Questions on the special mathematical functions implemented in Mathematica.

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1
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2answers
96 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. ...
3
votes
1answer
69 views

Interval arithmetic for DawsonF (or other special functions)

I am currently trying to estimate a complicated expression involving DawsonF using interval arithmetic. The interval arithmetic is partially supported by ...
34
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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 ...
17
votes
0answers
231 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 ...
9
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0answers
80 views

$\tt SphericalHarmonicY$ does not seem to be an eigenfunction of the spherical harmonic equation

I applied the spherical harmonic equation on the SphericalHarmonicY functions like this: ...
8
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0answers
794 views

Inverse Laplace transform not obtained

I can't seem to be able to compute the inverse Laplace transform of a Laplace transform: ...
6
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0answers
164 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
384 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
63 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 ...
5
votes
0answers
97 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 ...
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
73 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 ...
4
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0answers
45 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
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0answers
94 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) } ...
4
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0answers
57 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
317 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
179 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
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0answers
76 views

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

The function Pochhammer[1 + n, n] tends to infinity. We have ...
3
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0answers
84 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
61 views

Ramanujan's asymptotic formula for $p(n)$

The following expression is the exact formula of partition function. Ramanujan's asymptotic formula for $p(n)$ is following $$p(n)\sim\frac{1}{4n \sqrt{3}}e^{\pi \sqrt{2n/3}}$$ Can I use ...
2
votes
0answers
43 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
96 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
190 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
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0answers
260 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
129 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
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0answers
56 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
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0answers
34 views

Derivative wrt to order of MacDonald function

I'm trying to get the following result confirmed in Mathematica: $$ \left.\frac{\partial\mathop{K_{\nu}}\nolimits\!\left(x\right)}{\partial\nu}% ...
1
vote
0answers
90 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
82 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
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0answers
41 views

Problem on special functions

What does the symbol PolyLog^{(0,1)}(0,1/e) mean? I know the meaning of the Polylogarithm, but what is that exponent? It happens the same with the Lerch zeta function!! Thanks in advance.
1
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0answers
74 views

Odd plotting/math issue (could be a precision problem)

I've got a pretty odd error on a project I'm working on and was hoping to enlist some advice to fix it. The goal of this notebook is to show that I can eliminate the non-normalizable (blowing up part) ...
1
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0answers
67 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
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0answers
86 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
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0answers
80 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
159 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
226 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 ...
0
votes
0answers
59 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
79 views

Solving Fredholm equation of the 2nd kind

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

How to evaluate this integral

I am trying to evaluate the following integral in Mathematica: ...
0
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0answers
38 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
94 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
18 views

Not Simplify automatic some expression(values)

I try to calculate $$\text{res}\left(h(z) (\psi ^{(0)}(z)+\gamma )^2,\{z,-4\}\right)$$ the result is $$\frac{1}{6} \left(6 h'(-4)-25 h(-4)\right)$$ the result it is possible to write as $$2 \left(6 ...
0
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0answers
61 views

NSolve can not handle a PolyGamma equation

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

Does Mathematica support Laguerre Polynomials of Matrix Argument?

I am wondering whether Mathematica can also compute the Laguerre polynomials of matrix argument as the appear in multivariate statistics? For example, they appear in this paper here: ...
0
votes
0answers
87 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 WolframAlpha it gives back an explicit form, the one you would get if you were to do it by ...
0
votes
0answers
44 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 ...