Questions tagged [special-functions]

For questions about evaluating or implementing special functions, e.g. Bessel, hypergeometric, gamma, Lambert W.

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26 votes
4 answers
10k 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} K_{-\nu-5/2}(\lambda\sqrt{\kappa^2+...
  • 491
16 votes
2 answers
1k 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) = \sum_{k=0}^{\...
  • 3,333
15 votes
2 answers
2k 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 ...
15 votes
1 answer
335 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 ...
  • 3,333
10 votes
4 answers
4k 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}, t)\...
10 votes
1 answer
204 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 ...
10 votes
0 answers
311 views

Numerical integration using interval arithmetic, nowadays

Is there now a package for rigorous numerical integration that uses interval arithmetic and has access to a well-developed library of special functions? By "well-developed", I mean something that, at ...
9 votes
3 answers
3k 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 give ...
9 votes
3 answers
4k views

What is the fastest opensource implementation of Bessel functions computation?

I'm looking for an open-source (to use and learn from) software which computes Bessel functions of integer order of real argument to double precision the fastest among all such implementations. ...
  • 204
8 votes
2 answers
145 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 ...
  • 4,420
8 votes
3 answers
684 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 ...
  • 4,420
8 votes
1 answer
490 views

Accurate and efficient computation of the inverse Langevin function

The Langevin function $\mathcal{L}(x) = \mathrm{coth}(x) - \frac{1}{x}$ occurs in computations related to elastomers and paramagnetic materials. It is easily computed accurately and with high ...
  • 1,330
8 votes
1 answer
2k views

Newton iteration for cube root without division

It's a fairly well known trick to avoid division in calculating square-roots to apply Newton's method to finding $1/\sqrt{x}$, and probably better known, using Newton's method to find reciprocals ...
  • 3,199
8 votes
1 answer
199 views

Radial integration of expensive function with Bessel weights

I need to calculate the integral $$I = \int_0^R f(r)J_n\left(\frac{z_{nm}r}{R}\right)rdr$$ where $J_n$ is the $n^{\mathrm{th}}$ order Bessel functions of the first kind, $z_{nm}$ is its $m^{\mathrm{...
  • 81
8 votes
1 answer
441 views

What algorithm does (or did?) Excel use for Bessel functions that is discontinuous at x=8?

Writing this comment reminded me of something I noticed years ago about evaluating Bessel functions of the first kind $J_n(x)$ in Excel. (BESSELJ) I don't use Excel now but at the time I'd checked ...
  • 946
8 votes
1 answer
881 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 ...
  • 365
7 votes
3 answers
2k views

Finding the first N roots of transcendental equation

I need to find the first $n$ roots of the transcendental equation \begin{equation} F(k) = J_m'(kr)Y_m'(k)-J'_m(k)Y'_m(kr) \end{equation} for integer values of $m$ and any $r \in [0,1)$ where $J'$ ...
  • 873
6 votes
5 answers
293 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 ...
  • 61
6 votes
2 answers
3k 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 ...
6 votes
1 answer
231 views

Numerical integration of a hypergeometric function

The Task Let $z_1, z_2, z_3$ be positive real numbers and define $$ r(\mathbf{z}):= \int_0 ^\infty (t+z_1)^{-3/2}(t+z_2)^{-3/2}(t+z_3)^{-1/2}\text{d}t. $$ The task is to compute $r$ numerically in ...
6 votes
1 answer
109 views

Numerical computation of the complex elliptic integral $E(k)$ for medium $|k|$

I have implemented Carlson's algorithm for $E(k)$ from Numerical computation of real or complex elliptic integrals (available from ArXiv eprint, see also DLMF). It is essentially his formula (46) ...
6 votes
1 answer
600 views

Using Log Gamma function to avoid overflow

I have to do some numerical calculus using gamma functions. I am using the tgamma incluided in the C++ cmath library. The ...
5 votes
2 answers
1k views

I am searching for a C++ code implementing the complex polygamma function

I googled about some code on the complex polygamma function especially C++ code, but can't find anything. Does anyone know where to find such code? The complex digamma function does exist but not the ...
5 votes
0 answers
138 views

Best way to numerically compute elliptic integrals of the third kind with complex arguments?

I need to compute elliptic integrals of the third kind with complex arguments, preferably in C++. Is there code out there to do this? I have discovered the Arb library, but that does much more than I ...
4 votes
1 answer
225 views

Numerical evaluation of the Exponential Integral Ei by rational Chebyshev approximations fails

I am trying to evaluate the Exponential Integral $Ei(x)=-\int^{\infty}_{-x}\frac{e^{-t}}{t}dt$ for $x>0$ (interpreted as the Cauchy principal value) by using rational Chebyshev approximations, ...
  • 155
4 votes
2 answers
518 views

Interpolation with the roots of orthogonal polynomials & Spectral expansion

I'm a bit confused about the relationships between these two approximation methods mentioned in the title. Does this kind of interpolation also belongs to the field of spectral methods? Are the ...
  • 649
4 votes
2 answers
466 views

overflow upper incomplete gamma function

I want to calculate the following equation: $$\frac{\theta \Gamma \left(\kappa+1,\frac{o}{\theta }\right)-o \Gamma \left(\kappa,\frac{o}{\theta }\right)}{\Gamma (\kappa)}+o+s$$ with $s>0, o>0, ...
4 votes
1 answer
197 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 $0<x&...
  • 1,689
4 votes
2 answers
293 views

Evaluation of real-valued confluent hypergeometric function with specific complex arguments

In my C++ code, I need to evaluate the confluent hypergeometric function ${}_1F_1(a,b;z)$ with complex arguments in a special case. More precisely, I have to compute $$ e^{-i\phi}{}_1F_1(\ell+1+iZ,2(\...
4 votes
1 answer
86 views

Worst Case complexity of a search engine algorithm

Computer make it possible to find information in large databases. However, the results are often too large to be returned in their entirety to the user who requests them. Computer therefore sort the ...
user avatar
4 votes
1 answer
161 views

How do I develop numerical routines for the evaluation of my own special functions?

