math :: Y_\nu(z) = \lim_{\mu \to \nu} \frac{J_\mu(z) \cos(\pi \mu) - J_{-\mu}(z)}{\sin(\pi \mu)}, where $J_\mu(z)$ is the Bessel function of the first kind. It is a solution to Bessel's equation, and linearly independent from $J_\nu$. Examples ======== >>> from sympy import bessely, yn >>> from sympy.abc import z, n >>> b = bessely(n, z) >>> b.diff(z) bessely(n - 1, z)/2 - bessely(n + 1, z)/2 >>> b.rewrite(yn) sqrt(2)*sqrt(z)*yn(n - 1/2, z)/sqrt(pi) See Also ======== besselj, besseli, besselk References ========== .. [1] https://functions.wolfram.com/Bessel-TypeFunctions/BesselY/ c4UR(asUR(a[R$[U5RSLa[R$[U5R(a[R $U[R [R4;a[R$U[[R -:Xa0[[[-US--S- 5[R -$U[[R-:Xa1[[*[-US--S- 5[R -$UR(a8UR5(a"[RU*-[U*U5-$gg)NFr<rJ)rerrr#rrrrrrrr]rrrrBs r6rF bessely.evalFs 99zz)))B5((((Buu Q//0 066M !** qtR!V}Q'!**4 4 !$$$ $r"ub1f~a'(1::5 5 ==**,,}}s+GRCO;;- r:c URSLa?[[U-5[[U-5[ X5-[ U*U5- -$grS)r]rrrr_rs r6_eval_rewrite_as_besselj bessely._eval_rewrite_as_besseljZsD ==E !r"u:s2b5z'".87B3?JK K "r:c nUR"UR6nU(aUR[5$gr@)rr3rewriter`r5rDrErwajs r6r bessely._eval_rewrite_as_besseli^-  * *DII 6 ::g& & r:c |[SU-[- 5[U[R- UR 5-$r)r rrdrrr=rs r6_eval_rewrite_as_ynbessely._eval_rewrite_as_yncs,AaCF|baffdmm<<URupEURU5nURU5upxUR(aS[ - [ US- 5-[XE5-n UR(a US- U*-*[US- 5-[ - O[Rn US- U-*[ [U5-- [US-5[R- -n [XU /6RXS9nU$UR(aUS:XaSOUnXsU--n U R(dz[S5[![ U-S- U- [ S- -5*S[#[ U-S- U- [ S- -5-SU-- --[SU- 5-[[ 5- $U$[$[&U]SXUS9$![a Us$f=f) NrJr<rrrr)r3rrrrnrrr_rrrr( EulerGammarrpr rrrrr)r5rgrrrDrErrrterm_oneterm_two term_threerrLs r6rbessely._eval_as_leading_termfs  ##A&C$$Q' ==2s1Q3x6H>@=M=M1 }YrAv%66r9STSYSYHQ3)R " %56Q!,,8VWJJ78HHHVCJ ]] 1tD1W9D##AwRU1Wq[2a4%7!8 81SBq1rRStAS=T;TVWXYVY;Z Z[\`abcdad\eefjkmfnnnKWd9!T9RR'# K sG GGchURupUR(aUR(agggrr3r]rnrs r6rbessely._eval_is_extended_real$  ==Q]]+=r:cN>SSKJn URupgURU5upU R (Ga?UR(Ga-[X)- 5n [Xg5n S[- [US- 5-U -RXX45n //pU"X-U5nUS- RXX45R5nU[RLaU$[!US-5U-R5nU[R:aUU*-[#US- 5-[- nU R%U5 ['SU5H]nUU- U-nU[R:Xa UUU- -nOUUU- -n[!U5U-R5nU R%U5 M_ UU-[[#U5-- nU[)US-5[R*- -nUR%U5 ['SU S-S-5HbnUU*UUU--- -n[!U5U-R5nU[)UU-S-5[)US-5--nUR%U5 Md U [-U 6- [-U6- $[.[0U]3XX45$![[ 4a Us$f=fr)rrr3rrrrnr]rr_rrrrrrr rrrr(rrrr)r5rgrrrrrDrErrrbnrhbrrrrrrdenomprLs r6rbessely._eval_nseriessj -  ZZ]FA ???r}}}15>DBB$AaC#221DArqadAA1##A$5==?AAFF{!Q$!