Questions tagged [special-functions]

Questions on the special mathematical functions implemented in Mathematica.

Filter by
Sorted by
Tagged with
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
votes
1answer
42 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
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 ...
0
votes
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
votes
0answers
47 views

Indefinite Integral not Solving

I'm tring to solve the Indefinite Integral of the function: ...
1
vote
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 ...
6
votes
2answers
238 views

Plot of The RiemannSiegelZ Function

I would appreciate your help to visualize the of the following function RiemannSiegelZ[x] with this range. { 18154980120849865 , 18154980120849885 } I tried this: ...
0
votes
0answers
67 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 ...
4
votes
1answer
218 views

`QPochhammer` function simplification?

Consider the function $$(a;q)_n=\begin{cases} 1&n=0,\\ (1-a)(1-aq)\cdots(1-aq^{n-1})&n=1,2,\dots,\\ [(1-aq^{-1})(1-aq^{-2})\cdots(1-aq^n)]^{-1}&n=-1,-2,\dots \end{cases}$$ which in ...
3
votes
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 $...
13
votes
4answers
747 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 ...
6
votes
3answers
131 views

How to calculate Gamma functions values for Quaternions using Mathematica?

How can we calculate the values for the Gamma function using Quaternions on Mathematica? For example: ...
6
votes
2answers
383 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 ...
2
votes
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
votes
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
votes
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 ...
4
votes
1answer
112 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 ...
6
votes
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}}\...
2
votes
1answer
87 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
votes
0answers
75 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 ...
3
votes
2answers
264 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: <...
10
votes
2answers
521 views

Assumptions allowing to calculate an elliptic integral

When I feed Mathematica the following integral: Integrate[Sqrt[(A - x) (B - x)/x], {x, 0, B}] it spits it back out without evaluating it. However, it can ...
5
votes
2answers
927 views

Using RSolve correctly

I am having problems persuading Mathematica to solve even the simplest recurrence relations. As an example, how would you do the following? ...
1
vote
2answers
90 views

Simplifying sums and showing equality - limitations?

Is it possible to verify the following lhs,rhs involving the sums are equal, with Mathematica? I can verify it for individual values of $d$ variable: ...
1
vote
0answers
23 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
votes
1answer
126 views

Verification of a general solution to d'Alembert equation

I solved a nonlinear differential equation (d'Alembert one) by hand. Mathematica gives the same answer. But I am not able to get Mathematica to verify the solution due to branch cuts. Any one knows of ...
4
votes
0answers
80 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 ...
4
votes
2answers
682 views

How do I evaluate a symbolic integral involving Hermite polynomials?

I want to test a difficult integral : Integral on all reals of some complicated function involving the Hermitian polynomials, exponentials, squares, factorials, and being general considering any ...
3
votes
1answer
877 views

Integral over squared Hermite polynomial

I would like to calculate the uncertainty of the nth Eigenstate of a 1-dim harmonic oscillator. To obtain the result I have to compute the integral $$\int_{-\infty}^{\infty} \psi^* x^2 \psi \:dx\,,$$ ...
0
votes
0answers
51 views

Alternative to Coefficient List

I was using CoeffientList to get the probabilities of this generating function... ...
1
vote
0answers
50 views

Branch cut of generalized hypergeometric function (error in Mathematica documentation?)

I am trying to compute the discontinuity around $x=1$ (equivalently, the branch cut) of generalized Hypergeometric functions ${}_{q+1}F_q(a_1,\dots,a_{q+1};b_1,\dots,b_q;x)$. Two formulae are given in ...
2
votes
1answer
77 views

How to define a function the Dirichlet L-function $L(s,\overline{\chi(5,2)})$ in Mathematica?

In Mathematica: The Dirichlet L-function with character $\chi(5,2)$, $L(s,\chi(5,2))$, is expressed as DirichletL[5,2,s] Let $\overline{\chi(5,2)}$ be the complex ...
0
votes
1answer
51 views

Handling Errors in Slowly Converging Dirichlet Beta Infinite Sum

I have tried to calculate the following slowly converging double sum in Mathematica 11.3 $$\sum _{k=1}^{\infty } \left(\frac{ 1}{(2 k-1) (2 k+1)} \left(\sum _{n=1}^{\infty } \frac{(-1)^{n-1}}{(2 n-1)...
11
votes
1answer
292 views

Wrong Limit with LaguerreL

Bug introduced in 7.0 and fixed in 10.2.0 Limit[(n! LaguerreL[n,-1]/(n^(n+1/4)/Sqrt[2]/E^(n-2 Sqrt[n]+1/2))-1) Sqrt[n], n->∞] Mathematica (wrong) output <...
6
votes
0answers
66 views

What makes ListPlot better than N?

