Questions tagged [special-functions]

Questions on the special mathematical functions implemented in Mathematica.

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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
66 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
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9 votes
0 answers
415 views

Does anyone know what is this number? [migrated]

Answering this question, @user64494 came with a very nice answer. My problem is that the front factor is HypergeometricPFQ[{1/2, 1/2, 1, 1}, {3/4, 5/4, 3/2}, -1] ...
Claude Leibovici's user avatar
5 votes
4 answers
363 views

Is it possible to have the asymptotics of this function?

Working the problem of $$I_n=\int_0^1 \frac{\tan ^{-1}\left(x^n\right)}{\sqrt{1-x^n}} \, dx$$ which I have not be able to compute with Mathematica. A tedious work gave the result $$I_n=\frac{\sqrt{\pi ...
Claude Leibovici's user avatar
2 votes
1 answer
106 views

WeberE definition from Wikipedia doesn't match the definition from Wolfram Language?

According to Wikipedia - Struve function: Relation to other functions the WeberE function should return this, if the first argument is a negative integer: $$ \...
axelclk's user avatar
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2 votes
1 answer
110 views

Is it possible to express $\text{Li}_2(-\frac18(-i+\sqrt{15})+\text{Li}_2(\frac18 (i+\sqrt{15})$ as an explicit real expression (not numeric)? [closed]

I obtain this expression in my calculations, and numerically I am sure that it is a real number. ...
MsMath's user avatar
  • 175
0 votes
1 answer
58 views

HypExp package throwing errors when loading

I am trying to load the package HypExp by using the instructions given on the site. The package throws an error ...
QFTheorist's user avatar
2 votes
2 answers
163 views

Are there any commands besides NSolve for solving equations which involve product logarithm?

I have two complicated equations and NSolve cannot give the complete set of solutions: ...
A novice's user avatar
  • 405
0 votes
1 answer
54 views

Finding value of a function at limit zero

I want to evaluate the below function at limit zero. I tried with a function which has double differentiation for which it is giving me a finite value. For triple differentiation it is consistently ...
Anshul Bokade's user avatar
1 vote
0 answers
48 views

Is there a way of plotting elliptic rational function offline?

I'm studying the elliptic filter design and trying to plot the elliptic rational function, which is part of the filter equation. I have found out that there is an online way of plotting with the help ...
metroidman's user avatar
5 votes
1 answer
245 views

Quantum Field Regularization: Choice between ExpIntegralE and Gamma for Casimir Effect

For the Casimir effect with exponential regularization, we compute the vacuum energy between the plates (somewhat simplified) with: $$\omega = c\ \sqrt{q^2+k_z^2}$$ $$ E = \sum_{k_z=n\pi/a} 2 \int\...
Jos Bergervoet's user avatar
1 vote
0 answers
89 views

Differentiation of Parabolic Cylinder Functions and Hermite Polynomials w.r.t to Order

When I evaluated (ver. 12.3) parabolic functions below in the vanishing order limit: $u \rightarrow 0$, I got the following: ...
user91411's user avatar
  • 400
2 votes
0 answers
62 views

Abnormally long computation time using AppellF1 function

I am trying to use the AppellF1 function in Mathematica 13.3.1 on an Ubuntu machine with an i7 13700. The inbuilt function seems to be much slower in some cases ...
Arav BJ's user avatar
  • 66
2 votes
0 answers
99 views

What is the exact formula Mathematica uses for Riemann zeta function? [closed]

I'm trying to find the formula of the zeta function used to plot this graph: This is taken from Wikipedia Riemann Zeta Function. It was created in Mathematica by ...
zeynel's user avatar
  • 121
0 votes
0 answers
59 views

The Fourier-Legendre expansion of a plane wave

I want to perform a Fourier-Legendre expansion on the spatial part of a plane wave. $$ \mathrm{e}^{\mathrm{i}(\boldsymbol{k} \cdot R-\omega t)}=\mathrm{e}^{\mathrm{i}(k R \cos\theta-\omega t)} $$ $$ \...
lotus2019's user avatar
  • 2,131
2 votes
1 answer
68 views

FoxH function format implementation issue

I am trying to code a particular Fox-H function in Mathematica, $H_{2,2}^{2,0}\left(z\left| \begin{array}{c}(1,1),(1,\alpha)\\(1,1),(1,2)\\\end{array} \right.\right)$. I've been trying to use the ...
Math_fan_123's user avatar
2 votes
1 answer
67 views

Plotting real spherical harmonics with SphericalPlot3D - odd functions overlap themselves, even functions ok [duplicate]

I want to reproduce plots of real spherical harmonics as shown on Wikipedia. I used: ...
lixpas's user avatar
  • 65
3 votes
2 answers
159 views

