o Rh0 @sddlmZddlmZddlmZddlmZddlm 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>Gddde Z?Gddde?Z@Gddde?ZAGddde?ZBGddde?ZCGd d!d!e?ZDGd"d#d#e?ZEd$d%ZFGd&d'd'e?ZGd(d)ZHd*d+ZIGd,d-d-eGZJGd.d/d/eGZKGd0d1d1eGZLGd2d3d3eLZMGd4d5d5eLZNdKd8d9ZOGd:d;d;e ZPGdd?d?ePZRGd@dAdAePZSGdBdCdCePZTGdDdEdEe ZUGdFdGdGe ZVGdHdIdIe ZWdJS)Lwraps)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)mpworkprecc@s^eZdZdZeddZeddZeddZdd d Z d d Z d dZ ddZ ddZ dS) 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}``. cC |jdS)z( The order of the Bessel-type function. rargsselfr3`/home/air/sanwanet/backup_V2/venv/lib/python3.10/site-packages/sympy/functions/special/bessel.pyorder4 zBesselBase.ordercCr.)z+ The argument of the Bessel-type function. r/r1r3r3r4argument9r6zBesselBase.argumentcCsdSNr3clsnuzr3r3r4eval>szBesselBase.evalcCsN|dkr t|||jd||jd|j|jd||jd|jSNr?r7)r _b __class__r5r8_ar2argindexr3r3r4fdiffBs  zBesselBase.fdiffcCs*|j}|jdur||j|SdSNF)r8is_extended_negativerBr5 conjugater2r=r3r3r4_eval_conjugateHs zBesselBase._eval_conjugatecCsx|j|j}}||rdS|||sdS|||}|jr2t|ttt t t t fs-|j s2t|jStt|j |jgSrG)r5r8has_eval_is_meromorphicsubs is_integer isinstancebesseljbesselihn1hn2jnynis_zeror is_infiniter )r2xar<r=z0r3r3r4rMMs    zBesselBase._eval_is_meromorphiccKs|j|j|j}}}|jr_|djr7|j |j||d|d|j|d||d||S|djr_d|j|d||d|||j|j||d|S|SNr7r?) r5r8rBis_real is_positiverCrA_eval_expand_func is_negative)r2hintsr<r=fr3r3r4r_Zs & &zBesselBase._eval_expand_funccKsddlm}||S)Nr) besselsimp)sympy.simplify.simplifyrc)r2kwargsrcr3r3r4_eval_simplifyes zBesselBase._eval_simplifyNr?)__name__ __module__ __qualname____doc__propertyr5r8 classmethodr>rFrKrMr_rfr3r3r3r4r-$s     r-cfeZdZdZejZejZeddZ ddZ ddZ dd Z fd d Z d d Zdfdd ZZS)rQa4 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/ cCsR|jr,|jr tjS|jr|jdust|jrtjSt|jr&|jdur&tjS|j r,tj S|tj tj fvr7tjS| rK||| | t|| S|jrn| r^tj| t| |S|t}|rnt|t||S|jrt|}||kr~t||Sn|\}}|dkrtd|t|tt||St|}||krt||SdS)NFTrr?)rWrOnerOr"r^Zeror`ComplexInfinity is_imaginaryNaNInfinityNegativeInfinitycould_extract_minus_signrQ NegativeOneextract_multiplicativelyrrRr%extract_branch_factorrrr;r<r=newznnnur3r3r4r>s>    " z besselj.evalcKs(ttt|dt|tt |SNr?)rrrrRr$r2r<r=rer3r3r4_eval_rewrite_as_besseli(z besselj._eval_rewrite_as_besselicKs<|jdurtt|t| |tt|t||SdSrG)rOrrbesselyrrr3r3r4_eval_rewrite_as_bessely .z