#,,.AAFF{B3x "q& 11"4q"A!VQJE! %$TNQ.779DHHTN&2r)B-'(Agb1fo 45D HHTN1tax!m,aRAF_$a[1_--/'!b&1*-A>? - sAw;a( (Wd1!CCK/0 K sJJ$#J$rAr)r{r|r}r~rrrrMrKrrFrrrrrrrrrs@r6rrsV&P B B<<&L' =S2 /D/Dr:rc^\rSrSrSr\R *r\R r\ S5r S Sjr Sr Sr SrSrU4S jrSU4S jjrU4S jrS rU=r$)r`ia 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/ cUR(aUR(a[R$UR(aURSLd[ U5R (a[R $[ U5R(aURSLa[R$UR(a[R$[U5[R[R4;a[R $U[RLa[R$U[RLaSU-[R-$UR5(aX!-U*U*--[X*5-$UR(aUUR5(a [U*U5$UR!["5nU(a["U*-[%X*5-$UR(a['U5nX2:wa [X5$OCUR)5up4US:wa+[+SU-[,-U-["-5[X5-$['U5nX:wa [XR5$g)NFTrrJ)rerrr]r#rnrrprrrr$rrrr`rrr_r&rrrrs r6rF besseli.evals 99zzuu --BJJ%$7BrFURupEURU5nURU5upxUR(aXd-SU-[ US-5-- $UR (aEUS:XaSOUnXsU--n U R (d![U5[S[-U-5- $U$[[U]3XUS9$![a Us$f=f)NrJr<rr) r3rrrrnr'rprr rrr`rrs r6rbesseli._eval_as_leading_terms  ##A&C$$Q' ==7ArE%Q-/0 0 ]] 1tD1W9D##1vd1R46l**KWd9!T9RR# K sC CCc>SSKJn URupgURU5upU R (a[X)- 5n U"X-U5n US- RXX45R5n U [RLaU $[U S-5U -R5n X-[US-5- nU/n[SU S-S-5H=nXUUU--- -n[U5U -R5nURU5 M? [!U6U -$["[$U]#XX45$![[ 4a Us$f=fr)rrr3rrrrnrrrrrr r'rrrrr`rs r6rbesseli._eval_nseries2s< -  ZZ]FA ??15>DadAA1##A$5==?AAFF{!Q$!#,,.A5rAv&DA1tax!m,1b1f:& *335-7Q; Wd1!CC'/0 K sD22EEc .>SSKJn SSKJn USnU[R [R 4;aURup[U5V s/sH^n U"[SU-S- S5U 5U"[SU-S-S5U 5-SU -U [SU -S-S5--[U 5-- PM` sn U"SU [SU-S-S5-- U5/-n [U 5[S[-5- [U 6-$[T U]AXX45$s sn fNrrrr<rJ(sympy.functions.combinatorial.factorialsrrrrrrr3rrrrr rrr _eval_aseries r5rargs0rgrrrpointrDrErrrLs r6rbesseli._eval_aseriesQs+L,a QZZ!3!34 4IIEBGLQxQGO!"(1R4!8Q"7;OHUVWYUY\]U]_`LacdGOQTYZ[\]`hijklilopiprs`t\uZuwxTySz{Aq6$qt*$Q0 0w$Qq77 QsA%DrAr@r)r{r|r}r~rrrrMrKrrFrrrrrrrrrrrs@r6r`r`sb%N %%B B%#%#N*<' E S*D> 8 8r:r`c^\rSrSrSr\R r\R *r\ S5r Sr Sr Sr SrSrSS jrS rSU4S jjrU4S jrS rU=r$)besselki_a 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/ cUR(asUR(a[R$[U5RSLa[R$[U5R(a[R $U[R[ [R-[ [R-4;a[R$UR(a#UR5(a [U*U5$ggrS) rerrr#rrrrrr]rrrBs r6rF besselk.evals 99zzzz!B5((((Buu Qqzz\1Q-?-?+?@ @66M ==**,,sA&- r:c URSLa6[[[U-5-[U*U5[X5- -S- $g)NFrJ)r]rrr`rs r6r besselk._eval_rewrite_as_besselis@ ==E !c"R%j='2#q/GBN"BCAE E "r:c nUR"UR6nU(aUR[5$gr@)rr3rr_)r5rDrErwais r6r besselk._eval_rewrite_as_besseljrr:c nUR"UR6nU(aUR[5$gr@rrs r6r besselk._eval_rewrite_as_besselyrr:c nUR"UR6nU(aUR[5$gr@)rr3rrd)r5rDrErways r6rbesselk._eval_rewrite_as_yns,  * *DII 6 ::b> ! r:chURupUR(aUR(agggrrrs r6rbesselk._eval_is_extended_realrr:c VUR(a[U*5[X5-$gr@)rr_besselkrs r6r"besselk._eval_rewrite_as_tractables%  r78B?