This question was previously posted to Math.SE here and had received no answers at the time of this posting. When performing computational work, I often come across a univariate function, defined in ...
3 votes
1 answer
8k views

Python Vectorizing a Function Returning an Array

I have the following function that has been vectorized so that for every element in input array t, an array is output: ...
  • 33
3 votes
2 answers
123 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 \cos\theta)\...
3 votes
2 answers
474 views

What is an efficient way to calculate zeros of Bessel functions?

One approach is the brute force method of evaluating at all points at fixed intervals and when it nears zero write value, this can be combined with adaptive step size. Another approach is ...
3 votes
1 answer
109 views

Numerical evaluation of gaussian-like integral expressible as a recurrence relation

I'm looking to numerically evaluate $\log f_p(z)$ and its derivative $f^\prime_p(z)/f_p(z)$ accurately and efficiently in floating-point, where $$ f_p(z)=\int_0^\infty r^{p-1} \exp\left(-\tfrac{1}{2} ...
3 votes
2 answers
558 views

Bessel EVP and fzero

I am trying to solve the Eigenvalue problem $$ x^2 y''+ x y' + x^2 y = \lambda^2 y\,,\quad x\in(0,1)\,,\quad y(0)=0\,,\quad y'(1)=y(1) $$ The differential equation is the Bessel equation. The solution ...
3 votes
1 answer
123 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 ...
  • 4,420
3 votes
1 answer
177 views

What problems does softmax() solve and when should I think of using it - in simple terms

I just for the first time saw the function softmax() in this SO answer to How do I use a minimization function in scipy with constraints and was intrigued. Another way of weighting variables where ...
  • 946
3 votes
1 answer
80 views

Robust ways of evaluating $j_n(x+iy)/e^y$

For reasons to do with extending the range of applicability of this Mathematica package, I would like to have a good handle on the numerical evaluation of functions of the form $$ f(x) = \frac{j_n(\...
3 votes
3 answers
213 views

Hypergeometric function $_2F_1(z)$ with $|z| > 1$ in GSL

I need to evaluate the hypergeometric function $_2F_1$ with $|z| > 1$ as in Wolfram Language with GSL but the GSL documentation says the $_2F_1$ needs $|z| < 1$. Is there any way I can use GSL ...
  • 53
3 votes
1 answer
98 views

Robust ways to find zeros of the Tricomi confluent hypergeometric function as a function of its parameters

I'm solving a quantum mechanical problem, and the quantization condition requires me to solve the equation $$ U\left(\frac12(\ell+1-E), \ell+1, r^2\right) = 0, $$ where $U(a,b,z)$ is the confluent ...
2 votes
3 answers
276 views

Comparison of integrals with a function:

Consider the following integral: $$S(q)=\int_{x=2}^q\sin^2\left(\frac{π\Gamma(x)}{2x}\right)dx$$ And consider the functions : $$R(q)=\frac{q}{\log(q)}$$ $$T(q)=\int_2^q\frac{1}{\log(x)}dx$$ I ...
  • 129
2 votes
1 answer
608 views

Coding of the Legendre polynomial and the infinite sum using python

I'm looking to code the following function: $$|g(\theta)=\frac{1}{2k}\sum_{\ell=0}^{\infty} (2\ell +1)\sin(2\delta_{\ell})P_{\ell}\cos(\theta)|^2$$ ...
2 votes
2 answers
1k views

Kummer's confluent hypergeometric for complex arguments in C/C++?

I need to evaluate Kummer's confluent hypergeometric function for imaginary arguments: $$_1F_1(a,b;ix)$$ where $i$ is the imaginary unit, $a,b,x$ are real, and $a,b>0$. Is there a routine ...
  • 1,689
2 votes
1 answer
454 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.
  • 1,689
2 votes
2 answers
709 views

Integration of a diverge function in c++ GSL Library

I am trying to perform an Integral of Hypergeometric function 2F1(a,b,c,x) from -1 to 1 for some good values of $a,b,c$ (lets say $a=1,b=2,c=3$) . I did it in ...
2 votes
1 answer
333 views

Help with modified bessel functions

I'm trying to parse the following expression (Eq. 35 in The Magnetic Field in the Vicinity of Parallel and Twisted Three-Wire Cable Carrying Balanced Three-Phased Current) in Python and calculate its ...
2 votes
1 answer
155 views

Numerical evaluation of Fourier transform of a scaling function

Given a set of filter taps $\{h_n\}_{n=0}^{m-1}$, define a scaling function $\phi$ by $$\phi(x) = \sqrt{2}\sum_{n} h_n \phi(2x-n).$$ In keeping with the notation from Daubechies "Ten Lectures on ...
  • 2,063
2 votes
0 answers
154 views

Best way to compute given functional with accuracy:

I need to plot the following functional with accuracy: $$ I(x,s) =\int_0^\infty\mathrm dy \frac{F(x + \mathrm iy,s) − F(x −\mathrm iy,s)}{\mathrm e^{2πy}-1}, $$ Where $ F(z,s) = \dfrac{1}{z^s\Gamma(\...
  • 129
2 votes
0 answers
46 views

To use the confluent hypergeometric function or not to?

I am numerically computing the following integral as a function of positive $k$. $$I(k) := \int_0^\infty x^b(k+x)^{a-1} e^{-x} dx \tag1$$ It is shown in math.stackexchange.com that this can be ...
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