The conversion equation between BesselI and BesselJ

The conversion equation between BesselI and BesselJ is as follows: $$ I_v(x)=J_v(i x)/i^v $$ $$ I_{-v}(x)=i^{-v} J_{-v}(i x) $$ ...
lotus2019's user avatar
  • 2,131
2 votes
1 answer
42 views

Expressions in PlotLegends

When the order of the Bessel function is negative, it will automatically be evaluated in PlotLegends. ...
lotus2019's user avatar
  • 2,131
2 votes
2 answers
139 views

Circular function with arbitrary radius and center

I'm looking for an implementation in Mathematica of the generalized circular function $\text{circ}(x, y; R, c)$ of radius $R$ and centre $c=(c_1,c_2)$ such that $$\text{circ}(x, y; R, c)=\begin{cases}...
Noobgrammer's user avatar
3 votes
2 answers
72 views

Using FindSequenceFunction to solve the integral of the product of Legendre polynomials and power functions

The integral of the product of Legendre polynomials and power functions: $I=\int_{-1}^1 x^n \mathrm{P}_l(x) \mathrm{d} x$ The calculation result from the textbook is: $$ \begin{aligned} I & =0 \...
lotus2019's user avatar
  • 2,131
1 vote
1 answer
85 views

How to integrate Legendre polynomials with parameters?

The orthogonality of Legendre polynomials: $\int_{-1}^1 \mathrm{P}_l(x) \mathrm{P}_k(x) \mathrm{d} x=0, \quad k \neq l$ But ...
lotus2019's user avatar
  • 2,131
1 vote
1 answer
108 views

Find the range of Legendre polynomials

The range of Legendre polynomials in the Reals domain is [-1, 1]. How can we calculate it using FunctionRange or other MMA code? ...
lotus2019's user avatar
  • 2,131
0 votes
0 answers
65 views

Plot of a complicated function under double summation

I wrote the following inputs in mathematica to plot the following expression: ...
R. Bhattacharya's user avatar
1 vote
2 answers
141 views

Mathematica cannot solve this complicated integration

first-time here. I am trying to use Mathematica to evaluate a solution from Duhamel's principle, the integration looks like $\int^t_0\frac{e^{-ks-\frac{r^2}{2(2Ds+\sigma^2+2D_pt)}}}{2Ds+\sigma^2+2D_pt}...
CalBZ's user avatar
  • 11
4 votes
2 answers
168 views

How to compile inverse error function?

I have the following code: ...
John Taylor's user avatar
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1 vote
0 answers
84 views

Strange behavior of error function series expansion at infinity

Evaluating Normal[Series[1/ (1 + xi Erf[xi]), {xi, ∞, 2}]] Normal[Series[1/( xi (1/xi + Erf[xi])), {xi, ∞, 2}]] gives ...
ssskkkky's user avatar
1 vote
1 answer
101 views

Limit of Hypergeometric Functions

Mathematica (12) seems to have a lot of trouble with taking limits involving hypergeometric functions. A simple example I am interested in is the following ...
Rudyard's user avatar
  • 471
5 votes
1 answer
334 views

Solving third order DE from fluid dynamics

I am trying to use DSolve to solve a differential equation from G. Batchelor: An Introduction to Fluid Dynamics, eq. 5.12.4: [...] equation now reduces to $$\boxed{...
simon's user avatar
  • 47
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
  • 17.1k
1 vote
1 answer
131 views

What formula does Mathematica use for PolyGamma function of complex order?

For instance, PolyGamma function in Mathematica gives different values than the similar Psi function in Maple, which uses the ...
Anixx's user avatar
  • 3,585
0 votes
1 answer
151 views

PartitionsP for non-integer arguments [closed]

Is there a reason why the function PartitionsP[] does not return anything for non-integer arguments, although there is a way to calculate using Rademacher’s ‘exact’ ...
ftel's user avatar
  • 3
1 vote
0 answers
86 views

Heaviside function in NDSolve

I have: ...
user avatar
4 votes
1 answer
139 views

Strange result simplifying higher order BesselJ [duplicate]

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
63 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
  • 173
1 vote
2 answers
133 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
110 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
52 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
  • 1,343
1 vote
0 answers
72 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
117 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
70 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
87 views

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

...
Gopal Verma's user avatar
  • 1,055
0 votes
0 answers
81 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
82 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
  • 1,846
1 vote
1 answer
126 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
  • 960
0 votes
1 answer
77 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
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
0 votes
1 answer
100 views

Dirac Delta does not converge

I have trouble evaluating a simple integral in Mathematica. I have the code: ...
Nitaa a's user avatar
  • 738
1 vote
0 answers
37 views

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
  • 738
0 votes
0 answers
134 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
  • 585

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