(phD2SrSSKJrJr SSKrSSKJr SSKJ r J r SSK J r SSK r SSK J r JrJrJrJrJrJrJrJrJrJr SSKJr S rS rS rS rS rSr Sr!\"S:Xa\!"5 gg!\a N,f=f)zPrecompute coefficients of several series expansions of Wright's generalized Bessel function Phi(a, b, x). See https://dlmf.nist.gov/10.46.E1 with rho=a, beta=b, z=x. )ArgumentParserRawTextHelpFormatterN)quad)minimize_scalar curve_fit)time) EulerGammaRationalSSum factorialgamma gammasimppi polygammasymbolszeta)hornerc|Sn[S5upp4/n/n/n[X4-[U5- [X-U-5- US[R 45n[U5[ R"U5- U-n[SUS-5Hn URX5RUS5R5R5n U R[SU5S5R[S5n U SU --n URX-[U 5- 5 UR[!U 55 UR[!X- R555 M Sn U S- n [#/S QXVU/5H:up[[%U55HnU S U S US 3['X5-- n M M< U $) zATylor series expansion of Phi(a, b, x) in a=0 up to order 5. a b x krcgNrargss Z/var/www/html/venv/lib/python3.13/site-packages/scipy/special/_precompute/wright_bessel.py series_small_a..&Az=Tylor series expansion of Phi(a, b, x) in a=0 up to order 5. zAPhi(a, b, x) = exp(x)/gamma(b) * sum(A[i] * X[i] * B[i], i=0..5) )AXB [] = )rr r rr Infinitysympyexprangediffsubssimplifydoitrreplaceappendrziplenstr)orderabxkr$r%r& expressionntermx_partsnamecis rseries_small_arDs E#JA! A A AQT)A,&uQSU|3aAJJ5GHJq%))A,&3J1eAg q$))!Q/88:??A))IaOQ/79o6  2' il"#   ..012 IA MMAq 2s1vA 2dV1QCt$s14y0 0A3 Hr"c[S5nSU- [- [R"SU-[ U5-XS- --USUS-45-$)z|Symbolic expansion of digamma(z) in z=0 to order n. See https://dlmf.nist.gov/5.7.E4 and with https://dlmf.nist.gov/5.5.E2 r;r#r)rr r+ summationr)zr=r;s r dg_seriesrI7sU  A a4*  a$q')A!H4q!QqSkB CCr"cH[R"[XU-5X5$)z8Symbolic expansion of polygamma(k, z) in z=0 to order n.)r+r.rI)r;rHr=s r pg_seriesrKAs ::iQ3' ..r"c ^^Sm[S5upp#[S5upEn[U[U[S5U0n/n/n /n /n [ U5[ R "U5- [X#-[U5- [ X-U-5- US[R45-n [STS-5GHPmU RUT5RUS5R5R5n U R[!SU5S5R#[ S5nUST--nX- [ U5- nTS:atUR#[ UU4S j5nUR%USTS-T- S 9R'5R[!S S5S [S5-5R5nUR)UT-[T5- 5 U R)[+U55 U R)U5 GMS [ R,"U SRU5U5R/5n U R15 [[3U 55H%nU U[U5-R5U U'M' S nUS- nUS- nUS- nUS- nUS- nUS- n[5SS/X/5H>unn[[3U55H nUSUSUS3- nU[7UU5- nM" M@ [[3U 55HlnUSUS3- nU[7U U5- nUSUS3- nU[7U URU[U[U[S505R9S55- nMn US- nUS- nUS- n[;[TS- 5Vs/sHo1U-[U5- XS--PM sn5nUU S RU5- R5nUSU[S5:H3- nUS - n[;[TS - 5Vs/sHo1U-[U5- XS --PM sn5nUU SRU5- R5nUSU[S5:H3- nU$s snfs snf)!anTylor series expansion of Phi(a, b, x) in a=0 and b=0 up to order 5. Be aware of cancellation of poles in b=0 of digamma(b)/Gamma(b) and polygamma functions. digamma(b)/Gamma(b) = -1 - 2*M_EG*b + O(b^2) digamma(b)^2/Gamma(b) = 1/b + 3*M_EG + b*(-5/12*PI^2+7/2*M_EG^2) + O(b^2) polygamma(1, b)/Gamma(b) = 1/b + M_EG + b*(1/12*PI^2 + 1/2*M_EG^2) + O(b^2) and so on. rrzM_PI M_EG M_Z3rrcgrrrs rr(series_small_a_small_b..er!r"r#c(>[XTS-T-5$)Nr)rK)r;r:r=r7s rrrOms9Q57193Mr")r=rFzDTylor series expansion of Phi(a, b, x) in