a ¬Gd!d"„d"e:ƒZ?Gd#d$„d$e:ƒZ@d%d&„ZAGd'd(„d(e:ƒZBd)d*„ZCd+d,„ZDGd-d.„d.eBƒZEGd/d0„d0eBƒZFGd1d2„d2eBƒZGGd3d4„d4eGƒZHGd5d6„d6eGƒZIdHd9d:„ZJGd;d<„d„d>eKƒZLGd?d@„d@eKƒZMGdAdB„dBeKƒZNGdCdD„dDeKƒZOGdEdF„dFeƒZPdGS)Ié©Úwraps)ÚS)ÚAdd)Úcacheit)ÚExpr)ÚFunctionÚArgumentIndexErrorÚ_mexpand)Úfuzzy_orÚ fuzzy_not)ÚRationalÚpiÚI)ÚPow)ÚDummyÚWild)Úsympify)Ú factorial)ÚsinÚcosÚcscÚcot)Úceiling)ÚAbs)ÚexpÚlog)ÚsqrtÚroot)ÚreÚim)ÚgammaÚdigammaÚ uppergamma)Úhyper)Úspherical_bessel_fn)ÚmpÚworkprecc@s^eZdZdZedd„ƒZedd„ƒZedd„ƒZdd d „Z dd„Z d d„Zdd„Zdd„Z dS)Ú BesselBaseað Abstract base class for Bessel-type functions. This class is meant to reduce code duplication. All Bessel-type functions can 1) be differentiated, with the derivatives expressed in terms of similar functions, and 2) be rewritten in terms of other Bessel-type functions. Here, Bessel-type functions are assumed to have one complex parameter. To use this base class, define class attributes ``_a`` and ``_b`` such that ``2*F_n' = -_a*F_{n+1} + b*F_{n-1}``. cCs |jdS)z( The order of the Bessel-type function. r©Úargs©Úself©r-ún/Users/vegardjervell/Documents/master/model/venv/lib/python3.9/site-packages/sympy/functions/special/bessel.pyÚorder5szBesselBase.ordercCs |jdS)z+ The argument of the Bessel-type function. ér)r+r-r-r.Úargument:szBesselBase.argumentcCsdS©Nr-©ÚclsÚnuÚzr-r-r.Úeval?szBesselBase.evalécCsN|dkrt||ƒ‚|jd| |jd|j¡|jd| |jd|j¡S©Nr8r0)r Ú_bÚ __class__r/r1Ú_a©r,Úargindexr-r-r.ÚfdiffCs ÿzBesselBase.fdiffcCs*|j}|jdur&| |j ¡| ¡¡SdS©NF)r1Úis_extended_negativer;r/Ú conjugate©r,r6r-r-r.Ú_eval_conjugateIs zBesselBase._eval_conjugatecCsx|j|j}}| |¡rdS| ||¡s,dS| ||¡}|jrdt|ttt t ttfƒsZ|j sdt|jƒStt|j |jgƒƒSr@)r/r1ZhasÚ_eval_is_meromorphicÚsubsÚ is_integerÚ isinstanceÚbesseljÚbesseliÚhn1Úhn2ÚjnÚynÚis_zerorÚis_infiniter)r,ÚxÚar5r6Zz0r-r-r.rENs zBesselBase._eval_is_meromorphiccKsÂ|j|j|j}}}|jr¾|djrn|j|j||d|ƒ ¡d|j|d||d|ƒ ¡|S|djr¾d|j|d||d|ƒ ¡||j|j||d|ƒ ¡S|S©Nr0r8) r/r1r;Zis_realÚis_positiver<r:Ú_eval_expand_funcÚis_negative)r,Úhintsr5r6Úfr-r-r.rU[s &ÿ &ÿzBesselBase._eval_expand_funccKsddlm}||ƒS)Nr)Ú besselsimp)Zsympy.simplify.simplifyrY)r,ÚkwargsrYr-r-r.Ú_eval_simplifyfszBesselBase._eval_simplifyN)r8)Ú__name__Ú __module__Ú__qualname__Ú__doc__Úpropertyr/r1Úclassmethodr7r?rDrErUr[r-r-r-r.r(%s r(csdeZdZdZejZejZedd„ƒZ dd„Z dd„Zdd „Zddd „Z dd„Zd‡fdd„ Z‡ZS)rIa3 Bessel function of the first kind. Explanation =========== The Bessel $J$ function of order $\nu$ is defined to be the function satisfying Bessel's differential equation .. math :: z^2 \frac{\mathrm{d}^2 w}{\mathrm{d}z^2} + z \frac{\mathrm{d}w}{\mathrm{d}z} + (z^2 - \nu^2) w = 0, with Laurent expansion .. math :: J_\nu(z) = z^\nu \left(\frac{1}{\Gamma(\nu + 1) 2^\nu} + O(z^2) \right), if $\nu$ is not a negative integer. If $\nu=-n \in \mathbb{Z}_{<0}$ *is* a negative integer, then the definition is .. math :: J_{-n}(z) = (-1)^n J_n(z). Examples ======== Create a Bessel function object: >>> from sympy import besselj, jn >>> from sympy.abc import z, n >>> b = besselj(n, z) Differentiate it: >>> b.diff(z) besselj(n - 1, z)/2 - besselj(n + 1, z)/2 Rewrite in terms of spherical Bessel functions: >>> b.rewrite(jn) sqrt(2)*sqrt(z)*jn(n - 1/2, z)/sqrt(pi) Access the parameter and argument: >>> b.order n >>> b.argument z See Also ======== bessely, besseli, besselk References ========== .. [1] Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 9", Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables .. [2] Luke, Y. L. (1969), The Special Functions and Their Approximations, Volume 1 .. [3] https://en.wikipedia.org/wiki/Bessel_function .. [4] http://functions.wolfram.com/Bessel-TypeFunctions/BesselJ/ cCsf|jrX|jrtjS|jr"|jdus,t|ƒjr2tjSt|ƒjrL|jdurLtjS|j rXtj S|tjtjfvrntjS| ¡r–||||t||ƒS|jrÜ| ¡r¼tj|t||ƒS| t¡}|rÜt|t||ƒSddlm}|jr||ƒ}||krFt||ƒSn8| ¡\}}|dkrFtd|t|tƒt||ƒS||ƒ}||krbt||ƒSdS©NFTr)Ú unpolarifyr8)rOrÚOnerGrrTÚZerorVÚComplexInfinityÚis_imaginaryÚNaNÚInfinityÚNegativeInfinityÚcould_extract_minus_signrIÚNegativeOneÚextract_multiplicativelyrrJÚ$sympy.functions.elementary.complexesrcÚextract_branch_factorrr©r4r5r6ZnewzrcÚnZnnur-r-r.r7³s< " zbesselj.evalcKs4ddlm}ttt|dƒt||tƒ|ƒS©Nr)Ú polar_liftr8)rnrsrrrrJ©r,r5r6rZrsr-r-r.