Questions tagged [special-functions]

Questions on the special mathematical functions implemented in Mathematica.

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Heaviside function in NDSolve

I have: ...
Tim's user avatar
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4 votes
1 answer
115 views

Strange result simplifying higher order BesselJ

Consider the following integral: $$\int_0^1 Z_n^m(r)\ J_m(\rho r) r dr$$. The solution of this should contain a single Bessel function: $$(-1)^{(m-n)/2}\ J_{n+1} (\rho)/\rho$$ (see https://www.osti....
AstronomyGeek's user avatar
2 votes
0 answers
56 views

Expanding Pochammer symbols/Gamma function for simplifying expressions

TLDR: How to expand gamma functions or Pochammer symbols in an arbitrarily long product? Some context I am trying to find out a closed-form expression for $\langle r^\alpha\rangle$ for the non-...
Sanjana's user avatar
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1 vote
1 answer
84 views

Should expression evaluation depend on the choice of a variable name?

I am verifying the series representation of the Sonine polynomial or the associated Laguerre polynomial, which is $$ L_m^{(n)}(x)=\sum\limits_{l=0}^n\left(-1\right)^l\binom{m+n}{m-l}\frac{x^l}{l!}=S_{...
houzw's user avatar
  • 338
1 vote
1 answer
95 views

Mathieu Floquet solution

Mathematica provides the MathieuC[a,q,z] and MathieuS[a,q,z] functions - as well as some other Mathieu-related functions. Maple ...
Brian Cowan's user avatar
0 votes
0 answers
41 views

Euler angles and WignerD, a question of signs

Express a 3D point as a linear combination of Spherical Harmonics, then rotate that point to a new position and find the new expansion in SH : ...
Wouter's user avatar
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1 vote
0 answers
55 views

Elliptic theta function wont evaluate [closed]

It appears that Mathematica wont evaluate the Jacobi theta functions when the last argument has a magnitude greater than 1, for example: EllipticTheta[1, 1.5, 0.9] ...
Matt Majic's user avatar
2 votes
2 answers
106 views

Simplify inverse of function

This would be a noob question, but I need help simplifying the inverse of an expression ...
Zain Ahmad's user avatar
1 vote
1 answer
66 views

Solving a non-algebraic equation at the symbolic level [closed]

Versions 10, 11, 12, 13.0.0.0 and 13.2.0.0 solves the following system of equations ...
Vaclav Kotesovec's user avatar
2 votes
1 answer
81 views

Error in plotting the interefence of Laguerre-Gaussian (LG) beam

...
Gopal Verma's user avatar
  • 1,055
0 votes
0 answers
72 views

Time independent perturbation theory for solving coupled differential equations

The eqexact1 and eqexact2 are the coupled differential equation of motion with g lets say a repulsive factor, that I choose. ...
Pantelis Ashikkis's user avatar
1 vote
0 answers
78 views

Series expansion message with special functions

I am trying to expand the following expression in alp,eps. Since the expression involves Hypergeometric2F1, I am using HypExp. <...
BabaYaga's user avatar
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1 vote
1 answer
113 views

Can the Debye functions be implemented using built-in functions?

It is claimed in the comments here that the Debye functions can be implemented using built-in special functions. This is clearly true for some Debye functions, e.g., $D_n^{(1)}(x)$ for $n = 1, 2, 3$ (...
WillG's user avatar
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1 answer
67 views

Evaluate the time average of Mathieu functions

I defined a function composed of Mathieu's periodic functions: ...
ZHENGYAO HUANG's user avatar
3 votes
0 answers
56 views

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

During a more complicated calculation, I got an expression ...
Vaclav Kotesovec's user avatar
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1 answer
94 views

Dirac Delta does not converge

I have trouble evaluating a simple integral in Mathematica. I have the code: ...
Nitaa a's user avatar
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1 vote
0 answers
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Derivatives in the SpinWeightedSpheroidalHarmonics package

Hello I'm using the SpinWeightedSpheroidalHarmonics package from the Black Hole Perturbation Toolkit . This package includes the SpinWeightedSphericalHarmonicY, ...
Nitaa a's user avatar
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0 answers
124 views

Table of integrals involving modified Bessel function of the second kind

I need to compute integrals of the following form as accurately as possible (possibly with extended precision): $$ I_{nl}(\omega)=\int_0^{1/2}\left(1-x^2\right)^{1/4} \sin (2 \pi l x)\; \sin (2 \pi ...
user12588's user avatar
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0 answers
48 views

Plot of some complicated functions containing special functions under both Sum and Product

I am trying to plot the $|f(\theta,\phi)|^2$, where the expression of $f(\theta,\phi)$ is as under: $$ f =\sum_{n_j}\Pi_{j_x,j_y,j_z}P_{n_j}\sum_{l}A_s \text{e}^{-ia(l_1q_x+l_2q_y+l_3q_z)}\times \...
R. Bhattacharya's user avatar
1 vote
0 answers
25 views

Finding the coefficients of a decomposition of complicated expression into products of special functions

