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Questions tagged [special-functions]

Questions on the special mathematical functions implemented in Mathematica.

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Fast Spherical Harmonics radiative transfer

This is a rather specific question and I apologize for spamming you with some lengthy code. But it could be interesting for some reader and maybe you can help out, so please bear with me. I am using ...
Markus Roellig's user avatar
11 votes
0 answers
284 views

MacDonald formula for Modified Bessel Functions

How can I make Mathematica understand these two integrals? $$\int_0^{\infty} e^{-x \cosh{\xi}} d\xi = K_0(x)$$ $$\int_0^{\infty} e^{-\frac{1}{2} \Big( \frac{x y}{u} + u \frac{x^2+y^2}{x y} \Big) } K_{...
Kevin Driscoll's user avatar
10 votes
0 answers
152 views

Bug in SumConvergence

Bug introduced in 10.0.1 and fixed in 12.0.0 Version 11.2.0.0 on MacBook Pro: ...
Paul R.'s user avatar
  • 877
8 votes
0 answers
1k views

Inverse Laplace transform not obtained

I can't seem to be able to compute the inverse Laplace transform of a Laplace transform: ...
highBandWidth's user avatar
7 votes
0 answers
151 views

Buggy behavior of EllipticE[0,k] with arbitrary precision input

I am having an issue where if I provide the EllipticE function with a first argument of zero and a second argument with a precision lower than that of machine precision, the kernel crashes. For ...
Conor Dyson's user avatar
7 votes
0 answers
146 views

SiegelTheta gives misleading message when the dimensions don't match

Bug introduced in 6.0 and persisting through 13.2.0 SiegelTheta is new in 6.0 In order to test the SiegelTheta function, I ...
Semiclassical's user avatar
7 votes
0 answers
541 views

Reproducing the Integral Definition of the Modified Bessel function

I need to simplify some integral expressions in terms of special functions, such as the modified Bessel function of the first kind. See for example Eq. (5) on this page. Notice that the real ...
TriSSSe's user avatar
  • 533
6 votes
0 answers
75 views

Dedekind Zeta Function in Mathematica (at least for quadratic number field)

Does there exist some way to use Mathematica to compute the Dedekind Zeta function for an arbitrary algebraic number field? Or does there exist some package to do this? I am actually only interested ...
Mike Battaglia's user avatar
6 votes
0 answers
175 views

Bugs in hypergeometric functions with negative integer lower parameters

Bug introduced in 13.0 or earlier and persisting through 13.2.0 or later. If you were to evaluate these expressions, Mathematica returns the value shown. ...
Jeff L.'s user avatar
  • 61
6 votes
2 answers
658 views

Spherical harmonics and Laplace operator

The spherical harmonic function $Y_l^m(\theta,\phi)$ is defined to be an eigenfunction of the angular part of the Laplace operator with eigenvalue $-l(l+1)$. In other words, it solves the PDE: $$\...
Patrick.B's user avatar
  • 1,509
6 votes
0 answers
88 views

AppellF1 calculation hangs indefinitely

The built-in AppellF1 function seems generally useless. For example, AppellF1[3/4, 1/2, 1/2, 7/4, (7 + 4 Sqrt[3]), (7 - 4 Sqrt[3])] hangs indefinitely on my system....
Nickolas's user avatar
  • 121
6 votes
0 answers
73 views

What makes ListPlot better than N?

I wanted to numerically verify the validity of the formula for the first Stieltjes constant $$\gamma_1=-\frac12\sum_{n=0}^\infty\frac1{n+1}\sum_{k=0}^n\binom{n}{k}(-1)^k\log^2(k+1)$$ ...
Vaclav Kotesovec's user avatar
6 votes
0 answers
256 views

Why doesn't Log[Gamma[]] simplify to LogGamma[] where it could?

