All Questions
Tagged with mathematica hypergeometric-function
13
questions
2
votes
1
answer
67
views
Reduce $\frac{d^{n-1}}{dw^{n-1}}\frac{4^{-n/{\sqrt w}}}{\sqrt w}\Big|_1=\frac1{\sqrt\pi}G^{3,0}_{1,3}\left(^{3/2-n}_{0,1/2,1/2};(\ln(2)n)^2\right)$
In this
answer to Is there any valid complex or just real solution to $\sin(x)^{\cos(x)} = 2$?,
one must calculate $$\frac{d^{n-1}}{dw^{n-1}}\left.\frac{4^{-\frac{n}{\sqrt w}}}{\sqrt w}\right|_1=\...
- 12.5k
1
vote
0
answers
41
views
regularized incomplete beta function integration
Solve $\int_{0}^{1}\frac{I_{u^{\frac{1}{p}}}\left ( p+\frac{1}{a} ,1-\frac{1}{a}\right )}{u}du$ . In Mathematica, this integral does not converge but from an article, I got the answer to this integral ...
1
vote
1
answer
211
views
Solution to degenerate case of hypergeometric differential equation
I am trying to find two independent solutions of this differential equation: $$x(1-x)y''(x)+\left[\frac d2-\left(d+\frac12\right)x\right]y'(x)-\frac{(d-1)d}{4}y(x)=0,$$ for $0<x<1$.
This is a ...
- 171
0
votes
0
answers
83
views
What algorithm does mathematica use to compute the Gauss hypergeometric function?
I recently tried implementing Gauss Hypergeometric function with c++ in two different ways, but found that they each had some problems in certain parameter regions.
The first way uses the naive series ...
- 556
1
vote
0
answers
50
views
Mathematica Kampé de Fériet Notation
I'm looking at the definition of the Kampé de Fériet function in Mathematica's Notation Reference Document (p.36 here), and there is one part of their notation that I'm not quite sure about.
In the ...
- 11
1
vote
1
answer
71
views
Continuation of Hypergeometric Function when $a - b$ is natural number
I am currently implementing the 2F1 Gaussian hypergeometric function numerically, and need to know its continuation for $ |z| > 1 $.
I have researched this and found this nice formula in the ...
- 43
1
vote
1
answer
58
views
Hpergeometric Reduction with Mathematica
We know that:
$$\,_2F_1(a,b,b,z)=(1-z)^{-a}$$
But putting $b<0$ with $b$ integer Mathematica does not use the previous formula but generates the hypergeometric polynomial. For instance:
$$\,_2F_1(a,...
- 187
2
votes
1
answer
115
views
Understanding Mathematica's formula for $ \int_0^{\infty } x^a \exp \left(-\frac{c x^2+f x}{b}\right) \, dx $
My goal is to integrate the following function:
$$
\int_0^{\infty } x^a \exp \left(-\frac{c x^2+f x}{b}\right) \, dx
$$
where, $a, b, c > 0$ and $a, b, c, f \in \mathbb{R}$.
Mathematica gives me ...
- 157
1
vote
0
answers
81
views
Numerical Inversion of an incomplete beta function expressed as gauss hypergeometric function using Mathematica
I am currently working with this hypergeometric function ${_2}F_1$,
$\rho(r)=\frac{2b}{1-q}(1-(\frac{b}{r})^{1-q})^{\frac{1}{2}}{_2}F_1(\frac{1}{2},1-\frac{1}{q-1},\frac{3}{2},1-(\frac{b}{r})^{1-q})$
...
- 89
4
votes
1
answer
672
views
Integration over the Marchenko-Pastur distribution
Problem Statement:
I want to find closed form expression of the following definite integral
\begin{equation}
\int_{\alpha_{-}}^{\alpha_{+}} \ln x \, \frac{1}{2\pi \alpha x} \sqrt{ (\alpha_{+}- x )(x-...
- 1,005
3
votes
0
answers
66
views
Integrating a function times an exponential of a function plus an exponential yields hypergeometric result
During my studies I came across the following integral which is solvable by Mathematica. I tried to solve it by hand, but have not yet found a trick to get the desired result.
To make it even worse, ...
- 65
1
vote
0
answers
369
views
Convergence of a hypergeometric function
The hypergeometric function, ${}_{2}F_1(a,b,c;z)$ can be written in terms of a power series in $z$ as follows, $${}_{2}F_1(a,b,c;z) = \sum_{n=0}^{\infty} \frac{(a)_n (b)_n}{(c)_n} \frac{z^n}{n!}\,\,\,\...
- 2,850
3
votes
2
answers
861
views
Convergence of hypergeometric 2F1 with z=-1
Encountering the hypergeometric function $_2F_1(4n+3,\ m+1;\ m+3;\ -1)$ where $n\in\textbf{N}$ and $m\in\{2,4,6,\ldots,4n-2\}$ I'm a bit confused about its convergence.
According to Erdélyi's "...
- 916