Questions tagged [special-functions]

Questions on the special mathematical functions implemented in Mathematica.

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48 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 ...
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1answer
31 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 $M=1,2,3,4...$ ...
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1answer
44 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: ...
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0answers
39 views

How to find the equivalence Matlab code for Mathematica Meijer-G function in this case? [closed]

During my calculation, I end up receiving a strange function called the meijerG function with the Mathematica syntax ...
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0answers
38 views

Calculating a particular MeijerG function [closed]

I want to calculate this following MeijerG function...I used this command MeijerG[{{1/2 - s, 1 - s}, {-s}}, {{0}, {-s, 1/2}}, Abs[x]^2] but Mathematica can not ...
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1answer
87 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 ...
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4answers
376 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. ...
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2answers
37 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 ...
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2answers
88 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 <...
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1answer
97 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 ...
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1answer
44 views

How to programmatically Evaluate Notebook (all cells in an open notebook) rather than from Evaluation menu? [duplicate]

I am working on a DockedCells toolbar for common tasks like Save, ClearAll, scrolling, etc., ...
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2answers
50 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 ...
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0answers
32 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 ...
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1answer
44 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: ...
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1answer
102 views

Why does Integrate get this wrong?

Why does Integrate get this wrong? ...
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1answer
43 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+\...
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2answers
166 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 &...
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0answers
36 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 ...
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2answers
184 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 ...
3
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1answer
57 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)=...
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0answers
50 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? ...
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1answer
71 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 ...
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1answer
71 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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1answer
69 views

How to Integrate this expression?

All the parameters ($r, b,$ and $q$) are real and positive. Is it possible to do the below integration? ...
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1answer
57 views

How to DSolve this differential equation?

How can I find h[r] from this equation? I need h[r] or its r derivative. ...
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1answer
42 views

Non-Convergence In Creating Legendre Series

I'm trying to use Mathematica to create a Legendre-Fourier series using this Wikipedia article. Here is my code: ...
2
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1answer
50 views

Evaluate numerically derivatives of hypergeometric functions

I would like to evaluate numerically the coefficients of a series expansion. This is usually straightforward to do, however in this case I encounter terms of the following type: $$^{\phantom{0}}_2F_1^{...
4
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1answer
268 views

Evaluating a hard integral related to the two-fluid model

The following definite integral describing the density of the normal part of a superfluid equals to $$ \int_0^\infty dx\, x^4\, \frac{e^{x^2+a}}{\left(e^{x^2+a}-1\right)^2} = \frac{3\sqrt{\pi}}{8}Li_{...
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0answers
67 views

General Theta Functions in Mathematica?

Given a positive-definite or negative-definite matrix $A$ of size $r \times r$, one can define the corresponding theta function $$f_{A}(q) = \sum_{v \in \mathbb{Z}^{r}} q^{v^{t}Av} = 1 + a_{1}q + a_{2}...
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1answer
62 views

Mathematica Not Evaluating Small Input

So I'm a bit confused. I asked Mathematica to evaluate PolyLog[3, -4.900612445719819`*^-15 + 8.488109744191103`*^-15 I] and it refused to do so I assume it has ...
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1answer
47 views

Round-off error

Assuming $0<q<1$. I built these two functions A[k_,q_]:=Sum[PDF[BinomialDistribution[i, q], k]*PDF[ZipfDistribution[n], i], {i, k, Infinity}] and ...
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3answers
265 views

Problems with solutions involving Lambert W function of transcendental equation

Solve and Reduce fail here with rational parameter l, but succeed when I plug in a value <...
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2answers
126 views

Plotting a hypergeometric function

I am faced with the following expression $$ -\frac{(1 - a x^{2})^{b/2}}{b} {{}_2F_1} (1, \frac{b}{2}; \frac{c}{2}; 1 - a x^{2}) = - p t $$ where $ a, b, c, p $ are constant values. Also, $ {{}_2F_1} $ ...
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1answer
98 views

Simplifying this expression with Gamma

The Gamma function has the property that $\Gamma(z+1)=\Gamma(z)z$, so this expression: ...
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0answers
54 views

Analytic continuation of the prime zeta function

I want to plot the real part of the prime zeta function over the imaginary axis. This should be doable since the prime zeta function has an analytic continuation up to the imaginary axis. However, ...
2
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2answers
98 views

Different results with or without Assumptions in Integrate for an elliptic integral

Here are 2 examples I have examined. 1. Assumptions in Integrate. ...
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1answer
43 views

How to find expansion coefficients in Fourier-Legendre

I am trying to find the coefficients for the Fourier-Legendre expansion of a potential. My goal was to obtain the coefficients as expressions in terms of x and y. I followed the example given on the ...
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0answers
52 views

NIntegrate with highly oscilatory result

I want to evaluate a function defined by the following numerical integral: ...
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1answer
38 views

Comparing two equivalent definite integrals

Reading this question on Math.SE, I tried the following Mathematica instructions ...
9
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3answers
374 views

The numerics of FresnelS[]

Bug introduced in 12.1 or earlier and persisting through 12.1.1 or later [CASE:4615361] Note: A worse problem existed in 12.0 for inputs greater than 8 and of ...
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0answers
48 views

Indefinite Integral not Solving

I'm tring to solve the Indefinite Integral of the function: ...
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0answers
31 views

Taking the limit of the derivatives of the Beta function [closed]

I tried using Limit[D[Beta[a,b],{a,1},{b,1}],{a -> 1},{b -> 1}] and Limit[D[Beta[a,b],{a,1},{b,1}],{a , 1},{b , 1}] but it ...
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1answer
76 views

Is it possible to have a complex nu in ParabolicCylinderD?

I'm trying to recreate a graph from a paper, and the function that I'm plotting involves three separate parabolic cylinder functions that all have a complex Nu value. After getting it all typed out, I ...
3
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1answer
41 views

How can I remove vanishing derivatives of hypergeometric functions?

I have a lot of expressions containing derivatives of hypergeometric functions of the sort: $$_2F_1^{(0,0,1,0)} \left(\frac{1}{2} , 1 ; \frac{3}{2} ; 0 \right). \tag{1}$$ The last argument is always $...
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0answers
68 views

Strange behaviour with EllipticTheta functions

For some time I have been using EllipticTheta[3,-,-] functions, and to me it seems that Mathematica is really not handling them in the best way. Below are two of ...
2
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1answer
98 views

Numerical integration of MeijerG function with variables

I applied the integration, How to solve the function which is mentioned below. here, 'theta' is the only variable. q13, q1,q2,p,qi1 are all variables will take value from the loop ranges from 0 to 2, ...
2
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2answers
179 views

How to solve a nonlinear second order ODE

I want to solve this equation y''[x] + a + b y[x] + c y[x]^2 == 0, y[∞] == 0, y'[∞] == 0 where a, ...
0
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0answers
55 views

Why Mathematica doesn't give output for Integration of Special Function

I am trying to Integrate the expression given below. But, everytime I am not able to integrate it. Tried a lot but it's not working. Please any suggestion will help me a lot. I am trying to integrate ...
6
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2answers
384 views

Difficult Numerical Integral with special functions

Context I am trying to calculate some transport coefficients for a heat equation in confinement. The Boundaries are in the $x$ direction, and $y$ represents the parallel directions. This function ...
6
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2answers
102 views

Simplifying expression containing Gamma

According to this site this expression involving Gamma is valid: $$\prod_{k=0}^{n-1}\Gamma\left(\dfrac{k+z}{n}\right) = n^{\frac{1}{2}-z}(2\pi)^{\frac{n-1}{2}}\...

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