Ú_eval_rewrite_as_besseliØsz besselj._eval_rewrite_as_besselicKs<|jdur8tt|ƒt||ƒtt|ƒt||ƒSdSr@)rGrrÚbesselyr©r,r5r6rZr-r-r.Ú_eval_rewrite_as_besselyÜs z besselj._eval_rewrite_as_besselycKs"td|tƒt|tj|jƒS©Nr8)rrrMrÚHalfr1rwr-r-r.Ú_eval_rewrite_as_jnàszbesselj._eval_rewrite_as_jnNrcCsR|j\}}| |¡}||jvr:||d|t|dƒS| || |d¡¡SdS©Nr8r0r)r*Úas_leading_termÚfree_symbolsr!ÚfuncrF)r,rQÚlogxÚcdirr5r6Úargr-r-r.Ú_eval_as_leading_termãs zbesselj._eval_as_leading_termcCs|j\}}|jr|jrdSdS©NT©r*rGÚis_extended_real©r,r5r6r-r-r.Ú_eval_is_extended_realës zbesselj._eval_is_extended_realc s$ddlm}|j\}}z| |¡\}} WnttfyB|YS0| jrt|| ƒ} ||||ƒ}|d ||||¡ ¡}|t jurŒ|St|dƒ| ¡} ||t |dƒ}|g}td| ddƒD]4}|| |||9}t|ƒ| ¡}| |¡qÌt|Ž|Stt|ƒ ||||¡S©Nr)ÚOrderr8r0)Úsympy.series.orderrŠr*ÚleadtermÚ ValueErrorÚNotImplementedErrorrTrÚ _eval_nseriesÚremoveOrrer r!ÚrangeÚappendrÚsuperrI)r,rQrqr€rrŠr5r6Ú_rÚnewnÚoÚrÚtÚtermÚsÚk©r;r-r.rðs* zbesselj._eval_nseries)Nr)r)r\r]r^r_rrdr<r:rar7rurxr{rƒrˆrÚ __classcell__r-r-rœr.rIksD $ rIcsdeZdZdZejZejZedd„ƒZ dd„Z dd„Zdd „Zddd „Z dd„Zd‡fdd„ Z‡ZS)rva_ Bessel function of the second kind. Explanation =========== The Bessel $Y$ function of order $\nu$ is defined as .. math :: Y_\nu(z) = \lim_{\mu \to \nu} \frac{J_\mu(z) \cos(\pi \mu) - J_{-\mu}(z)}{\sin(\pi \mu)}, where $J_\mu(z)$ is the Bessel function of the first kind. It is a solution to Bessel's equation, and linearly independent from $J_\nu$. Examples ======== >>> from sympy import bessely, yn >>> from sympy.abc import z, n >>> b = bessely(n, z) >>> b.diff(z) bessely(n - 1, z)/2 - bessely(n + 1, z)/2 >>> b.rewrite(yn) sqrt(2)*sqrt(z)*yn(n - 1/2, z)/sqrt(pi) See Also ======== besselj, besseli, besselk References ========== .. [1] http://functions.wolfram.com/Bessel-TypeFunctions/BesselY/ cCsv|jr6|jrtjSt|ƒjdur&tjSt|ƒjr6tjS|tjtjfvrLtjS|jrr| ¡rrtj |t||ƒSdSr@)rOrrjrrfrhrirerGrkrlrvr3r-r-r.r7:s zbessely.evalcKs<|jdur8tt|ƒtt|ƒt||ƒt||ƒSdSr@)rGrrrrIrwr-r-r.Ú_eval_rewrite_as_besseljJs z bessely._eval_rewrite_as_besseljcKs|j|jŽ}|r| t¡SdSr2)ržr*ÚrewriterJ©r,r5r6rZZajr-r-r.ruNsz bessely._eval_rewrite_as_besselicKs"td|tƒt|tj|jƒSry)rrrNrrzr1rwr-r-r.Ú_eval_rewrite_as_ynSszbessely._eval_rewrite_as_ynNrc Cs¾|j\}}dtt|dƒt||ƒ}|djrP|d|t|dƒtntj}|d|tt|ƒt|dƒtj }t |||gŽ |¡} || jvr¢| S| || |d¡ ¡¡SdSr|)r*rrrIrTrrrer"Ú EulerGammarr}r~rrFÚcancel) r,rQr€rr5r6Zterm_oneZterm_twoZ term_threer‚r-r-r.rƒVs .* zbessely._eval_as_leading_termcCs|j\}}|jr|jrdSdSr„©r*rGrTr‡r-r-r.rˆas zbessely._eval_is_extended_realc s ddlm}|j\}}z| |¡\}} WnttfyB|YS0| jr |jr t|| ƒ} t ||ƒ}dt t|dƒ| ||||¡}gg} }||||ƒ}|d ||||¡ ¡}|tjurÌ|St|dƒ| ¡}|tjkrX||t|dƒt }| |¡td|dƒD]8}||||d|9}t|ƒ| ¡}| |¡q||t t|ƒ}|t|dƒtj}| |¡td| ddƒD]V}|||||9}t|ƒ| ¡}|t||dƒt|dƒ}| |¡qž|t| Žt|ŽStt|ƒ ||||¡Sr‰)r‹rŠr*rŒrrŽrTrGrrIrrrrrrer rdrr’r‘r"r¢rr“rv)r,rQrqr€rrŠr5r6r”rr•ZbnrRÚbÚcr–r—r˜r™r›Úprœr-r.rfsB $ zbessely._eval_nseries)Nr)r)r\r]r^r_rrdr<r:rar7ržrur¡rƒrˆrrr-r-rœr.rvs( rvc@sJeZdZdZejZejZedd„ƒZ dd„Z dd„Zdd „Zd d„Z dS) rJa Modified Bessel function of the first kind. Explanation =========== The Bessel $I$ function is a solution to the modified Bessel equation .. math :: z^2 \frac{\mathrm{d}^2 w}{\mathrm{d}z^2} + z \frac{\mathrm{d}w}{\mathrm{d}z} + (z^2 + \nu^2)^2 w = 0. It can be defined as .. math :: I_\nu(z) = i^{-\nu} J_\nu(iz), where $J_\nu(z)$ is the Bessel function of the first kind. Examples ======== >>> from sympy import besseli >>> from sympy.abc import z, n >>> besseli(n, z).diff(z) besseli(n - 1, z)/2 + besseli(n + 1, z)/2 See Also ======== besselj, bessely, besselk References ========== .. [1] http://functions.wolfram.com/Bessel-TypeFunctions/BesselI/ cCsb|jrX|jrtjS|jr"|jdus,t|ƒjr2tjSt|ƒjrL|jdurLtjS|j rXtj St|ƒtjtj fvrrtjS| ¡rš||||t||ƒS|jrØ| ¡r´t||ƒS| t¡}|rØt|t||ƒSddlm}|jr ||ƒ}||krBt||ƒSn8| ¡\}}|dkrBtd|t|tƒt||ƒS||ƒ}||kr^t||ƒSdSrb)rOrrdrGrrTrerVrfrgrhr rirjrkrJrmrrIrnrcrorrrpr-r-r.r7½s< " zbesseli.evalcKs4ddlm}ttt|dƒt||tƒ|ƒSrr)rnrsrrrrIrtr-r-r.ržâsz besseli._eval_rewrite_as_besseljcKs|j|jŽ}|r| t¡SdSr2©ržr*rŸrvr r-r-r.rxæsz besseli._eval_rewrite_as_besselycKs|j|jŽ t¡Sr2)ržr*rŸrMrwr-r-r.r{ëszbesseli._eval_rewrite_as_jncCs|j\}}|jr|jrdSdSr„r…r‡r-r-r.rˆîs zbesseli._eval_is_extended_realN)r\r]r^r_rrdr<r:rar7ržrxr{rˆr-r-r-r.rJ’s' $rJc@sReZdZdZejZejZedd„ƒZ dd„Z dd„Zdd „Zd d„Z dd „ZdS)Úbesselka Modified Bessel function of the second kind. Explanation =========== The Bessel $K$ function of order $\nu$ is defined as .. math :: K_\nu(z) = \lim_{\mu \to \nu} \frac{\pi}{2} \frac{I_{-\mu}(z) -I_\mu(z)}{\sin(\pi \mu)}, where $I_\mu(z)$ is the modified Bessel function of the first kind. It is a solution of the modified Bessel equation, and linearly independent from $Y_\nu$. Examples ======== >>> from sympy import besselk >>> from sympy.abc import z, n >>> besselk(n, z).diff(z) -besselk(n - 1, z)/2 - besselk(n + 1, z)/2 See Also ======== besselj, besseli, bessely References ========== .. [1] http://functions.wolfram.com/Bessel-TypeFunctions/BesselK/ cCsv|jr6|jrtjSt|ƒjdur&tjSt|ƒjr6tjS|tjttjttjfvrXtjS|j rr| ¡rrt||ƒSdSr@)rOrrirrfrhrrjrerGrkr©r3r-r-r.r7s zbesselk.evalcKs8|jdur4ttt|ƒt||ƒt||ƒdSdS)NFr8)rGrrrJrwr-r-r.ru-s z besselk._eval_rewrite_as_besselicKs|j|jŽ}|r| t¡SdSr2)rur*rŸrI)r,r5r6rZZair-r-r.rž1sz besselk._eval_rewrite_as_besseljcKs|j|jŽ}|r| t¡SdSr2r¨r r-r-r.rx6sz besselk._eval_rewrite_as_besselycKs|j|jŽ}|r| t¡SdSr2)rxr*rŸrN)r,r5r6rZZayr-r-r.r¡;szbesselk._eval_rewrite_as_yncCs|j\}}|jr|jrdSdSr„r¤r‡r-r-r.rˆ@s zbesselk._eval_is_extended_realN)r\r]r^r_rrdr<r:rar7ruržrxr¡rˆr-r-r-r.r©ôs% r©c@s$eZdZdZejZejZdd„ZdS)Úhankel1a£ Hankel function of the first kind. Explanation =========== This function is defined as .. math :: H_\nu^{(1)} = J_\nu(z) + iY_\nu(z), where $J_\nu(z)$ is the Bessel function of the first kind, and $Y_\nu(z)$ is the Bessel function of the second kind. It is a solution to Bessel's equation. Examples ======== >>> from sympy import hankel1 >>> from sympy.abc import z, n >>> hankel1(n, z).diff(z) hankel1(n - 1, z)/2 - hankel1(n + 1, z)/2 See Also ======== hankel2, besselj, bessely References ========== .. [1] http://functions.wolfram.com/Bessel-TypeFunctions/HankelH1/ cCs(|j}|jdur$t|j ¡| ¡ƒSdSr@)r1rAÚhankel2r/rBrCr-r-r.rDns zhankel1._eval_conjugateN© r\r]r^r_rrdr<r:rDr-r-r-r.rªFs$rªc@s$eZdZdZejZejZdd„ZdS)r«aÕ Hankel function of the second kind. Explanation =========== This function is defined as .. math :: H_\nu^{(2)} = J_\nu(z) - iY_\nu(z), where $J_\nu(z)$ is the Bessel function of the first kind, and $Y_\nu(z)$ is the Bessel function of the second kind. It is a solution to Bessel's equation, and linearly independent from $H_\nu^{(1)}$. Examples ======== >>> from sympy import hankel2 >>> from sympy.abc import z, n >>> hankel2(n, z).diff(z) hankel2(n - 1, z)/2 - hankel2(n + 1, z)/2 See Also ======== hankel1, besselj, bessely References ========== .. [1] http://functions.wolfram.com/Bessel-TypeFunctions/HankelH2/ cCs(|j}|jdur$t|j ¡| ¡ƒSdSr@)r1rArªr/rBrCr-r-r.rDs zhankel2._eval_conjugateNr¬r-r-r-r.r«ts%r«cstˆƒ‡fdd„ƒ}|S)Ncs|jrˆ|||ƒSdSr2)rGr‡©Úfnr-r.Úg¤szassume_integer_order..gr)r®r¯r-rr.Úassume_integer_order£sr°c@s*eZdZdZdd„Zdd„Zd dd„Zd S)ÚSphericalBesselBasea- Base class for spherical Bessel functions. These are thin wrappers around ordinary Bessel functions, since spherical Bessel functions differ from the ordinary ones just by a slight change in order. To use this class, define the ``_eval_evalf()`` and ``_expand()`` methods. cKstdƒ‚dS)z@ Expand self into a polynomial. Nu is guaranteed to be Integer. Z expansionN©rŽ©r,rWr-r-r.Ú_expand·szSphericalBesselBase._expandcKs|jjr|jfi|¤ŽS|Sr2)r/Ú is_Integerr´r³r-r-r.rU»sz%SphericalBesselBase._eval_expand_funcr8cCs:|dkrt||ƒ‚| |jd|j¡||jd|jSr9)r r;r/r1r=r-r-r.r?