Questions on the special mathematical functions implemented in Mathematica.

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41
votes
2answers
3k views

What is the difference between a few simplification techniques?

I am trying to understand the difference between Refine, Simplify and FullSimplify, and when ...
32
votes
3answers
801 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 ...
28
votes
1answer
962 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 ...
22
votes
8answers
5k 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 ...
20
votes
1answer
541 views

Incorrect result from Integrate

Bug introduced in 8.0 and fixed in 10.0 I attempted to calculate the following integral: ...
19
votes
1answer
872 views

Peirce's quincuncial projection

The Peirce quincuncial projection is the cartographic projection of a sphere onto a square. In short, I would like to see it implemented in Mathematica. Here is my code: ...
18
votes
2answers
2k views

Visualizing vector spherical harmonics

I have painstakingly derived the vector-spherical harmonics $\mathbf{V}_{J,\,M}^\ell(\theta, \phi)$, which are the generalization of ordinary spherical harmonics $Y_\ell^m(\theta, \phi)$ to vector ...
18
votes
4answers
472 views

Expansion of a hypergeometric function takes ages with Mathematica 9 and 10 (regression?)

Mathematica 8 (Linux version) can evaluate AbsoluteTiming[Series[Hypergeometric2F1[1, 1 - eps/2, 3 - eps, 1/2], {eps, 0, 0}]] in no time. On one of the ...
16
votes
2answers
294 views

How to improve performance of BesselJ to the level of GSL?

Consider the following code: zs = N /@ Range[0, 12, 10^-5]; AbsoluteTiming[bessels = BesselJ[1, #] & /@ zs;] Length@zs I've tried to measure only computation ...
16
votes
1answer
471 views

How to represent a continuous monotonic phase of Airy functions?

Note: In this question I am concerned only with real-valued variables and functions. DLMF, §9.8 Airy Functions, Modulus and Phase, formula $9.8.4$ defines the phase of Airy functions: ...
16
votes
0answers
201 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 ...
14
votes
4answers
531 views

How to plot Ramanujan's continued fraction in Mathematica?

I want to plot Ramanujan's continued fraction: $$R(q)=\cfrac{q^{1/5}}{1+\cfrac{q}{1+\cfrac{q^2}{1+\ddots}}}$$ but I do not know how to define this function in Mathematica. How do I define and plot ...
14
votes
1answer
356 views

How to enlarge Mathematica's knowledge about certain functions?

I'm often troubled with the following task. I need to carry out symbolical computations involving certain special functions. Let me take as an example Barnes gamma-function. It is included in ...
14
votes
1answer
591 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 ...
13
votes
2answers
418 views

What kind of hypergeometric function is it?

I found a formula for an integral of a product of three Bessel functions at The Wolfram Functions Site: I cannot understand what kind of hypergeometric function it is. The Mathematica code given ...
13
votes
1answer
314 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 ...
12
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.]. ...
12
votes
1answer
705 views

Numerically solve 2nd order differential equation with singularity

Consider a second order differential equation with a potential that diverges at some generic value in the variable. For example: $$-y^{\prime\prime}(s)+\frac1{\mathrm{cn}{(s\mid k^2)}}y(s)=0$$ where ...
11
votes
3answers
377 views

Find asymptotics of $\sum\limits_{i=0}^{n/3} 2^i \binom{n-i-1}{\frac{2n}{3}-1}$

I have an expression 2^n / Sum[ 2^i Binomial[ n - i - 1, 2n/3 - 1], { i, 0, n/3}] ...
10
votes
2answers
2k views

problem with coloring spherical harmonics

I want to color a spherical harmonics. So I write as follows. ...
10
votes
2answers
249 views

Asymptotics of $\frac{\sum _{i=0}^{\lfloor n/2 \rfloor} {2(n-2i) \choose n-2i} {n \choose 2i} {4i \choose 2i}}{2^{3n - 1}}$

I am fairly sure that asymptotically $$\frac{\sum _{i=0}^{\lfloor n/2 \rfloor} {2(n-2i) \choose n-2i} {n \choose 2i} {4i \choose 2i}}{2^{3n - 1}} \sim \frac{2}{\pi n}.$$ I tried ...
10
votes
1answer
798 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 ...
10
votes
1answer
665 views

Why can't I change the value of MaxRecursion in NIntegrate when integrating BesselJ?

Bug introduced in 8.0.4 or earlier and persists through 10.0.2. I am trying to evaluate this integral numerically $$ \int_0^{\infty } J_0(q R) \tanh(q) \, \mathrm{d}q $$ for large values of $R$. ...
9
votes
4answers
744 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 ...
9
votes
2answers
175 views

Integrating a BesselJ integrand to obtain the same result as Maple 16

I would like to check the following integration: Integrate[y*Integrate[1/x*BesselJ[1,x*Exp[I*π/4]]*BesselJ[1,x*Exp[-I*π/4]],{x,0,y}],{y,0,r}] Mathematica 9.0 is ...
9
votes
3answers
1k 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 < ...
9
votes
1answer
149 views

Eisenstein Series in Mathematica?