besselj._eval_rewrite_as_besselycK"td|tt|tj|jSr~)rrrUrHalfr8rr3r3r4_eval_rewrite_as_jn"zbesselj._eval_rewrite_as_jnc s|j\}}z||}Wn ty|YSw||\}}|jr0||d|t|dS|jr^|dkr9dn|}|||} | js\tdt|t d|ddtt |S|St t |j |||dS)Nr?r7rlogxcdir) r0as_leading_termNotImplementedErroras_coeff_exponentr^r&r`rrrsuperrQ_eval_as_leading_term r2rYrrr<r=argcesignrBr3r4rs   0zbesselj._eval_as_leading_termcC"|j\}}|jr |jrdSdSdSNTr0rOis_extended_realr2r<r=r3r3r4_eval_is_extended_real  zbesselj._eval_is_extended_realrc s"ddlm}|j\}}z ||\}} Wn ttfy!|YSw| jrt|| } ||||} |d|||| } | t j urE| St | d|  } | |t |d}|g}td| ddD]}|| |||9}t ||  }||qet|| Stt|||||SNrOrderr?r7)sympy.series.orderrr0leadterm ValueErrorrr^r _eval_nseriesremoveOrrpr r&rangeappendrrrQr2rYr|rrrr<r=_rnewnorttermskrr3r4rs,      zbesselj._eval_nseriesr)rhrirjrkrrorCrArmr>rrrrrr __classcell__r3r3rr4rQjsD # rQcrn)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/ cCs|jr|jr tjSt|jdurtjSt|jrtjS|tjtjfvr&tjS|ttjkrFs$  z bessely.evalcKs<|jdurtt|tt|t||t| |SdSrG)rOrrrrQrr3r3r4_eval_rewrite_as_besseljZrz bessely._eval_rewrite_as_besseljcK|j|j}|r |tSdSr9)rr0rewriterRr2r<r=reajr3r3r4r^  z bessely._eval_rewrite_as_besselicKrr~)rrrVrrr8rr3r3r4_eval_rewrite_as_yncrzbessely._eval_rewrite_as_ync sz|j\}}z||}Wn ty|YSw||\}}|jrkdtt|dt||} |jrD|d|  t|dtnt j } |d| tt|t |dt j } t | | | gj||d}|S|jr|dkrtdn|}|||} | jstdtt|d|td dtt|d|tdd|td|ttS|Stt|j|||dS) Nr?r7rrrr)r0rrrr^rrrQrrrpr' EulerGammarr`rrrrrr) r2rYrrr<r=rrrterm_oneterm_two term_threerrr3r4rfs(  ,, bzbessely._eval_as_leading_termcCrrr0rOr^rr3r3r4rrzbessely._eval_is_extended_realrc s.ddlm}|j\}}z ||\}} Wn ttfy!|YSw| jr |jr t|| } t ||} dt t |d|  ||||} gg} }||||}|d |||| }|tjurf|St|d| }|tjkr|| t|dt }| |td|D]'}|||}|tjkr|||9}n|||9}t|| }| |q||t t|}|t|dtj}||td| ddD]*}|| |||9}t|| }|t||dt|d}||q| t| t|Stt| ||||Sr)rrr0rrrr^rOrrQrrrrrrpr rrrr'rrrr)r2rYr|rrrr<r=rrrbnrZbrrrrrrdenomprr3r4rsJ    $           zbessely._eval_nseriesr)rhrirjrkrrorCrArmr>rrrrrrrr3r3rr4rs(  rcs~eZdZdZej ZejZeddZ dddZ ddZ d d Z d d Z d dZfddZdfdd ZfddZZS)rRa 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/ cCsv|jr,|jr tjS|jr|jdust|jrtjSt|jr&|jdur&tjS|j r,tj St |tj tj fvr9tjS|tj urAtj S|tj urMd|tj S|ra||| | t|| S|jr|rnt| |S|t}|rt| t|| S|jrt|}||krt||Sn|\}}|dkrtd|t|tt||St|}||krt||SdS)NFTrr?)rWrrorOr"r^rpr`rqrrrsr#rtrurvrRrxrrQr%ryrrrzr3r3r4r>sF       " z besseli.evalNcKs|jr t|t||SdSr9)rr_besselir2r<r=limitvarrer3r3r4_eval_rewrite_as_tractable