* * r:crURupEURU5nURU5upxUR(aUR (a*[ U5*[R- [ S5-n OKUR(a+[[U55US- [U5*--S- n O[SUS35eU RXS9$UR(a+[[5[U*5-[SU-5- $UR!XF5$![a Us$f=f)NrJz"Cannot proceed without knowing if z is zero or not.r)r3rrrrnrerrr is_nonzeror'r"rpr rrfunc) r5rgrrrDrErrrrs r6rbesselk._eval_as_leading_terms  ##A&C$$Q' ==zzAw-A6SW~qss2wh&779),NrdRb*cdd'''5 5 ]]8CI%d1S5k1 199R% %'# K sD'' D65D6cR>SSKJn URupgURU5upU R (GaAUS- RXX45R5n U [RLaU"Xv*-Xv--U5$U"X-U5n UR(Ga[X)- 5n [Xg5n SUS- -[US- 5-U -RXX45n//nn[U S-5nU[R:avX*-[!US- 5-S- nUR#U5 [%SU5H>nUUUU- U-- -n[U5U -R5nUR#U5 M@ X-SU--S[!U5-- nU['US-5[R(- -nUR#U5 [%SU S-S-5HanUUUUU--- -n[U5U -R5nU['UU-S-5['US-5--nUR#U5 Mc U[+U6-[+U6-U -$UR,(Ga[X&-U - 5n[X&- U - 5n//p[%US-S-5HSn[/U5U SU-U- --S[1SU- U5-[!U5-- nUR#[U55 MU [%US-S-5HTn[/U*5U SU-U---S[1US-U5-[!U5-- nUR#[U55 MV [+U6[+U6-U -$[ S5e[2[4U]XX45$![[ 4a Us$f=f)NrrrJrr<z4besselk expansion is only implemented for real order)rrr3rrrrnrrrrr]rr`rr rrrr(rr is_nonintegerr'rrr)r5rgrrrrrDrErrrrrrrhrrrrrrnewn_anewn_brLs r6rbesselk._eval_nseriess~,  ZZ]FA ???1##A$5==?AAFF{QX-q11adAA}}}qu~R^BF^C!H,R/>>qTP21QTN;s8Ib1f$55a7DHHTN"1b\AFA:. ( 2;;=* E2(NAimO4'"q&/ALL89q4!8a-0AAq2vJ'A!!q113Aga"fqj1GAENBCDHHTN 1 37{S!W,q00!!!!!$, !$,21q1}-A 9Q1R[0!OAbD!4L2LYWX\2YZDHHXd^,.q1}-A ":a!A#b&k11_RT15M3MiXYl3ZHHXd^,.Awa(1,,)*`aaWd1!CC{/0 K sNN&%N&c 0>SSKJn SSKJn USnU[R [R 4;aURup[U5V s/sH^n U"[SU-S- S5U 5U"[SU-S-S5U 5-SU -U [SU -S-S5--[U 5-- PM` sn U"SU [SU-S-S5-- U5/-n [U *5[[S- 5-[U 6-$[T U]AXX45$s sn fNrrrr<rJrrs r6rbesselk._eval_aseriess-L,a QZZ!3!34 4IIEBHMaRHP1"(1R4!8Q"7;OHUVWYUY\]U]_`Lacd>> 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/ cURnURSLa2[URR 5UR 55$grS)r=rThankel2r7rUrVs r6rWhankel1._eval_conjugateH> MM ! !U *4:://11;;=A A +r:rAN r{r|r}r~rrrrMrKrWrrAr:r6r7r7 s""H B BBr:r7cN\rSrSrSr\R r\R rSr Sr g)r9iNaz 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/ cURnURSLa2[URR 5UR 55$grS)r=rTr7r7rUrVs r6rWhankel2._eval_conjugatewr;r:rANr<rAr:r6r9r9Ns"#J B BBr:r9c0^[T5U4Sj5nU$)Nc:>UR(a T"XU5$gr@)r])r5rDrEfns r6gassume_integer_order..g~s ==d? " r:r)rBrCs` r6assume_integer_orderrE}s  2Y## Hr:c.\rSrSrSrSrSrSSjrSrg) SphericalBesselBaseia Base class for spherical Bessel functions. These are thin wrappers around ordinary Bessel functions, since spherical Bessel functions differ from the ordinary ones just by a slight change in order. To use this class, define the ``_eval_evalf()`` and ``_expand()`` methods. c [S5e)z?Expand self into a polynomial. Nu is guaranteed to be Integer. expansionrr5rqs r6_expandSphericalBesselBase._expands !