Questions tagged [special-functions]

Questions on the special mathematical functions implemented in Mathematica.

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61 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
40 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
52 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
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1answer
110 views

Why does Integrate get this wrong?

Why does Integrate get this wrong? ...
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1answer
59 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
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 &...
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0answers
39 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
208 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
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1answer
93 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
52 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
81 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
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1answer
75 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
71 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
59 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
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: ...
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1answer
73 views

Understanding the output of a calculation

I have a simple question about the output of a calculation. I have written a function which is a sum of 3j-symbols and Clebsch-Gordan coefficients in which I sum over various indices. ...
2
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1answer
67 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
277 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
72 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
69 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
48 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
331 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
132 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
131 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
58 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, ...
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2answers
106 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
52 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 ...
4
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3answers
249 views

Constant curvature surfaces. Revolution of the graphs of solutions to a nonlinear differential equation

I have the following differential equation: $$\pi\cdot\text{y}(x)^2=\sqrt{1+\text{y}'(x)^2}\tag1$$ With the initial condition $\text{y}(0)=1$. Now, I want to plot the solution in order to obtain 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: ...
1
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1answer
41 views

Comparing two equivalent definite integrals

Reading this question on Math.SE, I tried the following Mathematica instructions ...
10
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3answers
401 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
53 views

Indefinite Integral not Solving

I'm tring to solve the Indefinite Integral of the function: ...
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0answers
33 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
94 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
58 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 $...
1
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1answer
87 views

Problem plotting expression involving Generalized hypergeometric functions $_2F_2 \left(.,.,. \right)$

I'm trying to plot a graph for the following expectation $$\mathbb{E}\left[ a \mathcal{Q} \left( \sqrt{b } \gamma \right) \right]=a 2^{-\frac{\kappa }{2}-1} b^{-\frac{\kappa }{2}} \theta ^{-\kappa } \...
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0answers
83 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
104 views

Riemann Prime Counting Function correction/pairing terms by Mathematica

Riemann Prime Counting Function: $$f(x)=\operatorname{li}(x)-\sum_\rho\operatorname{li}(x^\rho)-\ln 2+\int_x^\infty \frac{\mathrm dt}{t(t^2-1)\ln t}$$ The second correction/paring terms: $$\sum_\rho\...
2
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1answer
113 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
215 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
57 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
402 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
108 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}}\...
14
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4answers
819 views

When is a symbol a Symbol? Is there an easy Mathematica way to test if an object is a symbol sort of like a SymbolQ?

Yes I know there is no built-in native function called SymbolQ (but JavaScript does). However, could one be simulated to work for most cases? I often rely on ...
4
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1answer
120 views

The NIntegrate cannot give a correct answer by Mathematica

Here I want to compute this integral using the Mathematica $$\int_0^\infty \int_{-\pi}^{\pi}s^{\frac{1}{2}+2} \exp (-s) \cos \left(\frac{f}{2}\right) \exp (-i k s) \exp \left(-\frac{\sqrt{8 s u}}{\cos ...
2
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1answer
164 views

Plotting Stability of damped mathieu equation

I am trying to create the stability diagram of the damped Mathieu equation using Mathematica.The Mathieu equation is $$D(y)+(a-2q \cos(2t))y=0$$ where D(y) is the ...
3
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2answers
281 views

Code that produces plot in V5 doesn't work in later versions

I have problem in plotting Integral function. I can compute/plot the graph of this integration below in Mathematica 5.0, but it is not possible to plot it in higher Mathematica versions. My code is: <...
3
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0answers
82 views

Problems with Erf for large arguments with small imaginary part [closed]

While performing some higher precision calculation, I noticed that Mathematica 12.1 killed the kernel and cleared all prior definitions. The problem was with Erf ...
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0answers
24 views

Is there a way to count quantity of evaluations (this preferred) or edits in a cell since last notebook save?

Abstract The goal is to use this to create an autoSave[evals_:5]:=Module[{}, save after set number of evaluations or edits have occured...]. I asked a similar ...
4
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0answers
83 views

AppellF1 unevaluated [duplicate]

I have very long symbolic expressions, which I have to evaluate numerically later on. They contain the AppellF1 function, which stays unevaluated for the specific ...

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