sz"besseli._eval_rewrite_as_tractablecKs(tt t|dt|tt|Sr~)rrrrQr$rr3r3r4r rz besseli._eval_rewrite_as_besseljcKrr9rr0rrrr3r3r4rrz besseli._eval_rewrite_as_besselycKs|j|jtSr9)rr0rrUrr3r3r4rzbesseli._eval_rewrite_as_jncCrrrrr3r3r4rrzbesseli._eval_is_extended_realc s|j\}}z||}Wn ty|YSw||\}}|jr0||d|t|dS|jrR|dkr9dn|}|||} | jsPt|tdt |S|St t |j |||dS)Nr?r7rr) r0rrrr^r&r`rrrrrRrrrr3r4rs   zbesseli._eval_as_leading_termrc s ddlm}|j\}}z ||\}} Wn ttfy!|YSw| jrt|| } ||||} |d|||| } | t j urE| St | d|  } | |t |d}|g}td| ddD]}|| |||9}t ||  }||qet|| Stt|||||Sr)rrr0rrrr^rrrrrpr r&rrrrrRrrr3r4r2s,      zbesseli._eval_nseriescsddlmddlm}|d}|tjtjfvrI|j\fddt|D|dt d|dd|g}t t dt t |St||||S)Nrrrr7cbg|]-}tddd|tddd|d|td|ddt|qSr?r7r r.0rrr<r=r3r4 X .$z)besseli._eval_aseries..r?(sympy.functions.combinatorial.factorialsrrrrrtrur0rr rrrrr _eval_aseriesr2r|args0rYrrpointrrrr4rQs    zbesseli._eval_aseriesr9r)rhrirjrkrrorCrArmr>rrrrrrrrrr3r3rr4rRs'  ' rRcseZdZdZejZej ZeddZ ddZ ddZ dd Z d d Z d d ZdddZddZdfdd ZfddZZS)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/ cCsz|jr|jr tjSt|jdurtjSt|jrtjS|tjttjttjfvr,tjS|j r9| r;t | |SdSdSrG) rWrrtr"rqrsrrurprOrvrr:r3r3r4r>s  z besselk.evalcKs8|jdurttt|t| |t||dSdS)NFr?)rOrrrRrr3r3r4rs *z besselk._eval_rewrite_as_besselicKrr9)rr0rrQ)r2r<r=reair3r3r4rrz besselk._eval_rewrite_as_besseljcKrr9rrr3r3r4rrz besselk._eval_rewrite_as_besselycKrr9)rr0rrV)r2r<r=reayr3r3r4rrzbesselk._eval_rewrite_as_yncCrrrrr3r3r4rrzbesselk._eval_is_extended_realNcKs|jr t| t||SdSr9)rr_besselkrr3r3r4rsz"besselk._eval_rewrite_as_tractablec Cs|j\}}z||}Wn ty|YSw||\}}|jrV|jr2t| tjtd} n|j rGt t ||dt | d} ntd|d| j||dS|j rht tt| t d|S|||S)Nr?z"Cannot proceed without knowing if z is zero or not.r)r0rrrr^rWrrr is_nonzeror&r!r`rrrfunc) r2rYrrr<r=rrrrr3r3r4rs"  $ zbesselk._eval_as_leading_termrc s,ddlm}|j\}}z ||\}} Wn ttfy!|YSw| jr|d||||} | t j urC||| |||S||||} |j r t || } t ||} d|dt|d| ||||}gg}}t| d}|t j kr| | t|dd}||td|D]}|||||9}t|| }||q| |d|dt|}|t|dt j}||td| ddD])}|||||9}t|| }|t||dt|d}||q|t|t|| S|jrt ||| }t ||| }gg}}t|ddD]#}t|| d||dtd||t|}|t|q,t|ddD]$}t| | d||dt|d|t|}|t|qXt|t|| Stdtt|||||S)Nrrr?rr7z4besselk expansion is only implemented for real order)rrr0rrrr^rrrrprOrrRrr rrrr'rr is_nonintegerr&rrr)r2rYr|rrrr<r=rrrrrrrZrrrrrrnewn_anewn_brr3r4rs^     (          24zbesselk._eval_nseriescsddlmddlm}|d}|tjtjfvrJ|j\fddt|D|dt d|dd|g}t  t t dt |St||||S)Nrrrr7cbg|]-}tddd|tddd|d|td|ddt|qSr?r7rrrr3r4rrz)besselk._eval_aseries..r?rrrrr4rs    zbesselk._eval_aseriesr9r)rhrirjrkrrorCrArmr>rrrrrrrrrrr3r3rr4r_s%  Erc@$eZdZdZejZejZddZdS)hankel1a Hankel function of the first kind. Explanation =========== This function is defined as .. math :: H_\nu^{(1)} = J_\nu(z) + iY_\nu(z), where $J_\nu(z)$ is the Bessel function of the first kind, and $Y_\nu(z)$ is the Bessel function of the second kind. It is a solution to Bessel's equation. Examples ======== >>> from sympy import hankel1 >>> from sympy.abc import z, n >>> hankel1(n, z).diff(z) hankel1(n - 1, z)/2 - hankel1(n + 1, z)/2 See Also ======== hankel2, besselj, bessely References ========== .. [1] https://functions.wolfram.com/Bessel-TypeFunctions/HankelH1/ cC(|j}|jdurt|j|SdSrG)r8rHhankel2r5rIrJr3r3r4rKH zhankel1._eval_conjugateN rhrirjrkrrorCrArKr3r3r3r4r s $ rc@r)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/ cCrrG)r8rHrr5rIrJr3r3r4rKwrzhankel2._eval_conjugateNrr3r3r3r4rNs % rcstfdd}|S)Ncs|jr |||SdSr9)rOrfnr3r4g~s zassume_integer_order..gr)rrr3rr4assume_integer_order}src@s*eZdZdZddZddZd ddZd S) 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. cKstd)z@ Expand self into a polynomial. Nu is guaranteed to be Integer. expansionrr2rar3r3r4_expandszSphericalBesselBase._expandcKs|jjr |jdi|S|SNr3)r5 is_Integerrrr3r3r4r_sz%SphericalBesselBase._eval_expand_funcr?cCs:|dkr t||||jd|j||jd|jSr@)r rBr5r8rDr3r3r4rFs  zSphericalBesselBase.fdiffNrg)rhrirjrkrr_rFr3r3r3r4rs  rcCs8t||t|tj|dt| d|t|SNr7)r*rrrwrr|r=r3r3r4_jns$rcCs8tj|dt| d|t|t||t|Sr)rrwr*rrrr3r3r4_yns$rc@sDeZdZdZeddZddZddZdd Zd d Z d d Z dS)rUa 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 cCsD|jr|jr tjS|jr|jrtjStjS|tjtjfvr tjSdSr9) rWrrorOr^rprqrurtr:r3r3r4r>szjn.evalcK ttd|t|tj|Sr~)rrrQrrrr3r3r4rs zjn._eval_rewrite_as_besseljcKs,tj|ttd|t| tj|Sr~)rrwrrrrrr3r3r4rs,zjn._eval_rewrite_as_besselycKstj|t| d|Sr)rrwrVrr3r3r4rszjn._eval_rewrite_as_yncKt|j|jSr9)rr5r8rr3r3r4rz jn._expandcC|jjr |t|SdSr9r5rrrQ _eval_evalfr2precr3r3r4rzjn._eval_evalfN) rhrirjrkrmr>rrrrrr3r3r3r4rUs9   rUc@s@eZdZdZeddZeddZddZdd Zd d Z d S) rVa 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 cKs0tj|dttd|t| tj|Sr\)rrwrrrQrrr3r3r4r3s0zyn._eval_rewrite_as_besseljcKrr~)rrrrrrr3r3r4r7s zyn._eval_rewrite_as_besselycKstj|dt| d|Sr)rrwrUrr3r3r4r;szyn._eval_rewrite_as_jncKrr9)rr5r8rr3r3r4r>rz