# Questions tagged [special-functions]

Questions on the special mathematical functions implemented in Mathematica.

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11k 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 ...
4k views

### Numerical underflow for a scaled error function

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

### Why does Expand not work within a function?

I'm writing this fairly simple function: ...
12k views

### What is the difference between a few simplification techniques?

I am trying to understand the difference between Refine, Simplify and FullSimplify, and when ...
421 views

### Integrating a real function I get a complex value, while after variable transformation the result is real. Bug?

I have the following integral: Integrate[1/Sqrt[0.7 + 0.3*(1 + z)^3], {z, 0, ∞}, Assumptions -> z ∈ Reals] >> -3.36354 - 3.85013 I The output is complex, ...
499 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 ...
700 views

### Is MathieuC for moderately large imaginary arguments broken?

Bug introduced in 3.0 and persisting through 12.0 (reported as CASE:3208982) I'm trying to plot MathieuC[-3,0.3,I x] for $x\in[0,10]$, and here's what I get even ...
1k 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 ...
446 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: ...
625 views

### is it possible to change/customize some conversions done by TeXForm?

I use TeXForm to convert output of some computation to Latex. I'd like to ask if there is a way to override/change/customize some of these conversions, related to just the name Mathematica assigns to ...
653 views

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] ...
1k 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 ...
2k 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: ...
355 views

### Fine tuning compiled code that computes dilogarithm function

As an exercise in 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 ...
1k 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 ...
6k views

### Plotting a Bifurcation diagram

I have the following system equation v'(t)=2*G*J1[v(t-τ)]cos(w*τ)-v(t) How do you plot the bifurcation diagram, τ in the x ...
270 views

### Why do these two different zetas produce the same value? [closed]

Zeta[-13] == Zeta[-1] == -1/12 Why do these two different zetas produce the same value?
855 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 ...
1k 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/...
2k views

690 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. ...
654 views

### Analytical approximation of indefinite integral on a given interval to a given precision

I'm looking for an analytical approximation of $\int_a^b e^{-x^2}\mathrm{erf}(x+c) dx$ that would be accurate to precision $\varepsilon$ for $a,b,c$ within a certain range. How do I ask Mathematica ...
896 views

### Implementing a compilable Faddeeva function of complex argument

For those, who are looking at Halirutan's answer and thinking "gee, I wish I was that good at LibraryLink, then I could really speed up my code!" I leave here the ...
664 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 ...
985 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 ...
4k views

### problem with coloring spherical harmonics

I want to color a spherical harmonics. So I write as follows. ...
330 views

### Taylor expansion of a function containing QPochhammer[q, q, n]

I want to get the following series expansion: Series[QPochhammer[q, q, 3], {q, 0,4}] but in Mathematica 11.0, I obtain the following gibberish: There weren't ...
878 views

4k views

### Creating a Mathieu stability diagram

I am attempting to re-create a Mathieu stability diagram like the one shown here: I expected that I could use MathieuC to generate this graph by assuming that ...
2k views

### Visualizing the Riemann zeta function

How to visualize in a nice geometric way (e.g., like in this YT video) the Riemann zeta function, $\zeta (z)$?
964 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)=\...
743 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 ...
506 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}] ...
555 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 $q$-...
389 views

### Backslide in NSolve in V11.1?

Bug introduced in 11.1.0 and persisting through 11.1.1 In V11.1, NSolve[BesselJ[0, x] == 0 && 0 < x < 20, x] returns no solutions: But in V11.0 (...
642 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 ...