+..r:c `URR(aUR"S0UD6$U$NrA)r7 is_IntegerrLrKs r6ro%SphericalBesselBase._eval_expand_funcs& :: <<(%( ( r:cUS:wa [X5eURURS- UR5XRS--UR- - $rI)r rLr7r=rNs r6rPSphericalBesselBase.fdiffsO q=$T4 4~~djj1ndmm< JJN #DMM 12 2r:rANrz) r{r|r}r~rrLrorPrrAr:r6rGrGs / 2r:rGc[X5[U5-[RUS--[U*S- U5-[ U5--$Nr<)r+rrrrrrEs r6_jnrWsK  %c!f , MMAE "#6rAvq#A A#a& H IJr:c[RUS--[U*S- U5-[U5-[X5[ U5-- $rU)rrr+rrrVs r6_ynrYsI MMAE "%8!a%C CCF J  %c!f , -.r:cF\rSrSrSr\S5rSrSrSr Sr Sr S r g ) rciaV 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 cTUR(acUR(a[R$UR(a1UR(a[R $[R $U[R[R4;a[R $gr@) rerrr]rnrrrrrBs r6rFjn.evalsa 99zzuu >>66M,,, ##QZZ0 066M 1r:c h[[SU-- 5[U[R-U5-$r)r rr_rrrs r6rjn._eval_rewrite_as_besseljs(B!H~QVV Q 777r:c [RU-[[SU-- 5-[ U*[R - U5-$r)rrr rrrrs r6rjn._eval_rewrite_as_besselys8}}b 4AaC>1GRC!&&L!4LLLr:c J[RU-[U*S- U5-$rU)rrrdrs r6rjn._eval_rewrite_as_yns"}}r"Ra^33r:c B[URUR5$r@)rWr7r=rKs r6rL jn._expand4::t}}--r:cURR(a$UR[5R U5$gr@r7rPrr_ _eval_evalfr5precs r6rhjn._eval_evalf. :: <<(44T: : !r:rAN) r{r|r}r~rrrFrrrrLrhrrAr:r6rcrcs68r  8M4.;r:rccJ\rSrSrSr\S5r\S5rSrSr Sr Sr g ) rdia% 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 [RUS--[[SU-- 5-[ U*[R - U5-$rl)rrr rr_rrs r6ryn._eval_rewrite_as_besselj3s<}}r!t$tB!H~5aff a8PPPr:c h[[SU-- 5[U[R-U5-$r)r rrrrrs r6ryn._eval_rewrite_as_bessely7s(B!H~QVV Q 777r:c P[RUS--[U*S- U5-$rU)rrrcrs r6ryn._eval_rewrite_as_jn;s&}}rAv&RC!GQ77r:c B[URUR5$r@)rYr7r=rKs r6rL yn._expand>rer:cURR(a$UR[5R U5$gr@)r7rPrrrhris r6rhyn._eval_evalfArlr:rAN) r{r|r}r~rrErrrrLrhrrAr:r6rdrdsA-\QQ888.;r:rdcR\rSrSr\S5r\S5rSrSrSr Sr Sr S r g ) SphericalHankelBaseiFc URn[[SU-- 5[U[R -U5U[ -[RUS---[U*[R - U5---$rI)_hankel_kind_signr rr_rrrrr5rDrErwhkss r6r,SphericalHankelBase._eval_rewrite_as_besseljHsq $$B!H~wrAFF{A6"1uQ]]RT%::7B3 >r:c URn[X5R[5U[-[X5--$r@)r{rcrrdrr|s r6r'SphericalHankelBase._eval_rewrite_as_ynZs3$$"y  $s1uRY66r:c URn[X5U[-[X5R [5--$r@)r{rcrrdrr|s r6r'SphericalHankelBase._eval_rewrite_as_jn^s4$$"y3q5B!2!22!6666r:c URR(aUR"S0UD6$URnURnURn[ X#5U[ -[X#5--$rO)r7rPrLr=r{rcrrd)r5rqrDrEr}s r6ro%SphericalHankelBase._eval_expand_funcbsY :: <<(%( (B A((Cb9s1uRY. .r:c URnURnURn[X#5U[-[ X#5--R 5$r@)r7r=r{rWrrYexpand)r5rqrrEr}s r6rLSphericalHankelBase._expandksE JJ MM$$A CE#a)O+3355r:cURR(a$UR[5R U5$gr@rgris r6rhSphericalHankelBase._eval_evalfzrlr:rAN) r{r|r}r~rErrrrrorLrhrrAr:r6ryryFsCUU>>77/ 6;r:ryc@\rSrSrSr\R r\S5r Sr g)raia 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 D[[SU-- 5[X5-$r)r rr7rs r6_eval_rewrite_as_hankel1hn1._eval_rewrite_as_hankel1B!H~gbn,,r:rAN) r{r|r}r~rrrr{rErrrAr:r6raras&.`--r:racB\rSrSrSr\R *r\S5r Sr g)rbia 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 D[[SU-- 5[X5-$r)r rr9rs r6_eval_rewrite_as_hankel2hn2._eval_rewrite_as_hankel2rr:rAN) r{r|r}r~rrrr{rErrrAr:r6rbrbs(.`--r:rbc 0^^^^^SSKJn TS:XatSSKJn SSKJn U"U5n[ SUS-5Vs/sHEn[R"U"[TS-5RU5[U55U5PMG sn$TS:XaSS K J m SS KJm UU4S jn O [%S5eUU4Sjn TU-n U "X5n U /n [ US- 5Hn U "XU-5n U R'U 5 M! U $s snf![ a SS KJm UU4S jn Nhf=f)a1 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)rsympy) besseljzero) dps_to_precr<g?scipy)newton) spherical_jnc>T"TU5$r@rA)rgrrs r6jn_zeros..)s ,q!,r:)sph_jnc">T"TU5SS$)NrrrA)rgrrs r6rr,s&A,q/"-r:Unknown method.c:>TS:Xa T"X5nU$[S5e)NrrrJ)rrrgr!methodrs r6solverjn_zeros..solver0s) W !>> 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<TcUR(aU[RLa[R$U[RLa[R$U[R La[R$UR (a6[RS[SS5-[[SS55-- $UR (a6[RS[SS5-[[SS55-- $g)NrrJ) is_Numberrrrrrrerrr'rCrs r6rF airyai.evals ==aee|uu  "vv ***vv uu8Aq> 1E(1a.4I IJJ ;;55Ax1~-hq!n0EEF F r:cTUS:Xa[URS5$[X5eNr<r) airyaiprimer3r rNs r6rP airyai.fdiff' q=tyy|, ,$T4 4r:c XUS:a[R$[U5n[U5S:aUSn[ S5U-U*-[ S5U-US---[ [ U[SS5-[SS5--5-[U5-[US- [SS5-5-[ [ U[SS5-[SS5--5[US-5-[US- [SS5-5-- U-$[RS[SS5-[ -- [U[R-[S5- 5-[ [SS5[ -U[R--5-[U5- [ S5U-U--$)Nrr<rrrJr) rrrlenrrrrrr'rrrgprevious_termsrs r6 taylor_termairyai.taylor_terms q566M A>"Q&"2&aqb)4719A*>>s2qRSUVGWZbcdfgZhGhCi?jjktuvkwwacHQN23458Qx1~=MPXYZ\]P^=^9_5`ajklopkpaq5qrwxyz{x{GHIKLMyMsN6NORSSTq(1a.034uagqt^7LLsS[\]_`SabdSdfghihmhmfmSnOoo!! %(,Q A~67r:c [SS5n[SS5n[U*[SS55n[U5R(a.U[ U*5-[ U*XE-5[ X4U-5--$gNr<rrJrrr#rpr r_r5rErwotttrhs r6rairyai._eval_rewrite_as_besseljsl a^ a^ HQN # a5  dA2h;'2#rt"4wra47H"HI I r:c f[SS5n[SS5n[U[SS55n[U5R(a-U[ U5-[ U*XE-5[ X4U-5- -$U[XS5[ U*XE-5-U[XS*5-[ X4U-5-- -$rrrr#rnr r`rs r6rairyai._eval_rewrite_as_besselis a^ a^ 8Aq> " a5  d1g:"bd!3gbQ$6G!GH Hs1z'2#rt"44qQ}WRTUQUEV7VVW Wr:c >[RS[SS5-[[SS55-- nU[ SS5[[SS55-- nU[ /[SS5/US-S- 5-U[ /[SS5/US-S- 5-- $)NrrJr< r)rrrr'r!r*r5rErwpf1pf2s r6_eval_rewrite_as_hyperairyai._eval_rewrite_as_hyperseeq(1a.(x1~)>>?41:eHQN334U2A/Aa883rHUVXYNK[]^`a]abc]cAd;dddr:c ^URSnURn[U5S:XGaUR5n[ SU/S9n[ SU/S9n[ SU/S9n[ SU/S9nUR XVXH--U--5n U bXnSU-R (aXnXnXnXdU--U-Xg-XGU---- n XVU--XGU---n [RU [R-[U 5-U [R- [S5- [U 5-- -$ggg Nrr<r)excludedmrr) r3 free_symbolsrpoprmatchr]rrrrr airybi r5rqrsymbsrErrrrMpfnewargs r6roairyai._eval_expand_funcs<iil   u:? AS1#&AS1#&AS1#&AS1#&A !qtVaK-(A}DaC##AAAd(Q!$qS/:BAXaC0F66b155j&.