yn._expandcCrr9)r5rrrrrr3r3r4rArzyn._eval_evalfN) rhrirjrkrrrrrrr3r3r3r4rVs.   rVc@sLeZdZeddZeddZddZddZd d Zd d Z d dZ dS)SphericalHankelBasecKsN|j}ttd|t|tj||ttj|dt| tj|Sr@)_hankel_kind_signrrrQrrrrwr2r<r=rehksr3r3r4rHs&z,SphericalHankelBase._eval_rewrite_as_besseljcKsJ|j}ttd|tj|t| tj||tt|tj|Sr~)rrrrrwrrrr r3r3r4rQs(z,SphericalHankelBase._eval_rewrite_as_besselycKs(|j}t||t|tt||Sr9)rrUrrVrr r3r3r4rZ"z'SphericalHankelBase._eval_rewrite_as_yncKs(|j}t|||tt||tSr9)rrUrrVrr r3r3r4r^r z'SphericalHankelBase._eval_rewrite_as_jncKsF|jjr |jdi|S|j}|j}|j}t|||tt||Sr)r5rrr8rrUrrV)r2rar<r=r r3r3r4r_bs z%SphericalHankelBase._eval_expand_funccKs2|j}|j}|j}t|||tt||Sr9)r5r8rrrrexpand)r2rar|r=r r3r3r4rks zSphericalHankelBase._expandcCrr9rrr3r3r4rzrzSphericalHankelBase._eval_evalfN) rhrirjrrrrrr_rrr3r3r3r4rFs   rc@s"eZdZdZejZeddZdS)rSa 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 cKttd|t||Sr~)rrrrr3r3r4_eval_rewrite_as_hankel1zhn1._eval_rewrite_as_hankel1N) rhrirjrkrrorrrr3r3r3r4rSs 0rSc@s$eZdZdZej ZeddZdS)rTa 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 cKr r~)rrrrr3r3r4_eval_rewrite_as_hankel2rzhn2._eval_rewrite_as_hankel2N) rhrirjrkrrorrrr3r3r3r4rTs 0rTsympyc sddlm}dkr*ddlmddlm}||fddtd|dDSd krZdd lmzdd l m fd d }Wnt yYddl m fdd }Ynwt dfdd}|}|||}|g} t|dD]} ||||}| |qw| S)a Zeros of the spherical Bessel function of the first kind. Explanation =========== This returns an array of zeros of $jn$ up to the $k$-th zero. * method = "sympy": uses `mpmath.besseljzero `_ * method = "scipy": uses the `SciPy's sph_jn `_ and `newton `_ to find all roots, which is faster than computing the zeros using a general numerical solver, but it requires SciPy and only works with low precision floating point numbers. (The function used with method="sympy" is a recent addition to mpmath; before that a general solver was used.) Examples ======== >>> from sympy import jn_zeros >>> jn_zeros(2, 4, dps=5) [5.7635, 9.095, 12.323, 15.515] See Also ======== jn, yn, besselj, besselk, bessely Parameters ========== n : integer order of Bessel function k : integer number of zeros to return r)rr) besseljzero) dps_to_preccs0g|]}ttdt|qS)g?)r _from_mpmathr _to_mpmathint)rl)rr|rr3r4r"szjn_zeros..r7scipy)newton) spherical_jncs |Sr9r3rY)r|rr3r4)s zjn_zeros..)sph_jncs|ddS)Nrrr3r)r|rr3r4r,sUnknown method.csdkr ||}|Std)Nrrr)rbrYr )methodrr3r4solver0s