Mathematica doesn't seem to have built-in tools to deal with the Eisenstein series: $$\begin{align*} E_{2}(\tau)&= 1-24 \sum_{n=1}^{\infty} \frac{n e^{2 \pi i n \tau}}{1-e^{2 \pi i n \tau}}\\ ...
9
votes
1answer
342 views

How to calculate this integral? Integrate[BesselJ[0, x - BesselJZero[0, 1]]/x, {x, -Infinity, Infinity}]

I tried to calculate the following integral, but it returned unevaluated. ...
9
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
1answer
184 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 ...
9
votes
1answer
194 views

Mathieu function periodicity problem

According to the documentation, the Mathieu characteristic function generates parameter a: The characteristic value Subscript[a, r] gives the value of the parameter a in y′′+(a-2q cos(2z))y=0 ...
9
votes
1answer
682 views

how to simplify large expression with lots of special functions in it (BesselY, Hypergeometric, MeijerG etc…)

I saw this DE in Maple forum. When solving it using Mathematica 9.01, even though the result was correct (both solutions gave the same numerical answer for some random values), Mathematica's answer ...
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] ...
8
votes
2answers
259 views

How to simplify an expression with special functions to zero

The following is a well-known Bessel function identity: $$J_{-n}(z)=(-1)^n J_n(z),\qquad n\in\mathbb Z$$ To check this, I used the following code and the result is as what I expected. ...
8
votes
1answer
774 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 ...
8
votes
1answer
240 views

Expanding derivatives of hypergeometric functions

Sometimes Mathematica expresses results of integration or summation in terms of symbolic derivatives of Hypergeometric2F1 function, and cannot further simplify ...
8
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 ...
8
votes
1answer
596 views

Numerical contour integrations in the complex plane - contour deformation gives different answer for analytic kernel

I am trying to do a contour integration in Mathematica numerically. In particular, I'm checking the identity: $$ H_m^{(1)}(z) =\frac{i^{-m}}{\pi}\int_{-\pi/2 + i \infty}^{\pi/2 - i \infty} \exp[i m ...
8
votes
0answers
659 views

Inverse Laplace transform not obtained

I can't seem to be able to compute the inverse Laplace transform of a Laplace transform: ...
7
votes
2answers
210 views

On the definition of the associated Legendre polynomials

Mathematica computes for n = 1,2,...: (-1)^n (LegendreP[n, -1, -3]/Sqrt[2]) -I, -3 I, -11 I, -45 I, -197 I, ... Maple ...
7
votes
1answer
631 views

Visualizing vector-spherical waves

This is a follow-up question to this one on visualizing vector-spherical harmonics. This time, I would like to visualize the vector spherical waves (including the radial dependence). The functions ...
7
votes
1answer
94 views

Sum over Binomials and Gammas

Given the function, ...
7
votes
2answers
339 views

Problem with NIntegrate when WorkingPrecision is specified

I am trying to evaluate this integral numerically: $$ \int_0^{\infty } m \exp (-m) J_1(m){}^2 \, dm $$ Everything is OK when only the integration method is specified: ...
7
votes
2answers
108 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: ...
7
votes
2answers
772 views

Mathematica 10 cannot solve definite integral [duplicate]

Bug introduced in 10.0 and fixed in 10.0.2 Mathematica 10 fails to solve the following integral, saying that it does not converge. ...
7
votes
1answer
102 views

How to define $\operatorname{Mc}$ and $\operatorname{Ms}$ Mathieu functions in terms of MathieuC and MathieuS?

Reading multiple books about Mathieu functions, I always come across notation like $\operatorname{ce}_r(z,q)$, $\operatorname{se}_r(z,q)$ for angular functions and $\operatorname{Ce}_r(z,q)$, ...
6
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 ...
6
votes
2answers
753 views

Why does Expand not work within a function?

I'm writing this fairly simple function: ...
6
votes
2answers
175 views

Number of divisors visualized with the QPochhammer function, how improve performance of code?

I have this code that is originally Jeffrey Stopple's code for the Riemann zeta function in the complex plane. Because I discovered yesterday that the number of divisors can be generated with the ...
6
votes
2answers
156 views

How does Mathematica calculate LaguerreL

About the function LaguerreL[n,a,x], the helping documents in Mathematica only say that this function satisfies equation $xy^{\prime\prime}+(a+1-x)y^\prime+ny=0$. ...