%@BJPTUVPWCWX^_eXfCf%fgg $  r:rANr<r{r|r}r~rnargs unbranchedrrFrP staticmethodrrrrrrorrAr:r6rr^sbVp EJ G G5   7  7JXe hr:rcl\rSrSrSrSrSr\S5rSSjr \ \ S55r Sr S rS rS rS rg )ri a The Airy function $\operatorname{Bi}$ of the second kind. Explanation =========== The Airy function $\operatorname{Bi}(z)$ is defined to be the function satisfying Airy's differential equation .. math:: \frac{\mathrm{d}^2 w(z)}{\mathrm{d}z^2} - z w(z) = 0. Equivalently, for real $z$ .. math:: \operatorname{Bi}(z) := \frac{1}{\pi} \int_0^\infty \exp\left(-\frac{t^3}{3} + z t\right) + \sin\left(\frac{t^3}{3} + z t\right) \mathrm{d}t. Examples ======== Create an Airy function object: >>> from sympy import airybi >>> from sympy.abc import z >>> airybi(z) airybi(z) Several special values are known: >>> airybi(0) 3**(5/6)/(3*gamma(2/3)) >>> from sympy import oo >>> airybi(oo) oo >>> airybi(-oo) 0 The Airy function obeys the mirror symmetry: >>> from sympy import conjugate >>> conjugate(airybi(z)) airybi(conjugate(z)) Differentiation with respect to $z$ is supported: >>> from sympy import diff >>> diff(airybi(z), z) airybiprime(z) >>> diff(airybi(z), z, 2) z*airybi(z) Series expansion is also supported: >>> from sympy import series >>> series(airybi(z), z, 0, 3) 3**(1/3)*gamma(1/3)/(2*pi) + 3**(2/3)*z*gamma(2/3)/(2*pi) + O(z**3) We can numerically evaluate the Airy function to arbitrary precision on the whole complex plane: >>> airybi(-2).evalf(50) -0.41230258795639848808323405461146104203453483447240 Rewrite $\operatorname{Bi}(z)$ in terms of hypergeometric functions: >>> from sympy import hyper >>> airybi(z).rewrite(hyper) 3**(1/6)*z*hyper((), (4/3,), z**3/9)/gamma(1/3) + 3**(5/6)*hyper((), (2/3,), z**3/9)/(3*gamma(2/3)) See Also ======== airyai: Airy function of the first kind. airyaiprime: Derivative of the Airy function of the first kind. airybiprime: Derivative of the Airy function of the second kind. References ========== .. [1] https://en.wikipedia.org/wiki/Airy_function .. [2] https://dlmf.nist.gov/9 .. [3] https://encyclopediaofmath.org/wiki/Airy_functions .. [4] https://mathworld.wolfram.com/AiryFunctions.html r<TcUR(aU[RLa[R$U[RLa[R$U[RLa[R $UR (a6[RS[SS5-[[SS55-- $UR (a6[RS[SS5-[[SS55-- $g)Nrr<rJ) rrrrrrrerrr'rs r6rF airybi.evalhs ==aee|uu  "zz!