Questions tagged [special-functions]
Questions on the special mathematical functions implemented in Mathematica.
968
questions
1
vote
3answers
54 views
When is a symbol a Symbol? Is there an easy mathematica function that tests if an object is a symbol something like SymbolQ?
Yes I know there is no built in function SymbolQ[...] (but JavaScript does) but could one be coded to work for most cases? I often rely on ...
2
votes
1answer
51 views
The NIntegral can not 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
votes
1answer
47 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
2answers
248 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
votes
0answers
74 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 ...
1
vote
0answers
17 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
0answers
74 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
1answer
115 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 ...
0
votes
0answers
45 views
Alternative to Coefficient List
I was using CoeffientList to get the probabilities of this generating function...
...
1
vote
0answers
48 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 ...
6
votes
0answers
62 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
75 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
63 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
31 views
Problem plotting finite sum involving HurwitzZeta function
I'm having problems regarding to the following code:
...
1
vote
1answer
62 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
315 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 ...
3
votes
1answer
50 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
55 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 $\...
0
votes
1answer
44 views
Simplify Jacobi Polynomials
How can I force Mathematica to use the identities satisfied by Jacobi polynomials
$$
(1-\cdot)P_n^{(\alpha+1,\beta)} = \frac{2}{2n+\alpha+\beta+2}\left((n+\alpha+1)P_n^{(\alpha,\beta)}-(n+1)P_{n+1}^{(\...
3
votes
1answer
27 views
Writing generalized hypergeometric function ${_nF}_{n-1}(\cdot)$ as a function of $n$
How can we code the hypergeometric function
$$
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)}
$$
as a function ...
1
vote
0answers
38 views
Dirichlet L-function associated to Kronecker symbol
The Fourier coefficients of the genus 2 Eisenstein series on the Siegel upper half-space are given by sums over Cohen functions. The Cohen function contains as a multiplicative factor a Dirichlet L-...
1
vote
1answer
66 views
Integration of a complicated oscillatory function
I've tried the answers in similar posts but they don't seem to work. As per title, I need to double integrate a complicated quickly oscillatory function. I've checked and there are no poles, the ...
0
votes
1answer
65 views
0
votes
1answer
67 views
Plotting a sum of Bessel functions
I would like to plot the Nth partial sum, given Ļ for different values of "n". I'm really new at this so i don't even know how to begin. I have to make a graph of Ī¾ vs Ļ and find the minimum of n so i ...
1
vote
1answer
85 views
How do you find the Inverse of Elliptic Integral of Second Kind when modulus is large
So I tried to take the inverse of EllipticE when modulus is large, in Mathematica, but the solution gives wrong answer.
...
3
votes
2answers
360 views
Why doesn't Integrate evaluate an elliptic integral?
My code is
Integrate[ 1/Sqrt[(x - a) (x - b) (x - c) (x - d)], {x, a, ā},
Assumptions -> 0 < d < c < b < a]
I know this can be ...
5
votes
2answers
171 views
Evaluate integrals with parameters
I'm new to Mathematica so I apologize if the answer to my question is trivial.
I need to calculate the following integral
$$ \int_{-\infty}^{0}\frac{dx}{\sqrt{(x-a)(x-b)(x-c)(x-d)}} $$
with
$$ 0 < ...
1
vote
1answer
42 views
Is there any way to prioritize Gamma function?
In Mathematica, if you type
Gamma[1, 0, -a]
It automatically simplifies to 1 - E^a. I wonder though if there is a way to stop ...
1
vote
0answers
31 views
Slow convergence of Fresnel integral evaluation
I am trying to solve a problem from optics, a so-called Fresnel integral. The argument of integration is defined with a function:
...
1
vote
1answer
49 views
difference equation and continued fractions
I'm interested in solving the following difference equation: $x[k-1]+(k^2+k+a)/x[k]=b$, $k=1,2,\ldots$, where $a,b$ are fixed positive numbers; let's say $x[1]=c>0$. Mathematica's ...
4
votes
1answer
63 views
How to Force the PolyGamma[0, x] function or the HarmonicNumber[ x ] function
I'm crunching some infinite summations, and sometimes Mathematica generates results that have the PolyGamma[0, x] function (which is the Digamma) and sometimes the <...
13
votes
2answers
542 views
Looping through all functions defined in Mathematica
Is there a way to loop through all the Functions (Elementary and Special functions) that exist in Mathematica?
I want to construct a table of some identities and maybe I can discover something ...
