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Questions tagged [special-functions]

Questions on the special mathematical functions implemented in Mathematica.

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Integration involving Piecewise function and DiracDelta function

I want to calculate an integration, which reads where $\delta\left(q_{23}^{01}\right)=\delta\left(1+q_1-q_2-q_3\right)$ and $\mathrm{min}(1,q_1,q_2,q_3)$ means the minimum of $(1,q_1,q_2,q_3)$. What ...
so_sure's user avatar
  • 391
3 votes
3 answers
136 views

Issue in HypergeometricPFQ function:

I have a solution from integral: A = Integrate[x^n*(1 + x)^n*Exp[-n*x^2] /. n -> 1, {x, 0, Infinity}] //Expand (*1/2 + Sqrt[π]/4*) %//N (*0.943113*) Then I ...
Mariusz Iwaniuk's user avatar
1 vote
1 answer
61 views

Mathematica can't handle expressions with Bessel functions in the limit of a large argument [duplicate]

Consider the following function: ...
John Taylor's user avatar
  • 5,863
0 votes
1 answer
91 views

Question about System`MeijerGDump`*

Steps Start a fresh kernel, then copy the below sentence, run it. ...
138 Aspen's user avatar
  • 1,541
2 votes
2 answers
116 views

How to transform this combination of $ _2F_1 $?

Following my previous question How to transform this MeijerG $G_{3,3}^{2,3}$ into a hypergeometric function $ _2F_1 $, I transformed a MeijerG function $G_{3,3}^{2,3}$ below into a combination of ...
Gallagher's user avatar
  • 859
0 votes
1 answer
35 views

How to transform this MeijerG $G_{3,3}^{2,3}$ into a hypergeometric function $ _2F_1 $

I want to transform the MeijerG function $G_{3,3}^{2,3}$ below into a hypergeometric function $ _2F_1 $: $$G_{3,3}^{2,3}\left(\frac{\kappa -z}{z},\frac{1}{2} | \begin{array}{c} \frac{1-\kappa }{2},2-\...
Gallagher's user avatar
  • 859
2 votes
2 answers
147 views

How to calculate this improper integral?

How to calculate the Integral involving MarcumQ function whose Integral interval is (0, +Infinity)? Any help (code or reference) would be greatly appreciated. ...
138 Aspen's user avatar
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2 votes
1 answer
65 views

Bugs with `Integrate` when dealing with complex situation?

I'm trying to calculate $\int_0^{\infty } \exp \left(t \left(-x^3+(1+i) x\right)\right) dx$ with mathematica, and different assumptions on t give different results: ...
Jie Zhu's user avatar
  • 1,761
6 votes
3 answers
444 views

Why do I get different results for the products of two identical expressions?

I have two expressions with the Gamma function that are identical: ...
Vaclav Kotesovec's user avatar
1 vote
1 answer
51 views

Laplace transform of special function

The Confluent hypergeometric function of first kind (aka Kummer's function) is defined as $${\mathbf{M}}\left(a,b,z\right)=\frac{1}{\Gamma\left(a\right)\Gamma\left(b-a% \right)}\int_{0}^{1}e^{zt}t^{a-...
K.K.McDonald's user avatar
4 votes
1 answer
168 views

Calculation time too long for FoxH

Clear["Global`*"]; FoxH[{{}, {}}, {{{0, 0.5}, {-3, 1}}, {}}, 0.2] Related: No result from FoxH
138 Aspen's user avatar
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2 votes
4 answers
118 views

Specify the Method for `NIntegrate` to evaluate a integral of special functions

I have a special function as $$f(x)=\frac{\Gamma \left(\frac{5}{3}\right) \, _2F_3\left(\frac{5}{12},\frac{11}{12};\frac{1}{3},\frac{1}{2},\frac{5}{6};-\frac{4 x^6}{729}\right)}{\pi }-\frac{x^2 \, ...
Jie Zhu's user avatar
  • 1,761
3 votes
2 answers
194 views

How to calculate the PDF of product of two random variables from generalized gamma distributions?

Namely, we want to find the explicit formula of PDF of double Generalized Gamma distribution. ...
138 Aspen's user avatar
  • 1,541
0 votes
2 answers
99 views

Computing a highly oscillating Fourier Transform

I'm trying to compute the Fourier Transform of a diverging and oscillating function:: ...
Syrocco's user avatar
  • 139
1 vote
0 answers
74 views

Simplification of expression with PolyLog[4,2]

I am trying to simplify the following expression. expr=1/2 \[Pi]^2 Log[2]^2-2/3 I \[Pi] Log[2]^3-4 PolyLog[4,2]+7/2 Log[2] Zeta[3] This ...
BabaYaga's user avatar
  • 1,897
1 vote
1 answer
63 views

All possible tuples satisfying conditions

Let $\beta = (\beta_1, \beta_2, \ldots)$ be any given sequence of non-negative integers with all but finitely many $\beta_i$ zero. I want to collect all possible tuples $\beta^{\prime}= (\beta_1^{\...
Anantadulal paul's user avatar
0 votes
3 answers
96 views

Function Definition without Arguments

I need a simple function that can be called without any arguments, myFunction[ ], that will create an element of a data structure. However, each element in the data structure has a "label" ...
ZGurdal's user avatar
2 votes
1 answer
85 views

Expressing elliptic $\lambda^*(r)$ with radicals [closed]

For some small $r$, the values of $\lambda^*(r)$ can be expressed symbolically, see Elliptic Lambda Function equations $(11)-(34)$ Is it possible to get some of these expressions using Mathematica? I ...
Vaclav Kotesovec's user avatar
3 votes
0 answers
53 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
80 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
5 votes
4 answers
407 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
109 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
  • 319
2 votes
1 answer
119 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
  • 195
0 votes
1 answer
62 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
164 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: ...
user avatar
0 votes
1 answer
55 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
52 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
253 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
95 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
73 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
119 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
63 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
  • 2,151
2 votes
1 answer
80 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
165 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
  • 2,151
2 votes
1 answer
43 views

Expressions in PlotLegends

When the order of the Bessel function is negative, it will automatically be evaluated in PlotLegends. ...
lotus2019's user avatar
  • 2,151
2 votes
2 answers
140 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
73 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
  • 2,151
1 vote
1 answer
91 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
  • 2,151
1 vote
1 answer
111 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
  • 2,151
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
147 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
171 views

How to compile inverse error function?

I have the following code: ...
John Taylor's user avatar
  • 5,863
1 vote
0 answers
88 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
113 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
351 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
  • 19.7k
1 vote
1 answer
138 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
  • 3,610
0 votes
1 answer
152 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
87 views

Heaviside function in NDSolve

I have: ...
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