I wanted to numerically verify the validity of the formula for the first Stieltjes constant $$\gamma_1=-\frac12\sum_{n=0}^\infty\frac1{n+1}\sum_{k=0}^n\binom{n}{k}(-1)^k\log^2(k+1)$$ ...
3
votes
0answers
112 views

rotation of spherical harmonics using Wigner D-matrix

A very stupid question as I am very confused: I have a surface charge density which is a function of spherical harmonics $\sigma_{l,m}=Y_{lm}$ (only the real part). Now I need to rotate the particle, ...
5
votes
1answer
71 views

Sine vs Sinc vs SphericalBesselJ in NIntegrate

I'm evaluating an oscillatory integral numerically, and ran into a weirdness with NIntegrate, which I've boiled down to a simple case for this question. Consider ...
0
votes
1answer
33 views

Problem plotting finite sum involving HurwitzZeta function

I'm having problems regarding to the following code: ...
1
vote
1answer
67 views

Find analytic solution for integral only defined for even integers

I would like to calculate the integral $$\int_0^{2\pi} dx \sin^6\left(\frac{x}{2}\right) F\left(\frac{4-n}{2}, \frac{4+n}{2}, \frac{1}{2}, \cos^2 \frac{x}{2} \right)^2$$ where $F$ is the ...
3
votes
3answers
345 views

How can I compute Erf of large numbers to more precision?

I would like to compute Erf[80/3] to enough precision to know the order of magnitude of 1 - Erf[80/3] How can I do that? I ...
1
vote
1answer
75 views

Integration of a complicated oscillatory function

I've tried the answers in similar posts but they don't seem to work. As per title, I need to double integrate a complicated quickly oscillatory function. I've checked and there are no poles, the ...
3
votes
1answer
53 views

Problem with a finite sum involving HurwitzZeta Function

I'm trying to reproduce some plots from the analytical expression: $f(\xi)=\left(\frac{2}{\beta^2}-1\right)+\left(\frac{\theta\,e^{-\beta}}{2}-\frac{1}{2}\right)\xi+\sum_{n=1}^{60}\left[\frac{(-\beta)...
2
votes
1answer
59 views

how to plot an approximate Riemann Zeta function beyond $t=180$?

Riemann zeta function $\zeta(s)$ is related to Riemann Xi function $\Xi(z)$ via: $$s=\frac12+ iz,\qquad \Xi(z):=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s),\tag{1}$$ We found the following function $\...
1
vote
1answer
55 views

Simplify Jacobi Polynomials

How can I force Mathematica to use the identities satisfied by Jacobi polynomials $$ (1-\cdot)P_n^{(\alpha+1,\beta)} = \frac{2}{2n+\alpha+\beta+2}\left((n+\alpha+1)P_n^{(\alpha,\beta)}-(n+1)P_{n+1}^{(\...
3
votes
1answer
33 views

Writing generalized hypergeometric function ${_nF}_{n-1}(\cdot)$ as a function of $n$

How can we code the hypergeometric function $$ f(n,k,z)={_nF}_{n-1}{\huge(}{1-k,\overbrace{2,\dots,2}^{n-1\ \text{times}} \atop \underbrace{1,\dots,1}_{n-1\ \text{times}}};z{\huge)} $$ as a function ...
2
votes
0answers
41 views

Dirichlet L-function associated to Kronecker symbol

The Fourier coefficients of the genus 2 Eisenstein series on the Siegel upper half-space are given by sums over Cohen functions. The Cohen function contains as a multiplicative factor a Dirichlet L-...
-1
votes
1answer
70 views

Finding all roots of a function within an interval [duplicate]

I have the following code: ...
9
votes
5answers
530 views

NDEigenvalues vs. FindRoot for finding the eigenvalues of Airy equation?

Say I am trying to find the first 5 eigenvalues of the differential equation $f''(x)=\lambda x f(x)$, on the interval [-1,0], with boundary conditions $f(-1)=f(0)=0$. I will try to do this 3 ways, ...

1
2
3 4 5
21