Questions on the special mathematical functions implemented in Mathematica.

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1
vote
4answers
2k views

How to take conjugate of a function?

Naïvely this is what happens and it obviously is not helpful! ...
4
votes
1answer
154 views

Find point at which equation stops having roots (if it exists)

I am interested in the roots of this function: f[M_, b_] := 1 - (2 M Gamma[2, 0, (1/M + b M)/Sqrt[b]])/(1/M + b M) for fixed values of b. In particular I want ...
5
votes
2answers
194 views

Why does Integrate return a solution that is not defined at a particular point when it actually is well defined at that point?

I am trying to compute Integrate[Sqrt[x^4 + (y - y^2)^2], {x, 0, y}] Mathematica 8 gives ...
7
votes
6answers
2k views

Series expansion in terms of Hermite polynomials

I am trying to expand a polynomial in terms of orthogonal polynomials (in my case, Hermite). Maple has a nice built-in function for this, ChangeBasis. Is there a ...
5
votes
2answers
458 views

Mathematica cannot calculate a limit

When I evaluate Limit[E^(-n)*Sum[n^k/(k!),{k,0,n}], n -> ∞] Mathematica gives me the result ...
1
vote
0answers
129 views

Getting poles of a Gamma functions [duplicate]

Why do the following 2 sequences give different answers? n = 1.5 Series[Gamma[0.5 - n - x], {x, 0, 2}] Series[Gamma[-1 - x], {x, 0, 2}] (..clearly the output from the second expression is ...
4
votes
1answer
321 views

Simplifying numerical expressions involving special functions

I've encountered the following problem. There is the identity (Legendre's relation) that the special functions EllipticK[x] and ...
3
votes
1answer
321 views

Strange behaviour of PolyLog Function

I discovered some strange behaviour of the PolyLog[] Function in Mathematica which seems to me like a bug in the function implementation. It looks like ...
1
vote
2answers
936 views

Irregular Confluent Hypergeometric Functions (Spherical Coulomb Wavefunctions)

I want to program in the regular and irregular spherical Coulomb wavefunctions $F_\ell(\gamma,kr)$ and $G_\ell(\gamma,kr)$, respectively, which are defined in terms of the regular and irregular ...
5
votes
0answers
115 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 ...
0
votes
2answers
642 views

Using Mathematica to find poles of Gamma functions

I am concerned about the expression on the RHS of equation A.5 (page 19) in this paper: $$\int\frac{d^d ...
6
votes
2answers
324 views

Cannot Get Numerical Results to Match

I try this numerical summation (in two parts) ...
8
votes
2answers
151 views

Erfc Not Returning Results Specified in Documentation

In the documentation for Erfc (under "Possible Issues"), the following command returns a number that is extremely close to 2: However, when I run this same command in a fresh kernel, I get: ...
10
votes
2answers
443 views

Compiling the VoigtDistribution PDF

According to List of compilable functions, Erf and Erfc are compilable functions. However, I want to make a compiled version ...
6
votes
0answers
374 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 ...
2
votes
1answer
366 views

Directional derivative of SiegelTheta

I'm working on a problem where I have to integrate both the Mathematica function SiegelTheta and some of its second order directional derivatives. Using the function works well but something goes ...
7
votes
1answer
601 views

Integrating over Bessel Function erroreous? (Hankel Transform)

Bug introduced in 8.0.4 or earlier and persists through 10.2. The Hankel Transform is given by Integrate[f[x] x BesselJ[0, x t], {x, 0, Infinity}] It is ...
1
vote
1answer
572 views

Plotting Fresnel function

I am trying to plot the partial sums and the Cesàro means of the function $\sqrt{|x|}$ and for $a_{n}$, I obtained the following code which contains FresnelS. ...
10
votes
2answers
2k views

problem with coloring spherical harmonics

I want to color a spherical harmonics. So I write as follows. ...
10
votes
1answer
2k views

solve an integral equation numerically

I am trying to find a numerical solution for an equation of the form: $$ f(t)=\int_{t_{min}}^{t} {\exp[2 (t^\prime - t)] E(t^\prime) f(t^\prime)} + \int_{t}^{0}{ \exp(t^\prime - t) E(t^\prime) ...
9
votes
3answers
2k views