Às ÿzSphericalBesselBase.fdiffN)r8)r\r]r^r_r´rUr?r-r-r-r.r±«sr±cCs8t||ƒt|ƒtj|dt|d|ƒt|ƒS©Nr0)r%rrrlr©rqr6r-r-r.Ú_jnÇs$ÿr¸cCs8tj|dt|d|ƒt|ƒt||ƒt|ƒSr¶)rrlr%rrr·r-r-r.Ú_ynÌs$ÿr¹c@sDeZdZdZedd„ƒZdd„Zdd„Zdd „Zd d„Z dd „Z dS)rMaõ Spherical Bessel function of the first kind. Explanation =========== This function is a solution to the spherical Bessel equation .. math :: z^2 \frac{\mathrm{d}^2 w}{\mathrm{d}z^2} + 2z \frac{\mathrm{d}w}{\mathrm{d}z} + (z^2 - \nu(\nu + 1)) w = 0. It can be defined as .. math :: j_\nu(z) = \sqrt{\frac{\pi}{2z}} J_{\nu + \frac{1}{2}}(z), where $J_\nu(z)$ is the Bessel function of the first kind. The spherical Bessel functions of integral order are calculated using the formula: .. math:: j_n(z) = f_n(z) \sin{z} + (-1)^{n+1} f_{-n-1}(z) \cos{z}, where the coefficients $f_n(z)$ are available as :func:`sympy.polys.orthopolys.spherical_bessel_fn`. Examples ======== >>> from sympy import Symbol, jn, sin, cos, expand_func, besselj, bessely >>> z = Symbol("z") >>> nu = Symbol("nu", integer=True) >>> print(expand_func(jn(0, z))) sin(z)/z >>> expand_func(jn(1, z)) == sin(z)/z**2 - cos(z)/z True >>> expand_func(jn(3, z)) (-6/z**2 + 15/z**4)*sin(z) + (1/z - 15/z**3)*cos(z) >>> jn(nu, z).rewrite(besselj) sqrt(2)*sqrt(pi)*sqrt(1/z)*besselj(nu + 1/2, z)/2 >>> jn(nu, z).rewrite(bessely) (-1)**nu*sqrt(2)*sqrt(pi)*sqrt(1/z)*bessely(-nu - 1/2, z)/2 >>> jn(2, 5.2+0.3j).evalf(20) 0.099419756723640344491 - 0.054525080242173562897*I See Also ======== besselj, bessely, besselk, yn References ========== .. [1] http://dlmf.nist.gov/10.47 cCsD|jr*|jrtjS|jr*|jr$tjStjS|tjtjfvr@tjSdSr2) rOrrdrGrTrerfrjrir3r-r-r.r7szjn.evalcKs ttd|ƒt|tj|ƒSry)rrrIrrzrwr-r-r.ržszjn._eval_rewrite_as_besseljcKs,tj|ttd|ƒt|tj|ƒSry)rrlrrrvrzrwr-r-r.rxszjn._eval_rewrite_as_besselycKstj|t|d|ƒSr¶)rrlrNrwr-r-r.r¡szjn._eval_rewrite_as_yncKst|j|jƒSr2)r¸r/r1r³r-r-r.r´"sz jn._expandcCs|jjr| t¡ |¡SdSr2©r/rµrŸrIÚ_eval_evalf©r,Úprecr-r-r.r»%szjn._eval_evalfN)r\r]r^r_rar7ržrxr¡r´r»r-r-r-r.rMÒs9 rMc@s@eZdZdZedd„ƒZedd„ƒZdd„Zdd „Zd d„Z dS) rNa¤ Spherical Bessel function of the second kind. Explanation =========== This function is another solution to the spherical Bessel equation, and linearly independent from $j_n$. It can be defined as .. math :: y_\nu(z) = \sqrt{\frac{\pi}{2z}} Y_{\nu + \frac{1}{2}}(z), where $Y_\nu(z)$ is the Bessel function of the second kind. For integral orders $n$, $y_n$ is calculated using the formula: .. math:: y_n(z) = (-1)^{n+1} j_{-n-1}(z) Examples ======== >>> from sympy import Symbol, yn, sin, cos, expand_func, besselj, bessely >>> z = Symbol("z") >>> nu = Symbol("nu", integer=True) >>> print(expand_func(yn(0, z))) -cos(z)/z >>> expand_func(yn(1, z)) == -cos(z)/z**2-sin(z)/z True >>> yn(nu, z).rewrite(besselj) (-1)**(nu + 1)*sqrt(2)*sqrt(pi)*sqrt(1/z)*besselj(-nu - 1/2, z)/2 >>> yn(nu, z).rewrite(bessely) sqrt(2)*sqrt(pi)*sqrt(1/z)*bessely(nu + 1/2, z)/2 >>> yn(2, 5.2+0.3j).evalf(20) 0.18525034196069722536 + 0.014895573969924817587*I See Also ======== besselj, bessely, besselk, jn References ========== .. [1] http://dlmf.nist.gov/10.47 cKs0tj|dttd|ƒt|tj|ƒSrS)rrlrrrIrzrwr-r-r.ržYszyn._eval_rewrite_as_besseljcKs ttd|ƒt|tj|ƒSry)rrrvrrzrwr-r-r.rx]szyn._eval_rewrite_as_besselycKstj|dt|d|ƒSr¶)rrlrMrwr-r-r.r{aszyn._eval_rewrite_as_jncKst|j|jƒSr2)r¹r/r1r³r-r-r.r´dsz yn._expandcCs|jjr| t¡ |¡SdSr2)r/rµrŸrvr»r¼r-r-r.r»gszyn._eval_evalfN) r\r]r^r_r°ržrxr{r´r»r-r-r-r.rN*s. rNc@sLeZdZedd„ƒZedd„ƒZdd„Zdd„Zd d „Zdd„Z d d„Z dS)ÚSphericalHankelBasecKsN|j}ttd|ƒt|tj|ƒ|ttj|dt|tj|ƒSr9)Ú_hankel_kind_signrrrIrrzrrl©r,r5r6rZÚhksr-r-r.ržns&ÿz,SphericalHankelBase._eval_rewrite_as_besseljcKsJ|j}ttd|ƒtj|t|tj|ƒ|tt|tj|ƒSry)r¿rrrrlrvrzrrÀr-r-r.rxws(ÿz,SphericalHankelBase._eval_rewrite_as_besselycKs(|j}t||ƒ t¡|tt||ƒSr2)r¿rMrŸrNrrÀr-r-r.r¡€sz'SphericalHankelBase._eval_rewrite_as_yncKs(|j}t||ƒ|tt||ƒ t¡Sr2)r¿rMrrNrŸrÀr-r-r.r{„sz'SphericalHankelBase._eval_rewrite_as_jncKsJ|jjr|jfi|¤ŽS|j}|j}|j}t||ƒ|tt||ƒSdSr2)r/rµr´r1r¿rMrrN)r,rWr5r6rÁr-r-r.rUˆsz%SphericalHankelBase._eval_expand_funccKs2|j}|j}|j}t||ƒ|tt||ƒ ¡Sr2)r/r1r¿r¸rr¹Úexpand)r,rWrqr6rÁr-r-r.r´‘s zSphericalHankelBase._expandcCs|jjr| t¡ |¡SdSr2rºr¼r-r-r.r» szSphericalHankelBase._eval_evalfN)r\r]r^r°ržrxr¡r{rUr´r»r-r-r-r.r¾ls r¾c@s"eZdZdZejZedd„ƒZdS)rKa“ Spherical Hankel function of the first kind. Explanation =========== This function is defined as .. math:: h_\nu^(1)(z) = j_\nu(z) + i y_\nu(z), where $j_\nu(z)$ and $y_\nu(z)$ are the spherical Bessel function of the first and second kinds. For integral orders $n$, $h_n^(1)$ is calculated using the formula: .. math:: h_n^(1)(z) = j_{n}(z) + i (-1)^{n+1} j_{-n-1}(z) Examples ======== >>> from sympy import Symbol, hn1, hankel1, expand_func, yn, jn >>> z = Symbol("z") >>> nu = Symbol("nu", integer=True) >>> print(expand_func(hn1(nu, z))) jn(nu, z) + I*yn(nu, z) >>> print(expand_func(hn1(0, z))) sin(z)/z - I*cos(z)/z >>> print(expand_func(hn1(1, z))) -I*sin(z)/z - cos(z)/z + sin(z)/z**2 - I*cos(z)/z**2 >>> hn1(nu, z).rewrite(jn) (-1)**(nu + 1)*I*jn(-nu - 1, z) + jn(nu, z) >>> hn1(nu, z).rewrite(yn) (-1)**nu*yn(-nu - 1, z) + I*yn(nu, z) >>> hn1(nu, z).rewrite(hankel1) sqrt(2)*sqrt(pi)*sqrt(1/z)*hankel1(nu, z)/2 See Also ======== hn2, jn, yn, hankel1, hankel2 References ========== .. [1] http://dlmf.nist.gov/10.47 cKsttd|ƒt||ƒSry)rrrªrwr-r-r.Ú_eval_rewrite_as_hankel1Øszhn1._eval_rewrite_as_hankel1N) r\r]r^r_rrdr¿r°rÃr-r-r-r.rK¥s0rKc@s$eZdZdZejZedd„ƒZdS)rLa‘ Spherical Hankel function of the second kind. Explanation =========== This function is defined as .. math:: h_\nu^(2)(z) = j_\nu(z) - i y_\nu(z), where $j_\nu(z)$ and $y_\nu(z)$ are the spherical Bessel function of the first and second kinds. For integral orders $n$, $h_n^(2)$ is calculated using the formula: .. math:: h_n^(2)(z) = j_{n} - i (-1)^{n+1} j_{-n-1}(z) Examples ======== >>> from sympy import Symbol, hn2, hankel2, expand_func, jn, yn >>> z = Symbol("z") >>> nu = Symbol("nu", integer=True) >>> print(expand_func(hn2(nu, z))) jn(nu, z) - I*yn(nu, z) >>> print(expand_func(hn2(0, z))) sin(z)/z + I*cos(z)/z >>> print(expand_func(hn2(1, z))) I*sin(z)/z - cos(z)/z + sin(z)/z**2 + I*cos(z)/z**2 >>> hn2(nu, z).rewrite(hankel2) sqrt(2)*sqrt(pi)*sqrt(1/z)*hankel2(nu, z)/2 >>> hn2(nu, z).rewrite(jn) -(-1)**(nu + 1)*I*jn(-nu - 1, z) + jn(nu, z) >>> hn2(nu, z).rewrite(yn) (-1)**nu*yn(-nu - 1, z) - I*yn(nu, z) See Also ======== hn1, jn, yn, hankel1, hankel2 References ========== .. [1] http://dlmf.nist.gov/10.47 cKsttd|ƒt||ƒSry)rrr«rwr-r-r.Ú_eval_rewrite_as_hankel2szhn2._eval_rewrite_as_hankel2N) r\r]r^r_rrdr¿r°rÄr-r-r-r.rLÝs0rLÚsympyécsddlm}ˆdkrTddlm‰ddlm}||ƒ‰‡‡‡fdd„td|dƒDƒSˆd kr¶dd lm‰zddl m ‰‡‡fdd „}Wq¾ty²ddl m‰‡‡fdd „}Yq¾0nt dƒ‚‡‡fdd„}ˆ|}|||ƒ}|g} t|dƒD]} ||||ƒ}| |¡qð| S)aª 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)rrÅ)Úbesseljzero)Údps_to_preccs0g|](}t ˆtˆdƒ ˆ¡t|ƒƒˆ¡‘qS)gà?)rÚ_from_mpmathrÚ _to_mpmathÚint)Ú.0Úl)rÇrqr½r-r.Ú Hsþÿÿzjn_zeros..r0Úscipy)Únewton)Úspherical_jncs ˆˆ|ƒSr2r-©rQ)rqrÑr-r.ÚOózjn_zeros..)Úsph_jncsˆˆ|ƒddS)Nréÿÿÿÿr-rÒ)rqrÕr-r.rÓRrÔúUnknown method.cs ˆdkrˆ||ƒ}ntdƒ‚|S)NrÏr×r²)rXrQr)ÚmethodrÐr-r.ÚsolverVszjn_zeros..solver)ÚmathrÚmpmathrÇZmpmath.libmp.libmpfrÈr‘Zscipy.optimizerÐZ scipy.specialrÑÚImportErrorrÕrŽr’)rqr›rØZdpsZmath_pirÈrXrÙrÚrootsÚir-)rÇrØrqrÐr½rÕrÑr.Újn_zeross2-þ rßc@s4eZdZdZdd„Zdd„Zddd„Zd d d „ZdS)ÚAiryBasezg Abstract base class for Airy functions. This class is meant to reduce code duplication. cCs| |jd ¡¡S©Nr)rr*rBr+r-r-r.rDqszAiryBase._eval_conjugatecCs|jdjSrá)r*r†r+r-r-r.rˆtszAiryBase._eval_is_extended_realTcKsL|jd}| ¡}|j}||ƒ||ƒd}t||ƒ||ƒd}||fS)Nrr8)r*rBrr)r,ÚdeeprWr6ZzcrXÚuÚvr-r-r.Úas_real_imagws zAiryBase.as_real_imagcKs&|jfd|i|¤Ž\}}||tjS)Nrâ)rårZ ImaginaryUnit)r,rârWZre_partZim_partr-r-r.