Questions on the special mathematical functions implemented in Mathematica.

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33
votes
8answers
7k 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 ...
5
votes
2answers
286 views

When does the real part of Zeta vanish on the critical line?

This seems to be a quite a simple problem but I cannot make it work. I am trying to find all values within a given range for which the real part of the Zeta ...
46
votes
2answers
5k views

What is the difference between a few simplification techniques?

I am trying to understand the difference between Refine, Simplify and FullSimplify, and when ...
6
votes
2answers
857 views

Why does Expand not work within a function?

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

Replacing gamma at half integers by double factorial

It is well-known that for any positive integer $n$ the equality $\Gamma(n+\frac12)=\sqrt\pi\,(2n-1)!!/2^n$ holds, where $!!$ stands for the double factorial. I am using ...
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.]. ...
24
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 ...
6
votes
1answer
225 views

Precompiling a Whittaker function

Is there a way to speed up the evaluation of special functions in Mathematica? I am particularly interested in the Whittaker W function. For instance, the following piece of code: ...
9
votes
3answers
293 views

Labeling solutions of an Eigenvalue equation involving Bessel functions

I'm solving the Schrödinger equation for a particle in an annular geometry with hard wall boundary conditions and I've reduced it to the following equation: $$J_m(k\,R_1)\,Y_m(k\,R_2) - J_m(k\,R_2)\,...
2
votes
3answers
234 views

Why do these two different zetas produce the same value?

Zeta[-13] == Zeta[-1] == -1/12 Why do these two different zetas produce the same value?
6
votes
0answers
175 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 ...
3
votes
2answers
415 views

Solve stiff system by shooting method

I'm trying to solve a second order differential equation with the shooting method but, it appears to be a stiff system. This is worst for a larger parameter $\mu$. I've tried different methods: ...
12
votes
1answer
336 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}}\\ E_{...
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) f(t^\...
8
votes
3answers
2k 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] ...
3
votes
1answer
648 views

FindRoot giving false roots with Bessel Functions

I have read in some places about the errors associated with FindRoot, but the closest thing I can find on this website seems to be due to the imaginary unit. I am dealing with what should be a ...
21
votes
3answers
459 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 ...
10
votes
1answer
354 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 ...
16
votes
1answer
571 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: $$\theta(x)=\...
9
votes
2answers
292 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. ...
11
votes
3answers
410 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}] ...
7
votes
6answers
3k 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 ...
20
votes
1answer
641 views

Incorrect result from Integrate

Bug introduced in 8.0 and fixed in 10.0 I attempted to calculate the following integral: ...
17
votes
0answers
235 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
1answer
655 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 ...
10
votes
1answer
939 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 ...
6
votes
2answers
334 views

Cannot Get Numerical Results to Match

I try this numerical summation (in two parts) ...
12
votes
2answers
829 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 10.4. I am trying to evaluate this integral numerically $$ \int_0^{\infty } J_0(q R) \tanh(q) \, \mathrm{d}q $$ for large values of $R$. ...
10
votes
2answers
3k views

problem with coloring spherical harmonics

I want to color a spherical harmonics. So I write as follows. ...
5
votes
1answer
261 views

How to plot the result of this singular integral?

Please I open a new post here after this one : http://mathematica.stackexchange.com/a/59203/10158 Now I want to plot the function $f(a,b)$ as a function of $b$ for different values of $a$ : $a=0.5$ , ...
0
votes
2answers
675 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 q}{(q^2)^{\nu_1}[(\vec{k}-\vec{q})^2]^{\nu_2}}=\frac{\Gamma(d/2-\nu_1)\Gamma(d/2-\nu_2)\Gamma(\...
0
votes
1answer
492 views

How to use `FindRoot` to solve an equation containing a parameter?

I'm trying to derive some of the results of the following paper: Electrodynamics of semiconductor-coated noble metal nanoshells, JT Manassah - Physical Review A In the paper there is matrix $\mathbf ...
-1
votes
1answer
743 views

Gamma Incomplete Function representation in Mathematica and Matlab

I am confused about incomplete gamma function calculation in Mathematica and MATLAB: For example, in Mathematica: Gamma[5,3] = 19.56 But in MATLAB: ...
11
votes
2answers
119 views

Why NIntegrate is badly-behaved on $J_{\frac{9}{2}}(x)$ by default?

A friend of mine showed me this example: Plot[BesselJ[9/2, x], {x, 0, 1}, PlotLabel -> Style["The integrand seems to be simple", 14]] ...
9
votes
1answer
907 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 ...
7
votes
2answers
199 views

Derivative of the Dedekind eta function fails to compute with errors I don't understand

When trying to understand better the question Eisenstein Series in Mathematica? I stumbled on the following: issuing Derivative[1][DedekindEta][.11 I] gives ...
4
votes
0answers
117 views

Puzzled by Assumptions [duplicate]

I don't know if this has already been discussed. Integrate[BesselJ[2 m + 1, x], {x, 0, ∞}, Assumptions -> m ϵ Integers] ...
4
votes
2answers
386 views

How do I evaluate a symbolic integral involving Hermite polynomials?

I want to test a difficult integral : Integral on all reals of some complicated function involving the Hermitian polynomials, exponentials, squares, factorials, and being general considering any ...
4
votes
2answers
333 views

Roots of Whittaker W function

I am interested in finding the roots $u$ of the equation $$ W_{1,\imath b}(a)=0, $$ where $W_{\kappa,\mu}(z)$ denotes the Whittaker $W$ function, $a>0$ is a fixed parameter, $\imath=\sqrt{-1}$ and $...
3
votes
1answer
108 views

Why is the Spherical Bessel Function acting strangely at this point?

I'm doing some computation that requires the use of Spherical Bessel Functions of the 1st kind, at high orders and values. So, I managed to find this, while running it over a wide range of values. I ...
3
votes
3answers
370 views

Hypergeometric function with a matrix argument

I am looking for the evaluation of a Hypergeometric function with a matrix argument as for example in Koev and Edelman or as showcased in this Wikipedia article. From what I understand from ...
3
votes
2answers
487 views

Assigning an analytical approximation to the error function erf(x)

Working with some iterative integral equations, I have Gaussian density functions involved therein. Integrating the Gaussian function I obtain the error function. When I take the second integration, ...
1
vote
3answers
251 views

Find all roots of a function with parabolic cylinder functions in a range of the variable

I want to find all roots of a function involving Parabolic Cylinder Functions. In what follows, I define 2 variables $\xi1$ and $\xi2$, which in turn depend on $\omega$. My function is then defined as ...
23
votes
2answers
1k 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: ...
34
votes
4answers
932 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 http://functions.wolfram.com/...
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 ...
8
votes
2answers
245 views

Number of divisors visualized with the QPochhammer function, how to 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 $q$-...
8
votes
1answer
822 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 ...
14
votes
1answer
461 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 ...
10
votes
2answers
327 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 Limit[n*Sum[...