***vv uu8Aq> 1E(1a.4I IJJ ;;55Ax1~-hq!n0EEF F r:cTUS:Xa[URS5$[X5er) airybiprimer3r rNs r6rP airybi.fdiffwrr:c pUS:a[R$[U5n[U5S:aUSn[ S5U-[ [ [SS5[-U[R--55-[U[R- [S5- 5-U[R-[ [[SS5[-U[R--55-[US- [S5- 5-- U-$[R[SS5[-- [U[R-[S5- 5-[ [ [SS5[-U[R--55-[U5- [ S5U-U--$)Nrr<rrrJr)rrrrrr"rrrrrrrr!r'rs r6rairybi.taylor_term}sw q566M A>"Q&"2&Q CHQN2,=q155y,I(J$KKiYZ]^]b]bYbdefgdhXhNiiaee)s3x1~b/@!aff*/M+N'OOR[]^ab]bdefgdh\hRiikmnoptAqz"}-q155y!A$6F0GG#cRZ[\^_R`acRcefijininenRoNpJqq!! %(,Q A~67r:c [SS5n[SS5n[U*[SS55n[U5R(a.[ U*S- 5[ U*XE-5[ X4U-5- -$grrrs r6rairybi._eval_rewrite_as_besseljsl a^ a^ HQN # a5  1:"bd!3gbQ$6G!GH H r:c [SS5n[SS5n[U[SS55n[U5R(a6[ U5[ S5- [ U*XE-5[ X4U-5--$[XS5n[XS*5n[ U5U[ U*XE-5-X-[ X4U-5---$rrr5rErwrrrhrrs r6rairybi._eval_rewrite_as_besselis a^ a^ 8Aq> " a5  747?grc24&872!t;L&LM MA AAs A8QwsBD11ACqD8I4IIJ Jr:c 8[R[SS5[[ SS55-- nU[SS5-[[ SS55- nU[ /[ SS5/US-S- 5-U[ /[ SS5/US-S- 5--$)NrrrJr<rr)rrr!r'rr*rs r6rairybi._eval_rewrite_as_hyperseetAqz%A"778Q lU8Aq>22U2A/Aa883rHUVXYNK[]^`a]abc]cAd;dddr:c ^URSnURn[U5S:XGaUR5n[ SU/S9n[ SU/S9n[ SU/S9n[ SU/S9nUR XVXH--U--5n U bXnSU-R (aXnXnXnXdU--U-Xg-XGU---- n XVU--XGU---n [R[S5[RU - -[U 5-[RU -[U 5---$gggr) r3rrrrrr]rrr rrrrs r6roairybi._eval_expand_funcs:iil   u:? AS1#&AS1#&AS1#&AS1#&A !qtVaK-(A}DaC##AAAd(Q!$qS/:BAXaC0F66T!Waeebj%9&.%HAEETVJX^_eXfKf%fgg $  r:rANrrrAr:r6rr sbXt EJ G G5   7  7I Ke hr:rcX\rSrSrSrSrSr\S5rSSjr Sr Sr S r S r S rS rg )riaI 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@UR(aFU[RLa[R$U[RLa[R$UR (a6[R S[SS5-[[SS55-- $g)Nrr<) rrrrrrerrr'rs r6rFairyaiprime.evalsh ==aee|uu  "vv ;;==Ax1~$5hq!n8M$MN N r:ctUS:Xa(URS[URS5-$[X5er)r3rr rNs r6rPairyaiprime.fdiff4 q=99Q<tyy| 44 4$T4 4r:cURSRU5n[U5 [R"USS9nSSS5 [ R "WU5$!,(df  N%=fNrr<) derivative)r3rr-r,rrrr5rjrEress r6rhairyaiprime._eval_evalf!Q IIaL # #D ) d^))A!,C  d++^ A A-c [SS5n[U*[SS55n[U5R(a$US- [ U*X4-5[ X3U-5- -$gNrJr)rrr#rpr_r5rErwrrhs r6r$airyaiprime._eval_rewrite_as_besselj's[ a^ HQN # a5  Q3'2#rt,wra4/@@A A r:c [SS5n[SS5nU[U[SS55-n[U5R(aUS- [ XE5[ U*U5- -$[U[SS55n[XT5n[XT*5nX1S-U-[ XDU-5-U[ U*XE-5-- -$r)rrr#rnr`r s r6r$airyaiprime._eval_rewrite_as_besseli-s a^ a^ QA' ' a5  Q3'".7B3?:; ;Ax1~&AA AAs AAaqD 11Agrc246H4HHI Ir:c .US-SS[SS5--[[SS55-- nS[SS5[[SS55-- nU[/[SS5/US-S- 5-U[/[SS5/US-S- 5-- $)NrJrr<r)rr'r!r*rs r6r"airyaiprime._eval_rewrite_as_hyper9sda8Aq>))%A*??@41:eHQN334U2A/Aa883rHUVXYNK[]^`a]abc]cAd;dddr:c ^URSnURn[U5S:XGaUR5n[ SU/S9n[ SU/S9n[ SU/S9n[ SU/S9nUR XVXH--U--5n U bXnSU-R (aXnXnXnXg-XHU---XdU--U-- n XVU--XHU---n [RU [R-[U 5-U [R- [S5- [U 5---$gggr) r3rrrrrr]rrrrr rrs r6roairyaiprime._eval_expand_func>s=iil   u:? AS1#&AS1#&AS1#&AS1#&A !qtVaK-(A}D aC##AAA$qS/aQ$h]:BAXaC0F66b155j+f2E%EaeeUYZ[U\H\]hio]pHp%pqq $  r:rANrr{r|r}r~rrrrrFrPrhrrrrorrAr:r6rrsKOb EJOO5 , B Je rr:rcX\rSrSrSrSrSr\S5rSSjr Sr Sr S r S r S rS rg )riXaR 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<TcUR(aU[RLa[R$U[RLa[R$U[RLa[R $UR (a%S[SS5-[[SS55- $UR (a%S[SS5-[[SS55- $g)Nrr<r) rrrrrrrerr'rs r6rFairybiprime.evals ==aee|uu  "zz!