I have the following expression which arises while studying minimal models in CFT. The 4-point amplitude of scalars is given by $$G(z,\overline{z}) = ((1-z)(1-\overline{z}))^{\frac{m-1}{m+1}}\mathcal{...
QFTheorist's user avatar
0 votes
1 answer
64 views

How to get the definite product of Sin squared? [closed]

I want to solve this for different values of D using mathematica. But I don't know how to write the program for this calculation. Please help. Edit: I tried to use the "Definite product" ...
Bibhash Das's user avatar
2 votes
2 answers
161 views

Show Factorial instead of Gamma in the result of Integrate

...
lotus2019's user avatar
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1 vote
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Complicated expressions involving hypergeometric functions

This is in relation to my last post which was unanswered but I think I may have found somewhat of a workaround. I have the following preliminary definitions ...
QFTheorist's user avatar
2 votes
1 answer
134 views

Simplification of integration of product of Bessel functions

I have been trying to evaluate the following integral involving modified Bessel functions: \begin{align} \int r I_1 (r) K_1(r) dr \end{align} This integral has an explicit expression given in the ...
Aaron's user avatar
  • 153
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0 answers
42 views

Manipulating Dirichlet characters and L functions

I read some basic knowledge about characters and L functions, and would like to play around with them in MMA. I tried to do the following things, but ending in minor success. (MMA notation used) ...
Po1ynomial's user avatar
5 votes
1 answer
120 views

Implementing recurrence relation for an integral

I would like to implement the following recurrence relation, $$I_{n+1}=-\log(2)I_n-\sum_{k=1}^n(-1)^k\left(1-\frac{1}{2^k}\right)\frac{n!\zeta(k+1)}{(n-k)!}I_{n-k}$$ with initial conditions, $$I_0=\...
bob's user avatar
  • 153
1 vote
4 answers
168 views

Evaluation of an integral using Mathematica or otherwise

I need to find a closed form (in terms of known functions) using Mathematica or otherwise of $$\Re\left(\int_{\frac{1}{2}}^{1}\frac{\tan^{-1}\left(\frac {1-x}{\sqrt{-i-x^2}}\right)}{\sqrt{-i-x^2}}\ dx\...
Max's user avatar
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11 votes
2 answers
398 views

Can you give a faster implementation with Mathematica for these q-analog functions?

Now(2023), Mathematica has some QFunctions for q-anlaog. See Official Documentation. e.g.: QPochhammer QFactorial ...
138 Aspen's user avatar
  • 897
3 votes
1 answer
100 views

Inconsistent behaviour of Integrate involving EllipticK?

...
Lacia's user avatar
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1 vote
1 answer
108 views

A request on the kernel of the function ”SpheroidalEigenvalue“ in Mathemitica 11

I want to understand the algorithm of the function "SpheroidalEigenvlaue" in Mathematica 11, especially the code of the following command. Then I will try to use this algorithm to reproduce ...
amon xu's user avatar
  • 31
0 votes
2 answers
120 views

Complex integral with branch cuts

I am struggling with a complex double integral with multiple branch cuts. Even the single variable complex integral I find quite complicated due to the branch cut and the special functions involved in....
Physics Moron's user avatar
1 vote
1 answer
119 views

Calculate $\lim_{n\to\infty} \left(\frac{e}{3}\right)^{3n} a_n^4$

I need to calculate the limit $$\lim_{n\to\infty} \left(\frac{e}{3}\right)^{3n} a_n^4 $$ where $a_n=\sum_{r=0}^{n}\left(\binom{n}{r}\binom{n+r}{r}\right)^2$ and $e$ is the natural base of logarithm. ...
Max's user avatar
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0 votes
0 answers
80 views

Is EllipticK defined correctly in Mathematica? [duplicate]

I needed to calculate F[a]=Integrate[1/(Sqrt[x^2+1]Sqrt[x^2+a^2]),{x,0,Infinity}] so I fed it to Mathematica. The result I got was: ...
Kari Karhi's user avatar
2 votes
0 answers
82 views

Mathematica 13.3 forgot integral defining Bessel functions [duplicate]

I installed the newest version of Mathematica 13.3.0.0 on Mac. It looks like it forgot how to compute a simple integral Integrate[Cos[n t - z Sin[t]], {t, 0, Pi}] ...
nukeyid's user avatar
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1 vote
0 answers
121 views

Mathematica function B

I came across this page on the Wolfram functions site: The conformal mapping from the triangle to the half plane. It describes a conformal map from an equilateral triangle to the upper half-plane ...
Gragarian's user avatar
0 votes
0 answers
86 views

How is the analytical continuation for the HurwitzZeta function implemented?

