Questions on the special mathematical functions implemented in Mathematica.

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2
votes
2answers
159 views

How to convolve the unit box function and the modified Bessel function of the second kind in 2D?

In 1D the convolution of the unit box function and the modified Bessel function of the second kind $K_0(x)$ works very well. ...
0
votes
1answer
73 views

Solving and Plotting Infinite sums over solutions to transcendental equation involving Bessel functions

I am trying to use Mathematica to plot the following equation $E(q,\Delta)$ vs $qa$, for set values of $ D \Delta/a^2$ and $\bar\rho a/D$. I have looked at other stackexchange posts which deal with ...
0
votes
1answer
87 views

How to solve algebra equations containing integration and parameters?

I'm trying to solve two nonlinear algebra equations for two unknown parameters, U and Tf. Since some terms in these equations contain integration, and the integration also contains U and Tf. The main ...
34
votes
0answers
1k views

Fast Spherical Harmonics radiative transfer

This is a rather specific question and I apologize for spamming you with some lengthy code. But it could be interesting for some reader and maybe you can help out, so please bear with me. I am using ...
18
votes
0answers
245 views

Is MathieuC for moderately large imaginary arguments broken?

I'm trying to plot MathieuC[-3,0.3,I x] for $x\in[0,10]$, and here's what I get even with arbitrary precision arithmetic (here I use ...
8
votes
0answers
841 views

Inverse Laplace transform not obtained

I can't seem to be able to compute the inverse Laplace transform of a Laplace transform: ...
6
votes
0answers
100 views

SiegelTheta throws errors from calling Range with complex arguments

Bug introduced in 10.4 or earlier. 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 Code Golf with the following code:...
6
votes
0answers
186 views

Fine tuning compiled code that computes dilogarithm function

As an exercise of writing a good Compile function, I want to do the simple task of coding a routine that outputs the real part of the dilogarithm function ...
6
votes
0answers
389 views

Reproducing the Integral Definition of the Modified Bessel function

I need to simplify some integral expressions in terms of special functions, such as the modified Bessel function of the first kind. See for example Eq. (5) on this page. Notice that the real ...
5
votes
0answers
79 views

SiegelTheta gives misleading message when the dimensions don't match

Bug introduced in 6.0 and persisting through 10.4 SiegelTheta is new in 6.0 In order to test the SiegelTheta function, I ...
5
votes
0answers
110 views

MacDonald formula for Modified Bessel Functions

How can I make Mathematica understand these two integrals? $$\int_0^{\infty} e^{-x \cosh{\xi}} d\xi = K_0(x)$$ $$\int_0^{\infty} e^{-\frac{1}{2} \Big( \frac{x y}{u} + u \frac{x^2+y^2}{x y} \Big) } K_{...
5
votes
0answers
118 views

Expansion of $E(i c \mid m)$ at $c\to\infty$?

Currently, I am using a Windows machine with Mathematica 8. I noticed a difference in a series expansion of the function EllipticE[] in comparison with a result ...
4
votes
0answers
69 views

Huge difference after changing a fraction to decimal

I have a limit to calculate. With[{p = 1/2, q = 0.1}, Limit[a^q Integrate[Sin[x]/x^p, {x, a, Infinity}], a -> Infinity]] gives correct result ...
4
votes
0answers
86 views

Bug in Series[Pochhammer[1+n, n], {n,Infinity,1}]?

The function Pochhammer[1 + n, n] tends to infinity. We have ...
4
votes
0answers
58 views

Integral Form of Modified Bessel Function of the Second Kind

Why can't Mathematica integrate r = Integrate[Exp[-x Cosh[t]], {t, 0, Infinity}]; r = Assuming[Element[x, Reals], Simplify[r]]; Together[r] From Wikipedia, it ...
4
votes
0answers
60 views

Example in Help File does not evaluate as claimed

In the help file, under BellB, I read at "Properties and Relations": Sum can give results involving ...
4
votes
0answers
327 views

Spherical harmonic derivative

Consider the following substitution Derivative[2, 0][S][th, ph] /. S -> Function[{th, ph}, SphericalHarmonicY[3, 0, th, ph]] which gives correct answer. While ...
4
votes
0answers
183 views

Why doesn't Log[Gamma[]] simplify to LogGamma[] where it could?