zjn_zeros..solver)mathrmpmathrmpmath.libmp.libmpfrrscipy.optimizer scipy.specialr ImportErrorrrr) r|rr dpsmath_pirrbr!r rootsir3)rr r|rrrrr4jn_zeross4 -         r,c@s4eZdZdZddZddZd ddZd d d Zd S) AiryBasezg Abstract base class for Airy functions. This class is meant to reduce code duplication. cCs||jdSNr)rr0rIr1r3r3r4rKKzAiryBase._eval_conjugatecCs |jdjSr.)r0rr1r3r3r4rNs zAiryBase._eval_is_extended_realTcKsL|jd}|}|j}||||d}t||||d}||fS)Nrr?)r0rIrr)r2deeprar=zcrbuvr3r3r4 as_real_imagQs zAiryBase.as_real_imagcKs$|jdd|i|\}}||tS)Nr0r3)r4r)r2r0rare_partim_partr3r3r4_eval_expand_complexYs zAiryBase._eval_expand_complexN)T)rhrirjrkrKrr4r7r3r3r3r4r-Cs  r-c@^eZdZdZdZdZeddZdddZe e dd Z d d Z d d Z ddZddZdS)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 r7TcCs|jr/|tjur tjS|tjurtjS|tjurtjS|jr/tjdtddt tddS|jrCtjdtddt tddSdS)Nrr?) is_NumberrrsrtrprurWror r&r;rr3r3r4r>s   ""z airyai.evalcC |dkr t|jdSt||Nr7r) airyaiprimer0r rDr3r3r4rF z airyai.fdiffcGs6|dkrtjSt|}t|dkrj|d}td|| td||dtt|tddtddt|t |dtddtt|tddtddt|dt |dtdd|Stj dtddtt |tj tdttddt|tj t|td||S)Nrr7rrr?r) rrprlenrrrr rr&ror|rYprevious_termsrr3r3r4 taylor_terms" L@Hzairyai.taylor_termcKs`tdd}tdd}t| tdd}t|jr.|t| t| ||t|||SdSNr7rr?r rr"r`rrQr2r=reotttrZr3r3r4r   ,zairyai._eval_rewrite_as_besseljcKstdd}tdd}t|tdd}t|jr,|t|t| ||t|||S|t||t| |||t|| t|||SrDr rr"r^rrRrFr3r3r4rs   *<zairyai._eval_rewrite_as_besselicKs~tjdtddttdd}|tddttdd}|tgtddg|dd|tgtddg|ddS)Nrr?r7 r)rror r&r r)r2r=repf1pf2r3r3r4_eval_rewrite_as_hypers"@zairyai._eval_rewrite_as_hyperc Ks"|jd}|j}t|dkr|}td|gd}td|gd}td|gd}td|gd}||||||} | dur| |}d|jr| |}| |}| |}|||||||||} ||||||} tj| tj t | | tj t dt | SdSdSdS Nrr7r)excludedmr|r) r0 free_symbolsr@poprmatchrOrrror9rairybi r2rarsymbsr=rrRrSr|Mpfnewargr3r3r4r_*   $2zairyai._eval_expand_funcNr7rhrirjrknargs unbranchedrmr>rF staticmethodrrCrrrOr_r3r3r3r4r9^sX     r9c@r8)rWa 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 r7TcCs|jr/|tjur tjS|tjurtjS|tjurtjS|jr/tjdtddt tddS|jrCtjdtddt tddSdS)Nrr7r?) r:rrsrtrurprWror r&r;r3r3r4r>hs   ""z airybi.evalcCr<r=) airybiprimer0r rDr3r3r4rFwr?z airybi.fdiffcGs|dkrtjSt|}t|dkrW|d}td|tttddt|tj t |tj td|tj tt tddt|tj t |dtd|Stj t ddtt|tj tdtttddt|tj t |td||S)Nrr7rrr?rc)rrprr@rr!rr rrorrrr r&rAr3r3r4rC}s @<Hzairybi.taylor_termcKs`tdd}tdd}t| tdd}t|jr.t| dt| ||t|||SdSrDrErFr3r3r4rrIzairybi._eval_rewrite_as_besseljcKstdd}tdd}t|tdd}t|jr.t|tdt| ||t|||St||}t|| }t||t| ||||t|||SrDrJr2r=rerGrHrZrrr3r3r4rs   .  