a=0 and b=0 up to order 5.z9 Phi(a, b, x) = exp(x) * sum(A[i] * X[i] * B[i], i=0..5) z B[0] = 1 z&B[i] = sum(C[k+i-1] * b**k/k!, k=0..) z M_PI = piz M_EG = EulerGammaz M_Z3 = zeta(3)r$r%r'r(r)z # C[z C[z/ Test if B[i] does have the assumed structure.z" C[i] are derived from B[1] alone.z: Test B[2] == C[1] + b*C[2] + b^2/2*C[3] + b^3/6*C[4] + ..z test successful = z- Test B[3] == C[2] + b*C[3] + b^2/2*C[4] + ..)rrr rrr+r,r r r r*r-r.r/r0r1rr2seriesremoveOr3rPolycoeffsreverser5r4r6evalfsum)r8r9r:r;M_PIM_EGM_Z3c_subsr$r%r&Cr<r>r?pg_partrCr@rArBtestr=r7s @@rseries_small_a_small_braFsi E#JA!/0D$ D$q'4 8F A A A A q%))A,& AD1 eACEl *Q1::,>?@J1eAg q!$))!Q/88:??A))IaOQ/79o6  2'+eAh& 6ooi&MOG~~aeAgai~8 Yq!_baj9    Ail"#   +0  1Q499V$a(//1AIIK 3q6]!y|#--/! OA FFAA 22AA A ASzA6*as1vA 2dV1QCt$ $A QqTNA+ 3q6] vaS  S1Y tA3d^ S1D*dBd1gFG%)   <\rSrSrSrSr\U4Sj5rSrg)asymptotic_series..gzHelper function g according to Wright (1935) g(n, rho, v) = (1 + (rho+2)/3 * v + (rho+2)*(rho+3)/(2*3) * v^2 + ...) Note: Wright (1935) uses square root of above definition. rMc>US:d [S5eUS:XagT"US- X#5[[US-U-5[US-5- 5[[SU-5[S5- 5- X1---$)Nrzmust have n >= 0rrFrM) ValueErrorrr)clsr=rhovgs reval!asymptotic_series..g.evals6 !344a1c~c!eAguSU| ;<ac 58 34556T:::r"rN __name__ __module__ __qualname____firstlineno____doc__nargs classmethodrm__static_attributes__rlsrrlres!   :  :r"rlc4>\rSrSrSrSr\U4Sj5rSrg)!asymptotic_series..coef_CzCalculate coefficients C_m for integer m. C_m is the coefficient of v^(2*m) in the Taylor expansion in v=0 of Gamma(m+1/2)/(2*pi) * (2/(rho+1))^(m+1/2) * (1-v)^(-b) * g(rho, v)^(-m-1/2) rMcr>US:d [S5e[S5nSU- U*-T"SU-X$5U*[SS5- --nURUSU-5R US5[ SU-5- nU[ U[SS5-5S[-- SUS-- U[SS5----nU$)Nrzmust have m >= 0rkrrF)rhrr r.r/r rr)rimrjbetarkr<resrls rrm&asymptotic_series..coef_C.evals6 !344 AA#$!AaC.A2hq!n;L*MMJ//!QqS)..q!4y1~ECq8Aq>12ad;s1uIXa^);<=>CJr"rNrorxsrcoef_Crzs!     r"rz xa b xap1rz!Asymptotic expansion for large x z.Phi(a, b, x) = Z**(1/2-b) * exp((1+a)/a * Z) z3 * sum((-1)**k * C[k]/Z**k, k=0..6) zZ = pow(a * x, 1/(1+a)) zA[k] = pow(a, k) zB[k] = pow(b, k) zAp1[k] = pow(1+a, k) z#C[0] = 1./sqrt(2. * M_PI * Ap1[1]) rzC[z ] = C[0] / (z * Ap1[z]) z] *= z Nz xa\*\*(\d+)zA[\1]z b\*\*(\d+)zB[\1]xap1zAp1[1]xar8z (\d{10,})z\1.)r+Functionrr-r0rUrV denominatorlcmcollectfactorxreplacer6recompilesubr2)r7rrr9rC0r@rCexprr:rrre_are_b re_digitsrls @rasymptotic_seriesrs  E:ENN:(,+&KB4 2 B,A ::A @@A ))A A A ##A //A 1eAg qa B"qyL1;;=+0::d+;+B+B+DE+Da--/+DE6" '')11!U\\B}}bdD\* r!Ls$ 77 r!E#d)D )) ::n %D 1A ::m $D 1A &(#A $A <(I fa A H#FsHc j^^^^ ^ ^ Sm SU 4Sjjm/SQm/SQm/SQm [R"TTT 5ummm TR5TR5T R5smmm /n[T R5H2m UR [ UUUU U 4SjSSS S 0S 9R5 M4 [R"U5nTTT US .nS