I have been playing with various equations involving amount of permutations in relatively large sets. Easiest way to look at these is something like Log[10, bignumber!] . Often expressions, even ...
3
votes
0answers
43 views

Series expansion in Infinity issue with Zeta(s) function

With this code: Series[Zeta[s], {s, Infinity, #}] & /@ Range[10] // MatrixForm Series expansion for the Zeta(s) ...
3
votes
0answers
64 views

Are there any Mathematica programs on or about the LMFDB Archive?

I would like to explore the LMFDB Archive with L-functions using Mathematica. Is there any Mathematica sample code available to get me started?
3
votes
0answers
89 views

Orthogonality relations of Hermite polynomials

The Hermite polynomials are orthogonal. $$ \int_{-\infty}^\infty H_m(x) H_n(x) e^{-x^2}\, \mathrm{d}x = \sqrt{ \pi} 2^n n! \delta_{nm} $$ Does Mathematica not use this relationship? Because running <...
2
votes
0answers
44 views

Unexpected imaginary term in the asymptotic expansion of DawsonF

FullSimplify[Series[2 DawsonF[x], {x, ∞, 8}]] (* -I Exp[-x^2] √π + (1/x + 1/(2 x^3) + 3/(4 x^5) + 15/(8 x^7) + O[1/x]^9) *) What is the reason the term ...
2
votes
0answers
97 views

Nasty integral advice

I have a pretty ferocious integral to solve, and since it doesn't seem I'll be able to do much analytically, I've taken to Mathematica to get some information. Mainly, I want to see if there are any ...
2
votes
0answers
205 views

Mathematica not evaluating q derivative of Jacobi theta function

Jacobi theta functions, $\theta_a(u,q)$ for $a=1,2,3,4$ are defined in the unit disk $|q|<1$. For some reason that I would like to understand, Mathematica does not evaluate numerically the $q$ ...
2
votes
0answers
282 views

How to prevent simplification of hypergeometric functions resulting from integrations?

Definite integrals from 0 to Infinity over a product of two hypergeometric (including exponential, trigonometric, hyperbolic, ...
2
votes
0answers
131 views

RSolve and incomplete gamma function

I am interested in how Mathematica uses the incomplete gamma function to solve difference equations. For example, if we have this inhomogeneous, 2nd order equation and use ...
1
vote
0answers
39 views

How to invert an Elliptic function where the elliptic nome is a function of an independent variable?

I have a Jacobian elliptic function as a function of two independent variables $x$ and $y$. The elliptic parameter $m=m(y)$, $0 \leq m \leq 1$, is also a function of the variable $y$, and thus the ...
1
vote
0answers
64 views

How to do this integral of Hypergeometric functions in Mathematica?

I've tried to integrate by part, but it seems that Mathematica is still not able to integrate. ...
1
vote
0answers
44 views

`QPochhammer` function simplification?

Consider the function which in Mathematica is denoted as QPochhammer[a,q,n], and its infinite product cousine: which in Mathematica reads ...
1
vote
0answers
57 views

Output in FunctionExpand for function of Gamma

I used of this code: α Gamma[α] // FunctionExpand and get output: Gamma[1 + α] Also I used of this code: ...
1
vote
0answers
104 views

Product of two Meijer's Function

I want to evaluate an integral $I_1$ defined in $Eq.(1)$ as \begin{align} I_1=\int_{0}^{\infty}\frac{x\exp(-\beta x)K_1(\alpha x)}{1+x}dx\tag{1} \end{align} Where $\alpha\geq0$, $\beta\geq0$, and $...
1
vote
0answers
85 views

How to understand the symbolic integral in a result returned by DSolve?