0
votes
0answers
63 views
Radial wave-function integration with mathematica
I'm new in using Mathematica. I wanted to do some integrations with the radial wave function for Hydrogen atom. The radial wave function is given by,
$$R_{nl}(r)=2^{l+1} e^{-\frac{r}{a n}} \sqrt{\frac{...
1
vote
0answers
27 views
Help in Better contour plot
In the following piece of code, I am trying to plot a function: fun[x,y] for some set of given parameters. but on the y axis of contour plot, after $\pm 4.5$, ...
12
votes
5answers
324 views
Possible bug in finite sum over inverse squares $\sum\limits_{i=1}^n \frac{1}{(x (n-i)+i)^2}$
Revisiting the problem Limit of partial sums involving inverse squares I found another difficulty with Sum[]
Consider this sum
...
0
votes
0answers
63 views
Numerical comparison of two integrals and a function :
Consider the following integral:
$$S(q)=\int_{x=2}^q\sin^2\left(\frac{Ļ\Gamma(x)}{2x}\right)dx$$
And consider the functions :
$$R(q)=\frac{q}{\log(q)}$$
$$T(q)=\int_2^q\frac{1}{\log(x)}dx$$
I ...
1
vote
1answer
70 views
Pochhammer Simplification
Assumption:: m and n are integers and greater than zero.
$Assumptions = {Element[m, Integers] && m >= 0, Element[n, Integers] && n >= 0}
...
1
vote
2answers
50 views
How to do series expansion for functions which have symbolic parameters?
I would like to find the series expansion of
(E^(x^k/k!) Gamma[k, x])/Gamma[k]
for $k$ being a positive integer, up to the order of $x^{2k+1}$. Mathematica ...
2
votes
2answers
84 views
How to calculate Gamma functions values for Quaternions using Mathematica?
How can we calculate the values for the Gamma function using Quaternions on Mathematica?
For example:
...
3
votes
2answers
88 views
Two identities involving contour integrals in the presence of a branch point where the integrand explodes, and the Kummer function
I need to understand how to establish two identities. The first is
$$ \int_{C} z^{-1-q}(1-z)^{-1-\lambda } dz=\frac{2 \pi \Gamma (q+\lambda +1)}{\Gamma
(\lambda +1) \Gamma (q+1)}, q\geq 0, \...
1
vote
3answers
139 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 ...
0
votes
0answers
61 views
Understanding Working Precision - two example equations
I am trying to understand how WorkingPrecision works, and in general how very small/large numbers are to be handled.
Let me preface I probably do not have ...
1
vote
0answers
41 views
Change the PolyLog branch cut in Wolfram Mathematica
The problem is to change the branch cut of PolyLog[2,z] function from the default half line z=1+x, x>0 to the new one ...
1
vote
2answers
152 views
Using Convolve with functions that contain error functions, and iterative convolutions
I need to produce convolutions of convolutions to study some probability distributions. My starting distribution is the Rayleigh distribution, which I convolve with itself as:
...
0
votes
1answer
41 views
Substitute gives me different result
I'm working with Legendre polynomials (& associated ones). When I do the following calculation:
...
0
votes
0answers
36 views
Gamma and Pochhammer simplification
I want to perform some simplification on Gamma function and Pochhammer symbols. For example,
...
1
vote
0answers
36 views
Jacobi elliptic function JacobiSN[z,1/2] should be equal to JacobiSN[z , -1/2], But MATHEMATICA shows different values, Why?
Jacobi elliptic function JacobiSN[z,1/2] should be equal to JacobiSN[z , -1/2] by their properties(NIST Handbook of Mathematical functions, SN[z,k]=SN[z,-k]). But in MATHEMATICA they are giving ...
2
votes
2answers
73 views
Why is FindRoot throwing a “singular Jacobian” error when function has small values?
Running the code:
FindRoot[PolyLog[2, -E^-(10 + Sqrt[1 + x^2])] == 10^-30, {x, 10}]
Gives me the error:
...
0
votes
1answer
64 views
Integrating FindRoot
I have a certain function F[a_] := FindRoot[f[a,x],{x,0.5}] and I want to compute NIntegrate[F[t],{t,0,1}]. However, ...
2
votes
0answers
52 views
Replacing an intrinsic Mathematica function by some other function
I want to express the Hypergeometric2F1 series (which is an intrinsic function of Mathematica ) wherever it appears by its definition.
Where ever this function appears with whatever the arguments I ...