Ú_eval_expand_complexszAiryBase._eval_expand_complexN)T)T)r\r]r^r_rDrˆrårær-r-r-r.ràis ràc@s^eZdZdZdZdZedd„ƒZddd„Ze e dd „ƒƒZd d„Zdd „Z dd„Zdd„ZdS)Úairyaia” The Airy function $\operatorname{Ai}$ of the first kind. Explanation =========== The Airy function $\operatorname{Ai}(z)$ is defined to be the function satisfying Airy's differential equation .. math:: \frac{\mathrm{d}^2 w(z)}{\mathrm{d}z^2} - z w(z) = 0. Equivalently, for real $z$ .. math:: \operatorname{Ai}(z) := \frac{1}{\pi} \int_0^\infty \cos\left(\frac{t^3}{3} + z t\right) \mathrm{d}t. Examples ======== Create an Airy function object: >>> from sympy import airyai >>> from sympy.abc import z >>> airyai(z) airyai(z) Several special values are known: >>> airyai(0) 3**(1/3)/(3*gamma(2/3)) >>> from sympy import oo >>> airyai(oo) 0 >>> airyai(-oo) 0 The Airy function obeys the mirror symmetry: >>> from sympy import conjugate >>> conjugate(airyai(z)) airyai(conjugate(z)) Differentiation with respect to $z$ is supported: >>> from sympy import diff >>> diff(airyai(z), z) airyaiprime(z) >>> diff(airyai(z), z, 2) z*airyai(z) Series expansion is also supported: >>> from sympy import series >>> series(airyai(z), z, 0, 3) 3**(5/6)*gamma(1/3)/(6*pi) - 3**(1/6)*z*gamma(2/3)/(2*pi) + O(z**3) We can numerically evaluate the Airy function to arbitrary precision on the whole complex plane: >>> airyai(-2).evalf(50) 0.22740742820168557599192443603787379946077222541710 Rewrite $\operatorname{Ai}(z)$ in terms of hypergeometric functions: >>> from sympy import hyper >>> airyai(z).rewrite(hyper) -3**(2/3)*z*hyper((), (4/3,), z**3/9)/(3*gamma(1/3)) + 3**(1/3)*hyper((), (2/3,), z**3/9)/(3*gamma(2/3)) See Also ======== airybi: Airy function of the second kind. airyaiprime: Derivative of the Airy function of the first kind. airybiprime: Derivative of the Airy function of the second kind. References ========== .. [1] https://en.wikipedia.org/wiki/Airy_function .. [2] http://dlmf.nist.gov/9 .. [3] http://www.encyclopediaofmath.org/index.php/Airy_functions .. [4] http://mathworld.wolfram.com/AiryFunctions.html r0TcCsŠ|jr^|tjurtjS|tjur&tjS|tjur6tjS|jr^tjdtddƒt tddƒƒS|jr†tjdtddƒt tddƒƒSdS)Nér8) Ú is_NumberrrhrirerjrOrdr r!©r4r‚r-r-r.r7às "zairyai.evalcCs$|dkrt|jdƒSt||ƒ‚dS©Nr0r)Úairyaiprimer*r r=r-r-r.r?îszairyai.fdiffcGsJ|dkrtjSt|ƒ}t|ƒdkrà|d}dtddƒ||dtddƒ||dtt|tddƒtddƒƒt|ƒt|dtddƒƒtt|tddƒtddƒƒt|dƒt|dtddƒƒ|Stj dtddƒtt|tj tdƒƒtdt|tj tdƒƒt|ƒt ddƒ||SdS)Nrr0rÖrèr8é)rrerÚlenr rrrr!rdr©rqrQZprevious_termsr§r-r-r.Útaylor_termôs"Xÿ@ÿÿJÿÿzairyai.taylor_termcKs`tddƒ}tddƒ}t|tddƒƒ}t|ƒjr\|t|ƒt|||ƒt|||ƒSdS©Nr0rèr8©r rrrVrrI©r,r6rZÚotÚttrRr-r-r.ržs zairyai._eval_rewrite_as_besseljcKs˜tddƒ}tddƒ}t|tddƒƒ}t|ƒjrX|t|ƒt|||ƒt|||ƒS|t||ƒt|||ƒ|t||ƒt|||ƒSdSrñ©r rrrTrrJrór-r-r.ru s *zairyai._eval_rewrite_as_besselicKs~tjdtddƒttddƒƒ}|tddƒttddƒƒ}|tgtddƒg|ddƒ|tgtddƒg|ddƒS)Nrèr8r0é rí)rrdr r!rr$©r,r6rZZpf1Zpf2r-r-r.Ú_eval_rewrite_as_hypers"zairyai._eval_rewrite_as_hypercKs |jd}|j}t|ƒdkr| ¡}td|gd}td|gd}td|gd}td|gd}| |||||¡} | dur| |}d|jr| |}| |}| |}|||||||||} ||||||}tj| tj t |ƒ| tj tdƒt|ƒSdS© Nrr0r¦)ÚexcludeÚdÚmrqrè) r*r~rîÚpoprÚmatchrGrrzrdrçrÚairybi©r,rWr‚Zsymbsr6r¦rürýrqÚMÚpfZnewargr-r-r.rUs$ $zairyai._eval_expand_funcN)r0©r\r]r^r_ÚnargsÚ unbranchedrar7r?ÚstaticmethodrrðržrurùrUr-r-r-r.rç„sX rçc@s^eZdZdZdZdZedd„ƒZddd„Ze e dd „ƒƒZd d„Zdd „Z dd„Zdd„ZdS)raè The Airy function $\operatorname{Bi}$ of the second kind. Explanation =========== The Airy function $\operatorname{Bi}(z)$ is defined to be the function satisfying Airy's differential equation .. math:: \frac{\mathrm{d}^2 w(z)}{\mathrm{d}z^2} - z w(z) = 0. Equivalently, for real $z$ .. math:: \operatorname{Bi}(z) := \frac{1}{\pi} \int_0^\infty \exp\left(-\frac{t^3}{3} + z t\right) + \sin\left(\frac{t^3}{3} + z t\right) \mathrm{d}t. Examples ======== Create an Airy function object: >>> from sympy import airybi >>> from sympy.abc import z >>> airybi(z) airybi(z) Several special values are known: >>> airybi(0) 3**(5/6)/(3*gamma(2/3)) >>> from sympy import oo >>> airybi(oo) oo >>> airybi(-oo) 0 The Airy function obeys the mirror symmetry: >>> from sympy import conjugate >>> conjugate(airybi(z)) airybi(conjugate(z)) Differentiation with respect to $z$ is supported: >>> from sympy import diff >>> diff(airybi(z), z) airybiprime(z) >>> diff(airybi(z), z, 2) z*airybi(z) Series expansion is also supported: >>> from sympy import series >>> series(airybi(z), z, 0, 3) 3**(1/3)*gamma(1/3)/(2*pi) + 3**(2/3)*z*gamma(2/3)/(2*pi) + O(z**3) We can numerically evaluate the Airy function to arbitrary precision on the whole complex plane: >>> airybi(-2).evalf(50) -0.41230258795639848808323405461146104203453483447240 Rewrite $\operatorname{Bi}(z)$ in terms of hypergeometric functions: >>> from sympy import hyper >>> airybi(z).rewrite(hyper) 3**(1/6)*z*hyper((), (4/3,), z**3/9)/gamma(1/3) + 3**(5/6)*hyper((), (2/3,), z**3/9)/(3*gamma(2/3)) See Also ======== airyai: Airy function of the first kind. airyaiprime: Derivative of the Airy function of the first kind. airybiprime: Derivative of the Airy function of the second kind. References ========== .. [1] https://en.wikipedia.org/wiki/Airy_function .. [2] http://dlmf.nist.gov/9 .. [3] http://www.encyclopediaofmath.org/index.php/Airy_functions .. [4] http://mathworld.wolfram.com/AiryFunctions.html r0TcCsŠ|jr^|tjurtjS|tjur&tjS|tjur6tjS|jr^tjdtddƒt tddƒƒS|jr†tjdtddƒt tddƒƒSdS)Nrèr0ér8) rérrhrirjrerOrdr r!rêr-r-r.r7Žs "zairybi.evalcCs$|dkrt|jdƒSt||ƒ‚dSrë)Úairybiprimer*r r=r-r-r.r?szairybi.fdiffcGs"|dkrtjSt|ƒ}t|ƒdkr¸|d}dtddƒ|ttdt|tjtdƒƒƒt |tjtdƒƒ|tjtt dt|tjtdƒƒƒt |dtdƒƒ|Stjtddƒtt |tjtdƒƒttdt|tjtdƒƒƒt |ƒtddƒ||SdS)Nrr0rÖrèr8r)rrerrîr rrrrdrrrzrr!rïr-r-r.rð£sH>ÿÿJÿÿzairybi.taylor_termcKs`tddƒ}tddƒ}t|tddƒƒ}t|ƒjr\t|dƒt|||ƒt|||ƒSdSrñròrór-r-r.rž²s zairybi._eval_rewrite_as_besseljcKs¨tddƒ}tddƒ}t|tddƒƒ}t|ƒjr\t|ƒtdƒt|||ƒt|||ƒSt||ƒ}t||ƒ}t|ƒ|t|||ƒ||t|||ƒSdSrñrö©r,r6rZrôrõrRr¥r¦r-r-r.ru¹s . zairybi._eval_rewrite_as_besselicKsztjtddƒttddƒƒ}|tddƒttddƒƒ}|tgtddƒg|ddƒ|tgtddƒg|ddƒS)Nrèrr8r0r÷rí)rrdrr!r r$rør-r-r.rùÄszairybi._eval_rewrite_as_hypercKs |jd}|j}t|ƒdkr| ¡}td|gd}td|gd}td|gd}td|gd}| |||||¡} | dur| |}d|jr| |}| |}| |}|||||||||} ||||||}tjt dƒtj | t|ƒtj | t|ƒSdSrú) r*r~rîrþrrÿrGrrzrrdrçrrr-r-r.rUÉs$ $zairybi._eval_expand_funcN)r0rr-r-r-r.r0sZ rc@sVeZdZdZdZdZedd„ƒZddd„Zdd „Z d d„Z dd „Zdd„Zdd„Z dS)rìa+ The derivative $\operatorname{Ai}^\prime$ of the Airy function of the first kind. Explanation =========== The Airy function $\operatorname{Ai}^\prime(z)$ is defined to be the function .. math:: \operatorname{Ai}^\prime(z) := \frac{\mathrm{d} \operatorname{Ai}(z)}{\mathrm{d} z}. Examples ======== Create an Airy function object: >>> from sympy import airyaiprime >>> from sympy.abc import z >>> airyaiprime(z) airyaiprime(z) Several special values are known: >>> airyaiprime(0) -3**(2/3)/(3*gamma(1/3)) >>> from sympy import oo >>> airyaiprime(oo) 0 The Airy function obeys the mirror symmetry: >>> from sympy import conjugate >>> conjugate(airyaiprime(z)) airyaiprime(conjugate(z)) Differentiation with respect to $z$ is supported: >>> from sympy import diff >>> diff(airyaiprime(z), z) z*airyai(z) >>> diff(airyaiprime(z), z, 2) z*airyaiprime(z) + airyai(z) Series expansion is also supported: >>> from sympy import series >>> series(airyaiprime(z), z, 0, 3) -3**(2/3)/(3*gamma(1/3)) + 3**(1/3)*z**2/(6*gamma(2/3)) + O(z**3) We can numerically evaluate the Airy function to arbitrary precision on the whole complex plane: >>> airyaiprime(-2).evalf(50) 0.61825902074169104140626429133247528291577794512415 Rewrite $\operatorname{Ai}^\prime(z)$ in terms of hypergeometric functions: >>> from sympy import hyper >>> airyaiprime(z).rewrite(hyper) 3**(1/3)*z**2*hyper((), (5/3,), z**3/9)/(6*gamma(2/3)) - 3**(2/3)*hyper((), (1/3,), z**3/9)/(3*gamma(1/3)) See Also ======== airyai: Airy function of the first kind. airybi: Airy function of the second kind. airybiprime: Derivative of the Airy function of the second kind. References ========== .. [1] https://en.wikipedia.org/wiki/Airy_function .. [2] http://dlmf.nist.gov/9 .. [3] http://www.encyclopediaofmath.org/index.php/Airy_functions .. [4] http://mathworld.wolfram.com/AiryFunctions.html r0TcCsR|jr&|tjurtjS|tjur&tjS|jrNtjdtddƒttddƒƒSdS)Nrèr0) rérrhrirerOrlr r!rêr-r-r.r76s zairyaiprime.evalcCs.