Questions tagged [special-functions]

Questions on the special mathematical functions implemented in Mathematica.

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3
votes
2answers
179 views

Is there a built-in for "circle function"?

Is there a built-in function in Mathematica for this function? \begin{equation} {\displaystyle \operatorname {Circ} (r)=\left\{{\begin{array}{rl}1,&{\text{if }}r<{1}\\{\frac {1}{2}},&{\text{...
2
votes
1answer
62 views

Verifying Summation form of Derivative of Hypergeometric1F1

First. Please read my code: ...
2
votes
0answers
71 views

How to do a fast numerical computation of an oscillatory integral including HeunC function using Mathematica?

I am trying to numerically compute the following integral in Mathematica ...
4
votes
2answers
96 views

Computing the sign of an expression

Can you please help with the following. I would like to compute the sign of an expression: ...
8
votes
1answer
310 views

why do these two Meijer G functions not cancel each other?

I encountered such expressions in Mathematica MeijerG[{{ }, {1, c + 1/2}}, {{0, c, c, c}, { }}, 1] + MeijerG[{{1}, {c + 1/2}}, {{c, c, c}, {0}}, 1] which in ...
3
votes
1answer
67 views

Too small to represent and Chop in inaccessible algorithm

Overview I want to apply a specified Chop function to every step in an function call in Mathematica (in my case LegendreP), ...
0
votes
2answers
73 views

Convert GiNaC harmonic polylogarithms to HPL packages' notation in Mathematica

A general output from GiNaC (https://www.ginac.de/) of harmonic polylogarithms is H(a,b,c...,x). We want to convert it to Mathematica format (https://www.physik....
1
vote
0answers
61 views

Plotting Radiation Pattern for Hermite-Gaussian beams for different modes

I want to plot the Radiation Pattern for Hermite-Gaussian beams (HG beam) as given in the Figures Ref Research Paper Fig 2, where u and v be the beam modes for the Hermite-Gaussian beams The coding ...
0
votes
0answers
68 views

How to simplifying the following integral that involves Bessel functions and Exponential integral function?

I have obtained the following as a solution of heat conduction equation of a semi-infinite model in cylindrical coordinates which is initially subject to non-homogenous initial condition and isolated ...
1
vote
1answer
88 views

How to simplify the following integral to be in terms of Bessel functions?

I have evaluated the following integration using Mathematica. I obtained a solution in terms of Meijer G function. I wonder if it can be simplified to be in terms of Bessel functions. ...
0
votes
0answers
52 views

How to evaluate the following integrals using Mathematica?

I have the following integrals obtained during solving heat diffusion equation for semi-infinite system that is subject to non-homogenous initial condition in Laplace domain. I want to simplify the ...
1
vote
3answers
229 views

How to prove the following integration identity?

I have the following integration that I want to evaluate it using Mathematica. ...
5
votes
1answer
116 views

How to show these two expressions are the same?

I think Can be simplified to $-x^2$ I tried Reduce, FullSimplify, FunctionExpand, and ...
1
vote
1answer
145 views

Spherical Harmonics Parity

In Mathematica's documentation, the Spherical Harmonics are said to be defined as follows, for $l \geq 0$: Furthermore, we know that $\cos(x)=\cos(-x)$, hence one can be led to believe that $Y_l^m(-\...
6
votes
1answer
121 views

Keeping Phase Factors in Sqrt

I am trying to plot certain holomorphic functions that contain square and higher roots. In the complex analysis sense, the function $f:z\mapsto z^\alpha$ for some $\alpha\in\mathbb C$ has a phase ...
2
votes
2answers
116 views

Plot[Zeta[x], {x, 2, 20}, ScalingFunctions -> "Log"] is not logarithmic [closed]

The code Plot[Zeta[x], {x, 2, 20}, ScalingFunctions -> "Log"] produces the following image, which is a plot of the ...
2
votes
1answer
71 views

Clebsch-Gordan coefficients: General Expression Does Not Match Specific Expression

The expression ClebschGordan[{2, 0}, {4, 0}, {2, 0}] yields the correct result of Sqrt[2/7]. However the expression ...
3
votes
2answers
472 views

Calculating a double integral

I want to calculate the following integral: $$\int^{10}_{0}\int^{\pi}_{0}\sqrt{(37-\frac{45\cdot37\cdot x^2}{74\cdot 150})^2\cdot \sin(t)^2-(40-\frac{27\cdot37\cdot x^2}{16\cdot 150})^2\cdot \cos(t)^2}...
0
votes
1answer
47 views

Integration of Bessel function including specific rings

Now, I am interested in the rings of Bessel function. So, basically, I used cylindrical coordinate to make it easy to figure out rings of Bessel function. ...
2
votes
1answer
112 views

Cannot understand the meaning of Derivative[1, 0][BesselK][-M, 2]?