n[[X!USSS9S5nSnUS- nUS- nUS- nUS- nUSRUVs/sHoUSPM sn5- nU$s snf)aFit optimal choice of epsilon for integral representation. The integrand of int_0^pi P(eps, a, b, x, phi) * dphi can exhibit oscillatory behaviour. It stems from the cosine of P and can be minimized by minimizing the arc length of the argument f(phi) = eps * sin(phi) - x * eps^(-a) * sin(a * phi) + (1 - b) * phi of cos(f(phi)). We minimize the arc length in eps for a grid of values (a, b, x) and fit a parametric function to it. c[R"SU-U*5nU[R"U5-X-U-[R"X-5-- S-U- $)zDerivative of f w.r.t. phi.g?r)nppowercos)epsr8r9r:phieps_as rfp$optimal_epsilon_integral..fpsOcA2&RVVC[ 155=266!'?#BBQFJJr"cX>^^^^[UUUUU4SjS[RUSS9S$)zCompute Arc length of f. Note that the arc length of a function f from t0 to t1 is given by int_t0^t1 sqrt(1 + f'(t)^2) dt c P>[R"ST"TTTTU5S--5$)NrrF)rsqrt)rr8r9rrr:s rr=optimal_epsilon_integral..arclength..s%BsAq!S,A1,D(D Er"rd)epsrellimit)rrr)rr8r9r:rrrs```` r arclength+optimal_epsilon_integral..arclength s. EEruu!../1 1r") MbP?g?g?g?rrFrrc)rrr ) rg?rFrr2rig@@g@g@c,>T"UTTTTTT5$Nr)rrdata_adata_bdata_xrCs rr*optimal_epsilon_integral..s #vay&)28))=r")riBoundedxatolr)boundsmethodoptions)r8r9r:rc DUSnUSnUSn X-[R"SU-5-[R"USSU-- [R"U 5--U[R"U*U-5-- US[R"Xg-5-- -5-$)z#Compute parametric function to fit.r8r9r:gr)rr,log) dataA0A1A2A3A4A5r8r9r:s rfunc&optimal_epsilon_integral..func*s I I Iq))&&a1q5kBFF1I55RVVRC!G_8LLRVVBF^!34566 7r"rtrf)rrz7Fit optimal eps for integrand P via minimal arc length zwith parametric function: zBoptimal_eps = (A0 * b * exp(-a/2) + exp(A1 + 1 / (1 + a) * log(x) z= - A2 * exp(-A3 * a) + A4 / (1 + exp(A5 * a))) z Fitted parameters A0 to A5 are: z, z.5g)g{Gz?r) rmeshgridflattenr-sizer3rr:arraylistrjoin) best_epsdfr func_paramsr@r:rrrrrrCs @@@@@@roptimal_epsilon_integralrsSK 15F F EF[[@FFF$nn.0@$nn.FFFH 6;;  ==#/#,wo GHIq   xx!H B 7y2e9UCAFGKBA &&A NNA JJA ,,A 4 1gJ 4 55A H5s D0 c"[5n[[[S9nUR S[ /SQSS9 UR 5nSSSS S.nURURS 5"5 [S [5U- S - S S35 g)N) descriptionformatter_classaction)rrFrMrzchose what expansion to precompute 1 : Series for small a 2 : Series for small a and small b 3 : Asymptotic series for large x This may take some time (>4h). 4 : Fit optimal eps for integral representation.)typechoiceshelpc([[55$r)printrDrr"rrmain..Ls ~/0r"c([[55$r)rrarr"rrrMs 578r"c([[55$r)rrrr"rrrNs 023r"c([[55$r)rrrr"rrrOs 79:r"c[S5$)NzInvalid input.)rrr"rrrQs E*:$;r"r'<z.1fz minutes elapsed. ) rrrtr add_argumentint parse_argsgetrr)t0parserrswitchs rmainr>s B ,@BF sLP    D083:F  JJt{{;<> B R$$7 89r"__main__)#rtargparserrnumpyrscipy.integraterscipy.optimizerrrr+r r r r r rrrrrrsympy.polys.polyfuncsr ImportErrorrDrIrKrarrrrprr"rrs : 5 BBBB,  DC/ V rV rC L:. zFI   s$A,,A54A5