|dkr |jdt|jdƒSt||ƒ‚dSrë)r*rçr r=r-r-r.r?Aszairyaiprime.fdiffcCsR|jd |¡}t|ƒtj|dd}Wdƒn1s<0Yt ||¡S©Nrr0)Z derivative)r*rÊr'r&rçrrÉ©r,r½r6Úresr-r-r.r»Gs ,zairyaiprime._eval_evalfcKsPtddƒ}t|tddƒƒ}t|ƒjrL|dt|||ƒt|||ƒSdS©Nr8rè)r rrrVrI©r,r6rZrõrRr-r-r.ržMs z$airyaiprime._eval_rewrite_as_besseljcKs¬tddƒ}tddƒ}|t|tddƒƒ}t|ƒjrP|dt||ƒt||ƒSt|tddƒƒ}t||ƒ}t||ƒ}||d|t|||ƒ|t|||ƒSdSrñ)r rrrTrJr r-r-r.ruSs z$airyaiprime._eval_rewrite_as_besselicKs„|dddtddƒttddƒƒ}dtddƒttddƒƒ}|tgtddƒg|ddƒ|tgtddƒg|ddƒS)Nr8rèr0ér÷)r r!rr$rør-r-r.rù_s(z"airyaiprime._eval_rewrite_as_hypercKs |jd}|j}t|ƒdkr| ¡}td|gd}td|gd}td|gd}td|gd}| |||||¡} | dur| |}d|jr| |}| |}| |}|||||||||} ||||||}tj| tj t |ƒ| tj tdƒt|ƒSdSrú) r*r~rîrþrrÿrGrrzrdrìrr rr-r-r.rUds$ $zairyaiprime._eval_expand_funcN)r0©r\r]r^r_rrrar7r?r»ržrurùrUr-r-r-r.rìásQ rìc@sVeZdZdZdZdZedd„ƒZddd„Zdd „Z d d„Z dd „Zdd„Zdd„Z dS)r a< 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] http://dlmf.nist.gov/9 .. [3] http://www.encyclopediaofmath.org/index.php/Airy_functions .. [4] http://mathworld.wolfram.com/AiryFunctions.html r0TcCs~|jrX|tjurtjS|tjur&tjS|tjur6tjS|jrXdtddƒttddƒƒS|jrzdtddƒttddƒƒSdS)Nrèr0r) rérrhrirjrerOr r!rêr-r-r.r7Õs zairybiprime.evalcCs.|dkr |jdt|jdƒSt||ƒ‚dSrë)r*rr r=r-r-r.r?åszairybiprime.fdiffcCsR|jd |¡}t|ƒtj|dd}Wdƒn1s<0Yt ||¡Sr)r*rÊr'r&rrrÉrr-r-r.r»ës ,zairybiprime._eval_evalfcKsRtddƒ}|t|tddƒƒ}t|ƒjrN|tdƒt||ƒt||ƒSdSrròrr-r-r.ržñs z$airybiprime._eval_rewrite_as_besseljcKs´tddƒ}tddƒ}|t|tddƒƒ}t|ƒjrT|tdƒt||ƒt||ƒSt|tddƒƒ}t||ƒ}t||ƒ}t|ƒ|t|||ƒ|d|t|||ƒSdSrñrör r-r-r.ru÷s " z$airybiprime._eval_rewrite_as_besselicKs||ddtddƒttddƒƒ}tddƒttddƒƒ}|tgtddƒg|ddƒ|tgtddƒg|ddƒS)Nr8rèrr0rr÷)rr!r r$rør-r-r.rùs$z"airybiprime._eval_rewrite_as_hypercKs |jd}|j}t|ƒdkr| ¡}td|gd}td|gd}td|gd}td|gd}| |||||¡} | dur| |}d|jr| |}| |}| |}|||||||||} ||||||}tjt dƒ| tj t|ƒ| tj t|ƒSdSrú) r*r~rîrþrrÿrGrrzrrdrìr rr-r-r.rUs$ $zairybiprime._eval_expand_funcN)r0rr-r-r-r.r ~sS r c@sFeZdZdZedd„ƒZddd„Zdd„Zd d „Zdd„Z d d„Z dS)Úmarcumqa– 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] http://mathworld.wolfram.com/MarcumQ-Function.html cCsZ|tjur@|tjur$|tjur$tjSt||dtjƒt|ƒS|tjurn|tjurnddt|dtjƒS||krò|tjur¦dt|dƒtd|dƒtjS|dkròtjtjt|dƒtd|dƒt|dƒtd|dƒS|jr,|jr|jrtjSt||dtjƒt|ƒS|jrV|jrVddt|dtjƒSdSr|) rrer#rzr!rrdrJrO)r4rýrRr¥r-r-r.r7Ss" &Dzmarcumq.evalr8cCsŽ|j\}}}|dkr6|t|||ƒtd|||ƒS|dkr€||||dt|d|ddƒt|d||ƒSt||ƒ‚dS)Nr8r0rè)r*rrrJr )r,r>rýrRr¥r-r-r.r?ks"Bz marcumq.fdiffcKsjddlm}| dtdƒ¡}|d||||t|d|ddƒt|d||ƒ||tjgƒS)Nr)ÚIntegralrQr0r8)Zsympy.integrals.integralsrÚgetrrrJrri)r,rýrRr¥rZrrQr-r-r.Ú_eval_rewrite_as_Integralts @ÿz!marcumq._eval_rewrite_as_IntegralcKsbddlm}| dtdƒ¡}t|d|ddƒ||||t|||ƒ|d|tjgƒS)Nr)ÚSumr›r8r0)Zsympy.concrete.summationsrrrrrJrri)r,rýrRr¥rZrr›r-r-r.Ú_eval_rewrite_as_Sumzszmarcumq._eval_rewrite_as_Sumcsšˆ|kr–|dkr4dtˆdƒtdˆdƒdS|jr–|dkr–t‡fdd„td|ƒDƒƒ}tjtˆdƒtdˆdƒdtˆdƒ|SdS)Nr0r8rcsg|]}t|ˆdƒ‘qS)r8)rJ)rÌrÞ©rRr-r.rÎ„rÔz4marcumq._eval_rewrite_as_besseli..)rrJrµÚsumr‘rrz)r,rýrRr¥rZršr-rr.rus$z marcumq._eval_rewrite_as_besselicCstdd„|jDƒƒrdSdS)Ncss|]}|jVqdSr2)rO)rÌr‚r-r-r.Ú ˆrÔz(marcumq._eval_is_zero..T)Úallr*r+r-r-r.Ú _eval_is_zero‡szmarcumq._eval_is_zeroN)r8)r\r]r^r_rar7r?rrrurr-r-r-r.r"s0 rN)rÅrÆ)QÚ functoolsrZ sympy.corerZsympy.core.addrZsympy.core.cacherZsympy.core.exprrZsympy.core.functionrr r Zsympy.core.logicrrZsympy.core.numbersr rrZsympy.core.powerrZsympy.core.symbolrrZsympy.core.sympifyrZ(sympy.functions.combinatorial.factorialsrZ(sympy.functions.elementary.trigonometricrrrrZ#sympy.functions.elementary.integersrrnrZ&sympy.functions.elementary.exponentialrrZ(sympy.functions.elementary.miscellaneousrrrr Z'sympy.functions.special.gamma_functionsr!r"r#Zsympy.functions.special.hyperr$Zsympy.polys.orthopolysr%rÛr&r'r(rIrvrJr©rªr«r°r±r¸r¹rMrNr¾rKrLrßràrçrrìr rr-r-r-r.Úsd F$bR./XB988 T-2%