Questions tagged [special-functions]

Questions on the special mathematical functions implemented in Mathematica.

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1answer
70 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 ...
1
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1answer
42 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
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1answer
82 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 ...
1
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2answers
84 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 <...
3
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2answers
134 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 &...
45
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9answers
11k views

About multi-root search in Mathematica for transcendental equations

I have some questions for multiroot search for transcendental equations. Is there any clever solution to find all the roots for a transcendental equation in a specific range? Perhaps ...
0
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1answer
39 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., ...
1
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1answer
55 views

Precompute Functions for use in NDSolve

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 ...
0
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2answers
47 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
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0answers
29 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
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1answer
42 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
98 views

Why does Integrate get this wrong?

Why does Integrate get this wrong? ...
7
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3answers
417 views

Integrating a real function I get a complex value, while after variable transformation the result is real. Bug?

I have the following integral: Integrate[1/Sqrt[0.7 + 0.3*(1 + z)^3], {z, 0, ∞}, Assumptions -> z ∈ Reals] >> -3.36354 - 3.85013 I The output is complex, ...
18
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1answer
4k views

Creating a Mathieu stability diagram

I am attempting to re-create a Mathieu stability diagram like the one shown here: I expected that I could use MathieuC to generate this graph by assuming that ...
0
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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 ...
0
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2answers
177 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
53 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
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0answers
49 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? ...
3
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1answer
69 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: ...
0
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1answer
68 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? ...
6
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2answers
967 views

Regarding obtaining a series expansion at infinity

I am trying to obtain an approximate expression for the behaviour of the following function for large $x$ $$F(x)=x\frac{ \text{erf}(x)^2}{\text{erf}(2 x)}$$ I know that the $\lim_{x\rightarrow\infty}...
0
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1answer
56 views

How to DSolve this differential equation?

How can I find h[r] from this equation? I need h[r] or its r derivative. ...
3
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1answer
40 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: ...
0
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3answers
172 views

Solving equations involving integrals

I need to find the value of $z$ for a particular value of $D_c$ (eg. $500$), but $z$ is inside an integral, and I'm not able to use Solve since the integral is ...
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
votes
2answers
774 views

Exact solutions of a linear second order differential equation

DSolve[ y''[x] + (a - (b x^2 - c) x^2) y[x] == 0, y[x], x] I couldn't solve this equation, please help
4
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1answer
264 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_{...
33
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1answer
700 views

Is MathieuC for moderately large imaginary arguments broken?

Bug introduced in 3.0 and persisting through 12.0 (reported as CASE:3208982) I'm trying to plot MathieuC[-3,0.3,I x] for $x\in[0,10]$, and here's what I get even ...
8
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0answers
167 views

SiegelTheta throws errors from calling Range with complex arguments

Bug introduced in 8.0 or earlier and persists through 12.1 or later This may or may not be related to the bug reported in this question. I was trying to verify the results of this challenge over on ...
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0answers
66 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}...
2
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2answers
96 views

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

Here are 2 examples I have examined. 1. Assumptions in Integrate. ...
0
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1answer
61 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 ...
1
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1answer
46 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 ...
-1
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1answer
59 views

`Reduce` with an inequality that involves `ProductLog` is running forever

I'm trying to make a comparison between two sides, one of which involves Lambert $W$ function. See my code: ...
1
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3answers
259 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 <...
6
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2answers
190 views

Erf function discrepancy

I have very strange problem involving $\operatorname{erf}$ functions: ...
-2
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2answers
125 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} $ ...
4
votes
1answer
136 views

How can I inform Mathematica of an identity concerning Bessel functions?

I am doing some analytical work that includes the integral of $e^{i(n\, t - x \sin t)}$. I know the result of this integral is a Bessel function. $$J_n(x)=\frac1{2\pi}\int_{-\pi}^\pi e^{i(x\sin\tau-n\...
2
votes
1answer
93 views

Simplifying this expression with Gamma

The Gamma function has the property that $\Gamma(z+1)=\Gamma(z)z$, so this expression: ...
0
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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, ...
0
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2answers
114 views

Boundary Condition for Elliptic Integral $|k| > 1$

I was calculating a problem and came across elliptic integrals (complete, first and second kind) no problem here as I have dealt with these before. Once I achieved my final solution I moved the ...
4
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1answer
184 views

Elliptic integrals

In trying to reproduce results from one paper I stumbled upon a problem with definition of some elliptic integrals (this is my guess of what could be the problem). I will first present in a simplified ...
0
votes
1answer
131 views

Using Assumptions in Expressions that Evaluate to be Elliptic Integrals

I have the following integral that I am trying to evaluate in Mathematica: $\int \sqrt{\alpha + m g l \cos(q)} dq$. If $\alpha > m g l$, then the result is a complete elliptic integral of the ...
0
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1answer
41 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 ...
9
votes
3answers
368 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 ...
0
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0answers
52 views

NIntegrate with highly oscilatory result

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

Comparing two equivalent definite integrals

Reading this question on Math.SE, I tried the following Mathematica instructions ...
3
votes
1answer
277 views

Find RSolve solution reflecting special structure of DifferenceRoot[Function[{\[FormalY], \[FormalN]}

RSolve yields a large (multi-page) solution (LeafCount=25891) containing a number of 7F6 (and higher) hypergeometric functions when applied to ...
0
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0answers
47 views

Indefinite Integral not Solving

I'm tring to solve the Indefinite Integral of the function: ...
1
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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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