I have been playing with various equations involving amount of permutations in relatively large sets. Easiest way to look at these is something like Log[10, bignumber!] . Often expressions, even ...
kirma's user avatar
  • 19.1k
5 votes
0 answers
210 views

Possible bug with EllipticPi

I am calculating the incomplete elliptic integral of the third kind in Mathematica 11.3 using EllipticPi. Since my range of phi ...
Torben Frost's user avatar
5 votes
0 answers
171 views

Issues with the series expansion of Nielsen generalized polylogarithms in Mathematica 11.3

Bug introduced in 11.3. Fixed in 12.0. In Mathematica 9, 10.3 and 11.0 I can easily expand PolyLog[2, 2, x] around $x=1$ using ...
vsht's user avatar
  • 3,517
5 votes
0 answers
244 views

Integrating a product of three Spherical Harmonics

The following command is returned unevaluated. The answer is well known to be related to Wigner's 3j Symbol which is also a defined function in Mathematica. ...
Quasar Supernova's user avatar
5 votes
0 answers
112 views

Express special function (Hypergeometric1F1) in terms of another special function (HermiteH)

I have the following Mathematica expression: (2^k Gamma[1/2 + k] Hypergeometric1F1[-k, 1/2, t^2/2])/Sqrt[\[Pi]] where k is an integer and t is a real number, and ...
Otwin's user avatar
  • 51
5 votes
0 answers
574 views

Integral Form of Modified Bessel Function of the Second Kind

Why can't Mathematica integrate r = Integrate[Exp[-x Cosh[t]], {t, 0, Infinity}]; r = Assuming[Element[x, Reals], Simplify[r]]; Together[r] From Wikipedia, it ...
Alex Wang's user avatar
5 votes
0 answers
135 views

Expansion of $E(i c \mid m)$ at $c\to\infty$?

Currently, I am using a Windows machine with Mathematica 8. I noticed a difference in a series expansion of the function EllipticE[] in comparison with a result ...
Kagaratsch's user avatar
4 votes
0 answers
49 views

Why does an integer index inside a symbol become real when entering a ChebyshevT function with floating point arithmetic?

When I enter an indexed symbol into a ChebyshevT function with integer arithmetic like this: ChebyshevT[1, 3 * a[4]] I get the expected result: ...
Danel's user avatar
  • 173
4 votes
0 answers
94 views

Siegel modular forms in Mathematica

Is there a convenient way to work with Siegel modular forms in Mathematica? I am interested in doing analytic computations using the $\chi_{10}(\Omega)$ Siegel modular form, where $\Omega$ is the $2\...
Holomaniac's user avatar
4 votes
0 answers
95 views

Series expansion of Lerch transcendent still buggy?

This series expansion of a Lerch transcendent seems fixed in V12. However, the following still fails: From the definition of a Lerch transcendent, ...
Roman's user avatar
  • 47.5k
4 votes
0 answers
130 views

Another example where using FullSimplify gives different result than Simplify

I believe this question is very similar to Result of Series[expression] is different when I simplify the expression, however, due to my lack of Mathematica experience, I am reluctant to call it a bug. ...
Gabriel Nagaoka's user avatar
4 votes
0 answers
678 views

How can I do a faster integration?

I have this part of my code, which takes forever to run. Does anybody know how to make it faster? Using NIntegrate I face error: "NIntegrate::eincr: The global error of the strategy GlobalAdaptive ...
Delaram Nematollahi's user avatar
4 votes
0 answers
75 views

Series of LegendreP[a,b,x] takes ages in Mathematica 11

My institution just upgraded to Mathematica 11.3 (from v10) and I'm experiencing a problem that is absent in Mathematica 10. Namely, ...
Paolo's user avatar
  • 171
4 votes
0 answers
964 views

Wormhole embedding diagrams

I am trying to reproduce the embedding diagrams for the evolution of a Schwarzschild wormhole described in this paper. Following the paper notation, we denote the Kruskal coordinates by $(v,u)$. For a ...
ASM's user avatar
  • 41
4 votes
0 answers
568 views

