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2
votes
0answers
43 views

Accurate computation of $\frac{\mathrm{B}_{x,y}(\alpha + 1,\beta)}{\mathrm{B}_{x,y}(\alpha,\beta)}$ for large paramers?

I need to calculate these ratios: $$\frac{\mathrm{B}_{x,y}(\alpha + 1,\beta)}{\mathrm{B}_{x,y}(\alpha,\beta)} \tag{1}$$ where $\alpha,\beta > 0$ and $0\le x\le y \le 1$. Here ...
0
votes
0answers
16 views

Compute hypergeometric function ratio: $\frac{_{2}F_{1}(a+1,b;c;x)}{_{2}F_{1}(a,b;c;x)}$, for large positive $a,b,c$? [duplicate]

I need a numerically stable way to compute the following ratio: $$\frac{_{2}F_{1}(a+1,b;c;x)}{_{2}F_{1}(a,b;c;x)}$$ All the parameters are real positive numbers, with $0 < a,b,c$ and $0 < x ...
3
votes
1answer
93 views

Compute hypergeometric function ratio: $\frac{_{2}F_{1}(a+1,b;c;x)}{_{2}F_{1}(a,b;c;x)}$?

I need a numerically stable way to compute the following ratio: $$\frac{_{2}F_{1}(a+1,b;c;x)}{_{2}F_{1}(a,b;c;x)}$$ All the parameters are real numbers, with $a< 0$,$\ $ $b,c > 0$ and ...
1
vote
1answer
64 views

Appell function implementation in C++?

Is there a C++/C implementation of the Appell series? GSL and Boost do not seem to have this function.
5
votes
2answers
82 views

Second derivative of the Associated Legendre functions

I would like to compute, as part of the solution of the Laplace equation using the Fast Multipole Method, the second derivative of the associated legendre functions of the first kind . Specifically, I ...
3
votes
1answer
84 views

Efficient computation of tangent of fraction of angle

I want to compute $a = \tan(f \theta)$ for $f\in [0,1]$, given $g = \tan\theta$. Obviously, I can compute $a = \tan(f\tan^{-1}g)$, but I'm wondering if there's a more efficient way that avoids having ...
4
votes
2answers
654 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 ...
6
votes
5answers
237 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 ...
14
votes
1answer
221 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 ...
21
votes
4answers
2k 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} ...
9
votes
2answers
127 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 ...
11
votes
2answers
260 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) = ...
6
votes
3answers
328 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
92 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
174 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
663 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
195 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 ...
8
votes
1answer
137 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 ...