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Questions tagged [special-functions]

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0answers
69 views

Computation of a functional for large values

Consider the following function : $$f(x) = \sin^2(\frac{π\Gamma(x)}{2x})$$ Now consider the following functional : $$I(x)=\int_0^\infty \frac{f(x + iy) − f(x − iy)}{e^{2πy}-1} dy$$ I need values for ...
2
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0answers
31 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 ...
2
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3answers
252 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 ...
2
votes
1answer
121 views

Coding of the Legendre polynomial and the infinite sum using python

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

Using Spherical Bessel Functions in Python

My question relates to using spherical Bessel functions in Python. If my ODE contains a spherical Bessel function of the form $$j_\ell(tx)$$ and similarly $$y_\ell(tx)$$ for given values of $t$ and $...
0
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2answers
107 views

Calculations to constraint a value with only min and max functions

I have an X value that can go from 0.0000001 to infinite. I want, if this value is less than 0.8, get 0.8 And if that value is equal to or more than 0.8, get 0 How would you do that using only min ...
2
votes
2answers
105 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 ...
0
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2answers
83 views

Is expm1 the right primitive?

I'm writing some code to calculate $\int_0^1 e^{ax} \mathrm{d} x$. Annoyingly there does not seem to be a way of doing this without if statements: ...
0
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2answers
89 views

How can I calculate the exponential integral?

(I originally asked this in a different exchange.) I'm writing a program that uses the prime-counting function. Right now, I'm using x/log(x), but I want to ...
3
votes
1answer
45 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 ...
5
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0answers
111 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
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0answers
197 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 ...
7
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1answer
193 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 ...
2
votes
2answers
350 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 ...
7
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3answers
1k 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 $J'$ ...
1
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2answers
1k views

Continuum removal algorithm in python

I'm using python 2.7 (on jupyter notebook, win10 64 bit) to perform my analysis. I need to perform continuum removal (CR) on a reflectance spectrum data. I need it to be as described here. EDITED: ...
0
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1answer
127 views

Normalization of MATLAB HermiteH

I was wandering - what kind of normalization does Matlab use in hermiteH, its implementation of the Hermite polynomials? It is certainly not the case that they use ...
1
vote
1answer
289 views

Matlab symbolic differentiation of Legendre polynomials

Consider $f(x)= \sum\limits_{n=0}^N a_n p_n (x)$, where $p_n$ are the Legendre polynomials. If one wants to differentiate $f'$ symbolically, i.e. to compute $$f'(x) = \sum\limits_{n=0}^{N-1} b_n p_n (...
6
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1answer
201 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 ...
3
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1answer
94 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} ...
4
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1answer
169 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, ...
7
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0answers
251 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 ...
6
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1answer
91 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) ...
4
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2answers
343 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 ...
4
votes
2answers
619 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 ...
3
votes
1answer
6k 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: ...
4
votes
1answer
148 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 ...
4
votes
2answers
428 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, ...
2
votes
2answers
926 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 ...
6
votes
1answer
450 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 ...
8
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3answers
3k 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. ...
8
votes
1answer
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
votes
2answers
474 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 ...
2
votes
0answers
64 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 $\mathrm{B}_{x,y}(\...
0
votes
0answers
23 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 &...
4
votes
1answer
175 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&...
2
votes
1answer
374 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.
8
votes
1answer
158 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{...
8
votes
1answer
652 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
119 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 ...
9
votes
3answers
2k 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 ...
6
votes
5answers
281 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 ...
15
votes
1answer
317 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 ...
25
votes
4answers
9k 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+...
8
votes
2answers
144 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 ...
16
votes
2answers
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}^{\...
8
votes
3answers
571 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
115 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)\...
15
votes
2answers
1k 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 ...
10
votes
4answers
3k 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)\...