Simplify hypergeometric function

Theorem: Let $H_n$ be the $nth$ harmonic number. Then $$\sum_{n=1}^\infty \binom{2n}{n} H_n x^n=\frac{2}{\sqrt{1-4x}}\log\bigg(\frac{1+\sqrt{1-4x}}{2\sqrt{1-4x}} \bigg)$$ How can I simplify ...
vito's user avatar
  • 8,958
4 votes
0 answers
155 views

FunctionExpand does not yield known simplification

Consider: Cosh[a ArcSinh[z]] // FunctionExpand Cosh[a ArcSinh[z]] Which seems to suggest that there is no simplification ...
Kagaratsch's user avatar
4 votes
0 answers
279 views

Does Mathematica know that $\small\frac{\vartheta_3\left(0,\frac{1}{\sqrt[10000000000]{e}}\right)^2}{10000000000}$ not equal $\pi$

The following is not an identity but is correct to over 42 billion digits: $$\bigg(\frac{1}{10^5}\sum_{n=-\infty}^{\infty}e^{-\frac{n^2}{10^{10}}}\bigg)^2=\pi$$ I want to check this. I tried: <...
vito's user avatar
  • 8,958
4 votes
0 answers
129 views

Series expansion in Infinity issue with Zeta(s) function

With this code: Series[Zeta[s], {s, Infinity, #}] & /@ Range[10] // MatrixForm Series expansion for the Zeta(s) function ...
Mariusz Iwaniuk's user avatar
4 votes
0 answers
101 views

Huge difference after changing a fraction to decimal

I have a limit to calculate. With[{p = 1/2, q = 0.1}, Limit[a^q Integrate[Sin[x]/x^p, {x, a, Infinity}], a -> Infinity]] gives correct result ...
Nick's user avatar
  • 390
4 votes
0 answers
272 views

What am I missing in this highly oscillatory integral?

I want to numerically integrate this equation (in python without calling Mathematica): $\int_0^\infty {\rm d}k f(k) J_v(r k) J_n(s k)$ where $f(k)$ is a non-smooth function, $J_v$ are the Bessel ...
Jorge's user avatar
  • 141
4 votes
0 answers
488 views

Orthogonality relations of Hermite polynomials

The Hermite polynomials are orthogonal. $$ \int_{-\infty}^\infty H_m(x) H_n(x) e^{-x^2}\, \mathrm{d}x = \sqrt{ \pi} 2^n n! \delta_{nm} $$ Does Mathematica not use this relationship? Because running <...
Raksha's user avatar
  • 633
4 votes
0 answers
131 views

Is it possible to circumvent this overflow?

I'm trying to evaluate the following function numerically: $ f(A,B)=\frac{2A\pi ^{5/2} (-1)^B}{(A!)^2B!} \, _4\tilde{F}_3\left({\frac{1}{2},1-A,1-A,1-B\atop \frac{1}{2}-A,\frac{1}{2}-A,\frac{1}{2}-...
Ziofil's user avatar
  • 2,470
4 votes
0 answers
69 views

Example in Help File does not evaluate as claimed

In the help file, under BellB, I read at "Properties and Relations": Sum can give results involving ...
Wouter's user avatar
  • 1,343
4 votes
0 answers
597 views

Spherical harmonic derivative

Consider the following substitution Derivative[2, 0][S][th, ph] /. S -> Function[{th, ph}, SphericalHarmonicY[3, 0, th, ph]] which gives correct answer. While ...
mmal's user avatar
  • 3,508
4 votes
0 answers
648 views

Minimizing a functional takes forever

I need help with minimizing a functional in Mathematica. I have a function $V(\xi)=\sum_{i=1}^\infty C_{3_i}J_0(\xi/C_i)+C_{4_i}Y_0(\xi/C_i),~~~\Sigma\leq\xi\leq\xi_1$, and want to find such $\{C_{...
ApplMath's user avatar
3 votes
0 answers
44 views

Dirac delta identity using MeijerG

In https://functions.wolfram.com/14.03.26.0002.01, there's an identity given for DiracDeltausing MeijerG which even has a code ...
Confuse-ray30's user avatar
3 votes
0 answers
69 views

Generalized Lambert W Function?

