The special-functions tag has no wiki summary.
4
votes
2answers
180 views
How to use polylogarithm function in c++?
Is there any preprocessor directives that could be used to use the polylog function? Or is it included in cmath? If so, do you call it by Li or by polylog?
EDIT:
What I really am trying to do is ...
5
votes
5answers
181 views
Evaluate the sum
I want to evaluate the sum $$\sum_{k=1}^\infty \left(\frac{i+1}{\sqrt{2}}\right)^k\cdot k^{-\alpha}$$ where $i=\sqrt{-1}$ and $\alpha\in[\frac{3}{4},1]$ with 8 digits accuracy.
If I am willing to ...
13
votes
1answer
142 views
Does transforming $J_0(x)\to\int\cos(x\sin\theta)$ help with numerical integration?
I've heard anecdotally that when one is trying to numerically do an integral of the form
$$\int_0^\infty f(x) J_0(x)\,\mathrm{d}x$$
with $f(x)$ smooth and well-behaved (e.g. not itself highly ...
14
votes
3answers
369 views
Method for numerical integration of difficult oscillatory integral
I need to numerically evaluate the integral below:
$$\int_0^\infty \mathrm{sinc}'(xr) r \sqrt{E(r)} dr$$
where $E(r) = r^4 (\lambda\sqrt{\kappa^2+r^2})^{-\nu-5/2} ...
7
votes
2answers
113 views
Computing ratio of trigonometric functions
I have need to compute the functions:
$$ f(x) = \frac{\sin^{-1}x}{x}$$
and
$$ g(x) = \frac{\sin a x}{\sin x} $$
where $a\in[0,1]$ and $ x\in[0,\frac{\pi}{2}]$ and is often very small ($x\ll 1$). Are ...
9
votes
2answers
123 views
What are the efficient, accurate algorithms for evaluation of hypergeometric functions?
I'm curious to know what good numerical algorithms exist for evaluation of the generalized hypergeometric function (or series), defined as
$${}_pF_q(a_1,\ldots,a_p;b_1,\ldots,b_q;z) = ...
5
votes
3answers
171 views
Evaluating sine and cosine of an integer multiple of an angle
When evaluating cylindrical harmonics, one needs to evaluate trigonometric functions $\cos(m\theta)$ and $\sin(m\theta)$, potentially for large integer $m$ and $\theta\in[-\pi,\pi]$. What is the best ...
3
votes
2answers
87 views
Representing an integral as a special function
In my research I have come across the following integral
\begin{equation}
f = \int_0^{2\pi} \text{d}\theta \exp\left\{\frac{3}{2}(h_1 \cos^2\theta + h_2 \sin^2\theta + 2 h_{12} \sin\theta ...
5
votes
1answer
81 views
Open source implementation of rational approximation to a function
I am looking for some open source implementation (any of Python, C, C++, Fortran is fine) of rational approximation to a function. Something along the article [1].
I give it a function and it gives me ...
5
votes
3answers
207 views
Fast and accurate double precision implementation of incomplete gamma function
What is the state of the art way of implementing double precision special functions? I need the following integral:
$$
F_m(t) = \int_0^1 u^{2m} e^{-tu^2} d u
= {\gamma(m+{1\over 2}, ...
4
votes
1answer
123 views
An efficient way to numerically compute Stirling numbers of the second kind?
Is there an efficient way to numerically compute Stirling numbers of the second kind?
An approximate (not exact) method would suffice. Something similar to the connection between factorials and gamma ...
7
votes
1answer
116 views
Polynomials that are orthogonal over curves in the complex plane
Various important sets of polynomials (Legendre, Chebyshev, etc.) are orthogonal over some real interval with some weighting. Are there known families of polynomials that are orthogonal over other ...