When I do the following integration Integrate[(Log[x]/x)*x^M*Exp[-x-1/x],{x,0,\[Infinity]},Assumptions->Element[M,PositiveIntegers]] Mathematica return a very ...
10
votes
3answers
334 views

Numerically evaluating parameter derivatives of a hypergeometric function

I am unable to obtain the numerical value of the derivative of the hypergeometric function. Please note that the (2,4,0,0) is the derivative with respect to the first and second argument ...
1
vote
1answer
181 views

Mathematica definition of Hermite polynomials: fractional index

I am doing some calculations in Mathematica. I have solved a differential equation and noticed that the solution contains the Hermite polynomial $H$. The strange thing is that it appears as ...
2
votes
1answer
164 views

Integral for Bhattacharyya distance between two Cauchy distributions

I need to perform the following integral to calculate the Bhattacharyya distance between two Cauchy distributions: $$ I = \frac{\sqrt{b_+ b_-}}{\pi}\int_{-\infty}^{\infty}dx\,\frac{1}{\sqrt{\left[(x-1)...
0
votes
1answer
62 views

FindRoot[] with dependent interval [closed]

I would like to use FindRoot for a problem of this type: {f[x, y] == 0, g[x, y] == 0, 0 < y < x < 10} My problem is ...
8
votes
1answer
215 views

Solve`DirInf[] — Meaningful value or just a bug?

Bug introduced in 4.1 or earlier and fixed in 12.3 To the problem below, I get four independent, incomplete solutions, three in terms of Solve`DirInf[]. Since <...
1
vote
0answers
65 views

is the Fox-Wright Psi solution of the trinomial equation implemented in mathematica?

According to (13.4) in Belkic: the trinomial roots, their powers and logarithms from the Lambert series, Bell polynomials and Fox–Wright function https://d-nb.info/1176090631/34 the solution of ...
0
votes
0answers
22 views

Can't plot derivative of Hankel function [duplicate]

I am trying to plot the first derivative of $H^{(1)}_3 (ix)$ with respect to its argument $ix$, where $H^{(1)}_3 (ix)$ is the Hankel function of the first kind with order $3$, $x \in \mathbb{R}$ is a ...
0
votes
1answer
54 views

Problem evaluating numerical value of MeijerG[] function at some parameter?

I need to evaluate this sum: Where x = 33.6614 and $k,l,s$ are non-negative index of the three sum. For example $k,l,s=0,1,2,3,...$. Furthermore, $M$ is a positive integer that is $M=1,2,3,4...$ ...
0
votes
1answer
62 views

NSolve with Bessel and Hankel functions

I have a complicated function that I would like to solve for the roots. The function comprises Bessel and Hankel functions: ...
2
votes
2answers
275 views

Integral giving a Dirac delta

I have the following type of integral Integrate[ r BesselJ[n, a r] BesselJ[n, b r], {r, 0, Infinity} (where a and ...
3
votes
4answers
422 views

How to find the sum of that series related to Legendre functions of the second kind?

I mean $$\sum _{n=0}^{\infty } \frac{Q_n\left(\frac{\sqrt{2}}{2}\right)}{n+1}. $$ It's unclear to me whether the series under consideration converges. I have strong doubts concerning its closed form. ...
0
votes
2answers
40 views

Find the numerical value of the 2nd positive solution of a Bessel Function [closed]

A) Find the numerical value of the 2nd positive solution of $J_2 (3x)=0$ to at least 5 significant digits Note: $J_n(y)$ represents a Bessel function, which is written as ...
2
votes
1answer
145 views

How to simplify Elliptic integrals?