***vv (1a.(5!Q+@@@ ;;hq!n$uXa^'<< < r:ctUS:Xa(URS[URS5-$[X5er)r3rr rNs r6rPairybiprime.fdiffrr:cURSRU5n[U5 [R"USS9nSSS5 [ R "WU5$!,(df  N%=fr)r3rr-r,rrrrs r6rhairybiprime._eval_evalfrrc [SS5nU[U*[SS55-n[U5R(a)U*[ S5- [ U*U5[ X45--$grrr s r6r$airybiprime._eval_rewrite_as_besseljs^ a^ aR!Q( ( a5  2d1g:"a72>!AB B r:c [SS5n[SS5nU[U[SS55-n[U5R(a(U[ S5- [ U*U5[ XE5--$[U[SS55n[XT5n[XT*5n[ U5U[ U*XE-5-US-U-[ XDU-5---$rrr s r6r$airybiprime._eval_rewrite_as_besselis a^ a^ QA' ' a5  T!W9Q'". @A AAx1~&AA AAs A8q"bd!33ad1fWRA=N6NNO Or:c "US-S[SS5-[[SS55-- n[SS5[[SS55- nU[/[SS5/US-S- 5-U[/[SS5/US-S- 5--$)NrJrrr<r%r)r!r'rr*rs r6r"airybiprime._eval_rewrite_as_hypersdaQ l5!Q#8891aj5!Q00U2A/Aa883rHUVXYNK[]^`a]abc]cAd;dddr:c ^URSnURn[U5S:XGaUR5n[ SU/S9n[ SU/S9n[ SU/S9n[ SU/S9nUR XVXH--U--5n U bXnSU-R (aXnXnXnXg-XHU---XdU--U-- n XVU--XHU---n [R[S5U [R- -[U 5-U [R-[U 5---$gggr) r3rrrrrr]rrr rrrrs r6roairybiprime._eval_expand_funcs?iil   u:? AS1#&AS1#&AS1#&AS1#&A !qtVaK-(A}D aC##AAA$qS/aQ$h]:BAXaC0F66T!Wb155j%9+f:M%MQSVWV[V[Q[]hio]pPp%pqq $  r:rANrr)rAr:r6rrXsIQf EJ = =5 , C Pe rr:rcJ\rSrSrSr\S5rS SjrSrSr Sr Sr S r g ) marcumqia The Marcum Q-function. Explanation =========== The Marcum Q-function is defined by the meromorphic continuation of .. math:: Q_m(a, b) = a^{- m + 1} \int_{b}^{\infty} x^{m} e^{- \frac{a^{2}}{2} - \frac{x^{2}}{2}} I_{m - 1}\left(a x\right)\, dx Examples ======== >>> from sympy import marcumq >>> from sympy.abc import m, a, b >>> marcumq(m, a, b) marcumq(m, a, b) Special values: >>> marcumq(m, 0, b) uppergamma(m, b**2/2)/gamma(m) >>> marcumq(0, 0, 0) 0 >>> marcumq(0, a, 0) 1 - exp(-a**2/2) >>> marcumq(1, a, a) 1/2 + exp(-a**2)*besseli(0, a**2)/2 >>> marcumq(2, a, a) 1/2 + exp(-a**2)*besseli(0, a**2)/2 + exp(-a**2)*besseli(1, a**2) Differentiation with respect to $a$ and $b$ is supported: >>> from sympy import diff >>> diff(marcumq(m, a, b), a) a*(-marcumq(m, a, b) + marcumq(m + 1, a, b)) >>> diff(marcumq(m, a, b), b) -a**(1 - m)*b**m*exp(-a**2/2 - b**2/2)*besseli(m - 1, a*b) References ========== .. [1] https://en.wikipedia.org/wiki/Marcum_Q-function .. 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