I am trying to solve an ODE like $$\frac{dF}{dx}=\frac{a}{\ln^3 x+[c+d(ax+b)][2\ln x+c+d(ax+b)]\ln x},$$ where $a,b,c,d$ are constants. I guess it could contain a special function, say, the ...
1
vote
0answers
74 views

Use Solve[] with Bessel, gamma, and hypergeometric functions?

I need to find values of {a,b,c} such that the 0th, 2nd, and 4th order moments of f[x]=Exp[-ax^4 - bx^2 - c] will equal respectively {1,2,10}. I didn't really expect this to work, and it didn't: <...
1
vote
0answers
90 views

Improve speed of evaluating a sum over four indices

I am trying to implement the Fox-H function with several variables as sum of residues. I have arrived at the following function; ...
1
vote
0answers
110 views

Plotting the Schwarz D minimal surface

With the aid of the following paper I'm trying to plot the Schwarz D minimal surface: paper. So far I have followed all the instructions to the best of my abilities and have come up with the following ...
1
vote
0answers
184 views

Converting a hypergeometric function to a Bessel function

Is there any way to get a Bessel function equivalent to a Hypergeometric function? Would it be possible in general? From Maple integration (as discussed in Integrating a BesselJ integrand to obtain ...
1
vote
0answers
239 views

Minimizing a functional takes forever

I need help with minimizing a functional in Mathematica. I have a function $V(\xi)=\sum_{i=1}^\infty C_{3_i}J_0(\xi/C_i)+C_{4_i}Y_0(\xi/C_i),~~~\Sigma\leq\xi\leq\xi_1$, and want to find such $\{C_{...
0
votes
0answers
47 views

How to get ContourPlot3D to run involving elliptic functions?

I need to plot zeroes of a function of three variables that involves elliptic functions. For example, I have ...
0
votes
0answers
64 views

Another question about inverse Laplace transform

Given $r = 0.06;\quad \theta = 105;\quad \kappa = 1;\quad x_0 = 100;\quad K = 100;\quad \sigma = 0.10;\quad T = 0.25;$ Define $ \nu = -\kappa/\sigma^2 - 0.5;\quad p = \kappa*\theta/\sigma;\quad q = -\...
0
votes
0answers
103 views

Solving Fredholm equation of the 2nd kind

While I was using the code from here to solve this integral equation: ...
0
votes
0answers
79 views

How to evaluate this integral

I am trying to evaluate the following integral in Mathematica: ...
0
votes
0answers
40 views

Determine class of special function from algebraic constraints?

Consider vectors x_i in arbitrary dimension. Let's say I have an expression in six variables F[x_1,x_2,x_3][x_4,x_5,x_6], which ...
0
votes
0answers
44 views

Timing of associated Legendre polynomials

I encountered a strange issue with the associated Legendre polynomials implemented with LegendreP[l,m,z]. Quite simply, the time used for the numerical computation of those quantities depends on ...
0
votes
0answers
97 views

PolyLog does not simplify

How come mathematica can not simplify a simple expression involving the PolyLog function? An example is: You know that ...
0
votes
0answers
64 views

NSolve can not handle a PolyGamma equation

I would like to solve the following equations numerically: ...
0
votes
0answers
98 views

Spherical Hankel and Bessel in Explicit form

This is probably an easy question, when I type the spherical Hankel (first kind) and the Bessel function into Wolfram Alpha, it gives back an explicit form, the one you would get if you were to do it ...
0
votes
0answers
48 views

Should this be equal to Gamma function or not?

I define the following function: Nat2[s_] := InverseFourierTransform[FourierTransform[1/t, t, w] (-I w)^(s - 1), w, x]/Cos[Pi (s - 1)] /. x -> 1 I expected ...