Multi-Factorial and Series with Triple-factorial terms

Let $n!^{(k)}$ denote a multi-factorial which is defined by $$ n!^{(k)} = \begin{cases} 1 & n \leqslant 0, \\ n, & 0 < n < ...
3
votes
0answers
171 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 ...
9
votes
1answer
1k views

Implementation of Incomplete Fermi-Dirac Integral in Mathematica

I'm working on a special algorithm to implement a more accurate effective mass calculation for hole carriers in silicon in Mathematica. This rather involved algorithm uses incomplete Fermi-Dirac ...
4
votes
2answers
196 views

LevinRule and SphericalBessels

I'm currently looking at a simplified problem that approximates another problem I'm looking into. In this simplified problem I at least have an analytic integrand and can easily provide all info on ...
10
votes
1answer
894 views

Is my expression too complicated for FullSimplify or am I doing something wrong?

I have a messily defined function $v(h, w)$ with $h, w \in \mathbb{R}$ and with a removable singularity at $h=1/2$, and I am trying to prove some of its properties using Mathematica. In particular I ...
5
votes
2answers
251 views

Solve for $a$ as a function of $\beta$?

I am trying to solve this equation: $$\beta^{-a} \Gamma(a) \sin(a \pi) + e^\beta \beta^{2 a - 1} \Gamma(1 - a) \sin(a \pi) = 0$$ I tried the following: ...
31
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 ...
0
votes
1answer
357 views

Iterative way to find roots of confluent hypergeometric function

I am trying to find roots of confluent hypergeometric function and I wonder if I can choose the initial guess by the choice of $\beta$. ...
9
votes
4answers
776 views

Finding the roots of Hypergeometric1F1[]

I am trying to find the roots, λ, for this equation: Hypergeometric1F1[1/4 (2 -  λ /β), n + 1, β] for certain ...
13
votes
1answer
320 views

Extra factors appear when evaluating Euler integrals

Note: this is fixed in version 9. When I perform the double integral in Mathematica, Integrate[(x (1 - x))^z (y (1 - y))^z, {x, 0, 1}, {y, 0, 1}] which ...
3
votes
2answers
325 views
8
votes
0answers
729 views

Inverse Laplace transform not obtained

I can't seem to be able to compute the inverse Laplace transform of a Laplace transform: ...
8
votes
3answers
1k views

Checking if the roots of a function are real

I'm trying to determine if the roots of a function are real. How would you do that? (In particular I'm interested in verifying that the roots of LegendreP[6, x] ...
26
votes
8answers
6k views

About multi-root search in Mathematica for transcendental equations

I have some questions for multiroot search for transcendental equations. Is there any clever solution to find all the roots for a transcendental equation in a specific range? Perhaps ...
14
votes
1answer
623 views

Incorrect results for elementary integrals when using Integrate

There is a rather simple integral ($K_0$ is the 0-th order MacDonald function) $$\int_0^\infty e^{-x \cosh\xi}\, d\xi = K_0(x)$$ which mathematica cannot solve. This even though the documentation ...
6
votes
2answers
805 views

Why does Expand not work within a function?

I'm writing this fairly simple function: ...
44
votes
2answers
4k views

What is the difference between a few simplification techniques?

I am trying to understand the difference between Refine, Simplify and FullSimplify, and when ...
15
votes
4answers
2k views

Numerical underflow for a scaled error function

I calculate scaled error function defined as f[x_] := Erfc[x]*Exp[x^2] but it can not calculate f[30000.]. ...
8
votes
1answer
834 views

Hankel Transform integrals won't work in Mathematica

I'm trying to do this integral, which is shown on the Wikipedia page on the Hankel transformation: $$\int_0^{2\pi}\mathrm d\varphi\;e^{\mathrm im\varphi}e^{\mathrm ikr\cos(\varphi)}$$ The answer is ...
4
votes
2answers
233 views

Problem with EllipticE documentation

The complete elliptic integral of the second kind, EllipticE, is defined as, Integrate[Sqrt[1-m Sin[t]^2],{t,0,z}] According ...
34
votes
4answers
890 views

Is there a Mathematica API for the functions.wolfram site?

Is there a Mathematica API for the functions.wolfram site? If there's not, has anyone implemented a web scraper for it? For example it would be nice to be able to access ...