Following up on this question, I am trying to understand the implementation details of the HurwitzZeta[x,y] function in Mathematica, particularly when the first ...
stefan_chem's user avatar
1 vote
1 answer
215 views

Discrepancy with Hurwitz Zeta function

I've come across an issue while using Wolfram Mathematica that I don't quite understand. Consider the following symbolic computation: ...
stefan_chem's user avatar
5 votes
1 answer
131 views

Simplifying expression with BesselJ and BesselY and Gamma

I solved this ode by hand and got much simpler solution than Mathematica's. Both are correct. But I could not find a way to simplify Mathematica's solution to the simpler one. Could someone find a ...
Nasser's user avatar
  • 137k
3 votes
1 answer
163 views

No result from FoxH

Why the following code shows no result? FoxH[{{{0.5, 0.5}, {1, 1}}, {{-0.75, 0.75}, {0, 0.5}}}, {{{0, 1}, {0.5, 0.5}}, {{-0.75, 0.75}, {1., 0.5}}}, 1] Tested on <...
Dante's user avatar
  • 131
2 votes
0 answers
150 views

Closed form for a sum involving Bernoulli numbers

I need a closed form for the sum $$\sum_{n=1}^{\infty}\frac{(-1)^{n-1}(2^{2n}-1)\pi^{2n}B_{2n} {2n+4 \choose m}x^{2n+4}}{(2n)!}$$ where $0<x<1$, $B_{2n}$ denotes Bernoulli numbers , $m\in\mathbb{...
Max's user avatar
  • 115
6 votes
3 answers
251 views

How to prove an identity involving a hypergeometric function?

How to prove with the help of Mathematica the following statement? $$ {}_2F_1\left(\frac{2}{3},\frac{2}{3};\frac{5}{3};-1\right)=\frac{\pi -3 \sqrt{3} \log \left(\sqrt[3]{2}-1\right)-6 \tan^{-1}\left(\...
yarchik's user avatar
  • 17.8k
3 votes
1 answer
430 views

Evaluating ${}_5F_4\left(1,1,1,1,1;\frac{3}{2},2,2,2;-\frac{1}{4} \right)$

Using Mathematica, how can I find a closed-form expression (in terms of elementary functions) of $$ {}_5F_4\left ( 1,1,1,1,1;\frac{3}{2},2,2,2;-\frac{1}{4} \right ),$$ where ${}_5F_4$ represents the ...
Max's user avatar
  • 115
1 vote
1 answer
162 views

Evaluating $\int_{1/\phi^2}^{1}\frac{\log^2x}{1-x}\ \log^2\left(\frac{1}{x}\left(\frac{1-x}{1+x}\right)^2\right)\ \mathrm{d}x$

I want to calculate the following integral: $$I=\int_{1/\phi^2}^{1}\frac{\log^2x}{1-x}\ \log^2\left(\frac{1}{x}\left(\frac{1-x}{1+x}\right)^2\right)\ \mathrm{d}x$$ where $\phi=(\sqrt{5}+1)/2$ is the ...
Max's user avatar
  • 115
0 votes
1 answer
121 views

Inverse Laplace Transform of $\frac{\tan ^{-1}\left(\sqrt{s}\right)}{\sqrt{s}}$

How do I obtain the Inverse Laplace transform of the following expression? $$\frac{\tan ^{-1}\left(\sqrt{s}\right)}{\sqrt{s}}$$ Using Mariusz Iwaniuk recipes with Feynmann trick and "Mellin ...
Yaroslav Bulatov's user avatar
8 votes
2 answers
478 views

Why can't NSolve solve for the obvious zeros?

Bug introduced in 13.2 or earlier and persisting through 13.2.1 or later ...
Vancheers's user avatar
  • 736
2 votes
0 answers
65 views

How to verify the Riemann hypothesis up to a given height?

According to the Wikipedia article on Riemann hypothesis, The number of zeros of the zeta function with imaginary part between $0$ and $T$ is given by $$N(T)=\frac{...
japjap's user avatar
  • 21
5 votes
2 answers
344 views

Is there a way to convert functions to hypergeometric functions?

For example, I have a function f[x_] = x*Sqrt[1 - x^2] + ArcSin[x] Is there a way to convert such functions to (a sum of) hypergeometric functions? In this case $f$...
Andrew's user avatar
  • 2,513
0 votes
0 answers
32 views

Weird expression for function Series-Expansion with Gamma function for different values of gamma coefficient

I extract the function jin[r] by solving eqsynin, and then I develop the function's series (around zero) to generate an equation for m1in and m2in based on esyn and gamma, knowing that the function ...
Pantelis Ashikkis's user avatar
1 vote
2 answers
137 views

Solving an implicit equation involving Elliptic integral

I have the following equation: EllipticE[1/(1 + 16*(ja*me + jb*mo)^2)] + 16*(ja*me + jb*mo)^2*EllipticK[1/(1 + 16*(ja*me + jb*mo)^2)] == 0 I want to find values ...
Barry's user avatar
  • 113
6 votes
4 answers
406 views

Approximation of the Fabius function with a quotient of exponentials

Approximation of the Fabius function $f(x) = \text{FabiusF}[x+1]\cdot \text{HeavisideTheta}[1-x^2]$ - FabiusF[x] doesn't work in Wolfram-Alpha I am looking to figure out how well the displaced version ...
Joako's user avatar
  • 163

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