The Lambert W Function (defined as the inverse of $x e^x$) is implemented in Mathematica as ProductLog. Has anyone made any progress implementing the Generalized Lambert W Function defined as the ...
pdmclean's user avatar
  • 1,398
3 votes
0 answers
75 views

PrimeZetaP evaluation in different versions of Mathematica

PrimeZetaP was introduced in version 7.0. I suspect there were made some changes in the definition of this function in subsequent versions. Is there any user that ...
azerbajdzan's user avatar
  • 17.5k
3 votes
0 answers
60 views

Derivative[0, 1, 1][QPolyGamma] cannot be calculated numerically?

During a more complicated calculation, I got an expression ...
Vaclav Kotesovec's user avatar
3 votes
0 answers
80 views

Can I solve a differential equation with the (ir)regular Coulomb wavefunctions?

When I evaluate DSolve[w''[ρ]] + (1 - (2 η)/ρ) w[ρ] == 0, w[ρ], ρ] Mathematica returns me a solution in terms of the Hypergeometric1F1 and HypergeometricU ...
Yann's user avatar
  • 31
3 votes
0 answers
216 views

Using new FoxH function to solve a trinomial equation. Possible issue?

There is a possible issue on solving the following trinomial equation $$x^\alpha+x=y$$ for real $x,y,\alpha$. From Belkic [1], it is known that for $\alpha>1$ a solution is found using the ...
Jorge Zuniga's user avatar
3 votes
0 answers
98 views

FullSimplify gives erroneous answer

FullSimplify gives a wrong expression when simplifying Hypergeometric functions: f = Hypergeometric2F1[1/2 - n, -n, 1 - 2 n, -4 a]; g = FullSimplify[f]; The ...
Henry Lin's user avatar
3 votes
0 answers
115 views

Spherical harmonics Y (l,m,theta,phi) for general l, m

I am trying to solve integrals involving spherical harmonics Y(l,m, theta, phi) and their derivatives. I do not have any particular l,m, theta, phi values. I need to solve it for general l,m. When I ...
apk's user avatar
  • 307
3 votes
0 answers
129 views

Does anyone have a Mathematica implementation of the standard $\arg\zeta(s)$ function required to evaluate $S(T)$?

This question is related to my question Is there an elegant exact formula for the zeta zero counting function? on Math StackExchange. Question: Does anyone have a Mathematica implementation of the ...
Steven Clark's user avatar
3 votes
0 answers
83 views

Algorithm used by Mathematica for evaluating partial sums

Today, while using mathematica, I entered the command Sum[1/Factorial[n], {n, 0, x}] and found that: $$\sum_{x\geq n\geq0}\frac{1}{n!}=\frac{e\Gamma(x+1,1)}{\Gamma(...
user avatar
3 votes
0 answers
487 views

rotation of spherical harmonics using Wigner D-matrix

A very stupid question as I am very confused: I have a surface charge density which is a function of spherical harmonics $\sigma_{l,m}=Y_{lm}$ (only the real part). Now I need to rotate the particle, ...
Aa Aa's user avatar
  • 61
3 votes
0 answers
138 views

Dirichlet L-function associated to Kronecker symbol

The Fourier coefficients of the genus 2 Eisenstein series on the Siegel upper half-space are given by sums over Cohen functions. The Cohen function contains as a multiplicative factor a Dirichlet L-...
El Rafu's user avatar
  • 287
3 votes
0 answers
116 views

Evaluating Lauricella functions of Third kind numerically in Mathematica

Let's consider the Lauricella function of third kind, denote as $F_{C}(a,b;c_1,...,c_n;x_1,...x_n)$ in MathWorld. Is anyone aware of an algorithm that allows to numerically evaluate such a function ...
Alessandro Pini's user avatar

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