2zairybi._eval_rewrite_as_besselicKsztjtddttdd}|tddttdd}|tgtddg|dd|tgtddg|ddS)Nrrcr?r7rKr)rror r&r r)rLr3r3r4rOs@zairybi._eval_rewrite_as_hyperc Ks"|jd}|j}t|dkr|}td|gd}td|gd}td|gd}td|gd}||||||} | dur| |}d|jr| |}| |}| |}|||||||||} ||||||} tjt dtj | t | tj | t | SdSdSdSrP) r0rTr@rUrrVrOrrrror9rWrXr3r3r4r_r]zairybi._eval_expand_funcNr^r_r3r3r3r4rW sZ     rWc@VeZdZdZdZdZeddZdddZdd Z d d Z d d Z ddZ ddZ dS)r>a% 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 r7TcCsR|jr|tjur tjS|tjurtjS|jr'tjdtddttddSdS)Nrr7) r:rrsrtrprWrwr r&r;r3r3r4r>s  "zairyaiprime.evalcC*|dkr|jdt|jdSt||r=)r0r9r rDr3r3r4rF zairyaiprime.fdiffcCR|jd|}t|tj|dd}Wdn1swYt||SNrr7) derivative)r0rr,r+r9rrr2rr=resr3r3r4r!   zairyaiprime._eval_evalfcKsPtdd}t| tdd}t|jr&|dt| ||t|||SdSNr?r)r rr"r`rQr2r=rerHrZr3r3r4r's  &z$airyaiprime._eval_rewrite_as_besseljcKstdd}tdd}|t|tdd}t|jr(|dt||t| |St|tdd}t||}t|| }||d|t||||t| ||SrD)r rr"r^rRrer3r3r4r-s     2z$airyaiprime._eval_rewrite_as_besselicKs|dddtddttdd}dtddttdd}|tgtddg|dd|tgtddg|ddS)Nr?rr7rK)r r&r r)rLr3r3r4rO9s(@z"airyaiprime._eval_rewrite_as_hyperc Ks"|jd}|j}t|dkr|}td|gd}td|gd}td|gd}td|gd}||||||} | dur| |}d|jr| |}| |}| |}|||||||||} ||||||} tj| tj t | | tj t dt | SdSdSdSrP) r0rTr@rUrrVrOrrror>rrdrXr3r3r4r_>*   $2zairyaiprime._eval_expand_funcNr^rhrirjrkr`rarmr>rFrrrrOr_r3r3r3r4r>sQ   r>c@rf)rda6 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 r7TcCs~|jr,|tjur tjS|tjurtjS|tjurtjS|jr,dtddttddS|jr=dtddttddSdS)Nrr7rc) r:rrsrtrurprWr r&r;r3r3r4r>s   zairybiprime.evalcCrgr=)r0rWr rDr3r3r4rFrhzairybiprime.fdiffcCrirj)r0rr,r+rWrrrlr3r3r4rrnzairybiprime._eval_evalfcKsRtdd}|t| tdd}t|jr'| tdt| |t||SdSrorErpr3r3r4rs  $z$airybiprime._eval_rewrite_as_besseljcKstdd}tdd}|t|tdd}t|jr*|tdt| |t||St|tdd}t||}t|| }t||t| |||d|t|||SrDrJrer3r3r4rs   "  6z$airybiprime._eval_rewrite_as_besselicKs||ddtddttdd}tddttdd}|tgtddg|dd|tgtddg|ddS)Nr?rrcr7rqrK)r r&r r)rLr3r3r4rOs$@z"airybiprime._eval_rewrite_as_hyperc Ks"|jd}|j}t|dkr|}td|gd}td|gd}td|gd}td|gd}||||||} | dur| |}d|jr| |}| |}| |}|||||||||} ||||||} tjt d| tj t | | tj t | SdSdSdSrP) r0rTr@rUrrVrOrrrror>rdrXr3r3r4r_rrzairybiprime._eval_expand_funcNr^rsr3r3r3r4rdXsS   rdc@sFeZdZdZeddZdddZddZd d Zd d Z d dZ dS)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 .. 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