Context I would like to show that these 2 functions are identical ...
3
votes
0answers
66 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(...
0
votes
1answer
98 views

Partial derivative of an integral

I have the following function (it is the incomplete elliptic integral of first kind) $$ F(b,g) = \int_{0}^{b} \frac{dx}{\sqrt{(1-x^2)(1-gx^2)}} $$ I would like to compute $$\frac{\partial F}{\partial ...
1
vote
2answers
139 views

Why can't Mathematica evaluate this integral?

I want to work with the rectangle function, which I define by f[x_, m_] := Limit[1/((2*(x - m))^(2*k) + 1), k -> Infinity]; (I know that in theory I can use <...
4
votes
1answer
128 views

Integration of product of BesselJ and BesselY not giving correct results

I am trying to integrate a product of Bessel functions as shown below. Where z is real valued and positive. The integration yields MeijerG functions. Taking a ratio of the derivative of the MeijerG ...
2
votes
4answers
113 views

Integration of LegendreP

I am trying to integrate a product of 2 Legendre polynomials as follows: Integrate[LegendreP[1, x] LegendreP[2l+1, x], {x, -1, 1}] I get the result: ...
0
votes
2answers
65 views

expansion of hypergeometric series [closed]

I am completely new to mathematica. I am not sure how to calculate the expression 2F1. Is it possible to a closed form solution in terms of z of the hypergeometric series 2F1(-a, N/2-a, N/2; z) and ...
1
vote
0answers
44 views

Adding a module to teach Mathematica the Laplace transform of particular Mittag-Leffler functions

Mathematica 11.3 is not aware of a useful Laplace transform LaplaceTransform[t^(-a) MittagLefflerE[a, a, t^a], t, s] which for ...
0
votes
1answer
56 views

Plotting a Legendre Polynomial

currently I am trying to plot for the Legendre polynomial $P_n(x)$ for $n=0,1,2,3,4$ and $-1 \leq x \leq 1$ . To plot for it, I wrote the following code: ...
3
votes
1answer
111 views

Why does Integrate get this wrong?

Why does Integrate get this wrong? ...
1
vote
1answer
79 views

Optimizing a series expansion for high order in $x$

I would like to expand the following function at $x \sim 0$ up to some high $x_\text{max} = \Delta_\text{max}$: $$16 \sum_{\Delta=1}^{\Delta_\text{max}} \sum_{s=0}^{\Delta-2} f_{\Delta,s} \frac{(s+\...
3
votes
2answers
176 views

Why can't I evaluate this integral and obtain a closed-form solution?

I have the following spherical density distribution: $\rho(x, z) = \frac{1}{\sqrt{x^2 + z^2}\left(1+\sqrt{x^2+z^2}\right)^2}$ which I have broken into a "line of sight" dimension $z$ and a &...
0
votes
0answers
42 views

Series expansion of hypergeometric function with two variables

I have a function $g(x,y)$ that contains a product of hypergeometric functions, both involving the variables $x$ and $y$. I try to do a series expansion in the two variables as recommended in this ...
0
votes
2answers
221 views

Can the error function be expressed in terms of other special functions?

I obtained with Mathematica some results written in terms of the error function Erfi[x]. Is there is a way to transform the error function into other special functions e.g. Bessel functions or ...
4
votes
1answer
104 views

Hypergeometric Function and Elliptic Integral

In the wiki page, complete elliptic integrals of first and second kind are related to the Hypergeometric function via: $$ K(k)=\frac{\pi}{2} \mathstrut_{2}F_{1}(\frac{1}{2},\frac{1}{2};1;k^{2})\\ E(k)=...
2
votes
0answers
54 views

Exact value involving ProductLog

By manual manipulation, it's not too difficult to show that the value of the following expression is exactly $1.$ How to convince Mathematica to do the simplification? ...
1
vote
1answer
86 views

Precompute Functions for use in NDSolve [closed]

I have a very complicated pair of functions $F(x,y), G(x,y)$ that are used in a differential equation $0 = x'' + F(x,y)$, $0 =y '' + G(x,y)\,. $ $F$ and $G$ are roughly sums of products of the Bessel ...
3
votes
1answer
78 views

Error in Creating Orthogonal Polynomials

I'm trying to create my own set of polynomials orthogonal to weight $w(x)=x^{14}$ on $[-a,a]$. My code: ...

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