All Questions
Tagged with integration hypergeometric-functions
25 questions
4
votes
1
answer
228
views
A definite integral of a hypergeometric series related to the enumeration of fusenes
If my calculations are correct, the number of (not necessarily planar) fusenes of perimeter $2n$ grows asymptotically like $\mu^n$ where
\begin{equation}
\mu = \frac{6}{\int_0^{1/6} H(t)\mathrm{d}t} = ...
- 3,927
3
votes
2
answers
336
views
An Integral invoving products of modified bessel functions
I am a physicist working on a problem where the following integrals are concerned:
$$\int_0^\infty k^{l+1} e^{-p^2k^2}I_\mu(k)K_{l-\mu}(k) \, dk$$
$$\int_0^\infty k^{l+1} e^{-p^2k^2}(K_\mu(k))^2 \, dk,...
- 41
2
votes
1
answer
133
views
How to calculate this integral of squared Tricomi hypergeometric function
How to solve this integral
$$
\int_{0}^{\infty}r^2 e^{-\omega r^2}U(-\nu,\frac{3}{2},\omega r^2)^2 \mathrm{d}r
$$
where $\omega>0$ and $\nu \in \mathbb{R} \setminus \left \{ \frac{n-1}{2}\mid n \in ...
1
vote
1
answer
197
views
Derivation of indefinite integral involving hypergeometric function
I am doing a project on projectile motion and I ended up with this integral:
$$\int \frac{m \left(g - \left(\frac{1}{e^t - g^{\frac{m}{c}}}\right)^{\frac{m}{c}}\right)}{c} \, dt$$
where $g, c,$ and $m$...
- 21
4
votes
1
answer
114
views
Antiderivative of $f\left( \xi \right) =\xi^{a}\left( b\xi ^c+K\right) ^d$
I need to know the primitive function (Antiderivative) of this function:
$$f\left( \xi \right) =\xi^{a}\left( b\xi ^c+K\right) ^d$$
where
$K$ is an integration constant,
$d=-\frac{1}{2p}$ with $p<...
- 51
7
votes
1
answer
292
views
On a certain double integral appearing in the Fourier series coefficients of $\mathrm{SL}_2(\mathbb{C})$-Eisenstein series
The following integral appears naturally within the computation of the Fourier series coefficients of a real analytic $\mathrm{SL}_2(\mathbb{C})$-Eisenstein series:
\begin{align*}
\int_{-\infty}^{\...
- 7,609
5
votes
2
answers
466
views
About the integral $\int_{0}^{1}\frac{\log(x)}{\sqrt{1+x^{4}}}dx$ and elliptic functions
NOTE: I post this question on math.stackexchange but nobody answered, so I try here.
For a work we need to evaluate the following integral $$\int_{0}^{1}\frac{\log\left(x\right)}{\sqrt{1+x^{4}}}dx=\,-...
- 319
11
votes
1
answer
566
views
Integral representation of product of two Whittaker functions
Does anyone know anything about the following formula involving special functions:
$$\begin{multline*}
W_{\kappa,\mu}(z)W_{\lambda,\mu}(w)=\frac{e^{-(z+w)/2}(zw)^{\mu+1/2}}{\Gamma(1-\kappa-\lambda)}\...
- 373
2
votes
0
answers
136
views
Integral of Legendre's function
Is there any formula for computing the following integral $$\int_a^1(P^m_l)^2(x)\,dx \, ,\text{with} -1<a<1$$
where $P^m_l$ is the associated Legendre's function (of the first kind) of order $m\...
- 61
2
votes
1
answer
340
views
An integral of $|\sin(x)\cos(nx)^{-2/n}|$ from $-\pi$ to $\pi$
For an integer $n \geq 3$, define
$$A_n = \int\limits_{-\pi}^\pi\frac{|\sin(x)|}{|\cos(nx)|^{2/n}}dx.$$
It is a fact that $A_n$ is finite for all such $n$. I am interested in the behaviour of a ...
- 1,625
0
votes
2
answers
178
views
"Find a representation [using Mellin transform] of 𝑓(𝜇,𝛽) as Gauss hypergeometric function in variable 𝜇"
This is a follow-up to the first comment (by Nemo) to the posting Compute the two-fold partial integral, where the three-fold full integral is known . (I also just asked this as a comment to that ...
- 723
-1
votes
1
answer
149
views
Perform a univariate integral, involving a Gauss hypergeometric function
This is a follow-up question to the one posed in Compute the two-fold partial integral, where the three-fold full integral is known . (I hope that doing so is viewed as a legitimate step. If not so, I ...
- 723
3
votes
4
answers
847
views
Compute the two-fold partial integral, where the three-fold full integral is known
I have the following trivariate ($\rho_{11}, \rho_{22}, \mu$) function
\begin{equation}
4 \mu ^{3 \beta +1} \rho_{11}^{3 \beta +1} \left(-\rho_{11}-\rho_{22}+1\right){}^{3 \beta +1}
\rho_{22}^{3 \...
- 723
1
vote
0
answers
212
views
Euclidean volumes under the matrix norm of one-parameter subsets of unit balls of $2 \times 2$ matrices
Prove that the Hilbert-Schmidt volume (normalized to equal 1 at $\varepsilon=1$) of the subset of the unit ball in operator norm (http://mathworld.wolfram.com/OperatorNorm.html) of the $2\times 2$ ...
- 723
3
votes
1
answer
237
views
integral involving hypergeometric function of matrix argument
This conjecture comes from an observation on simulations of the matrix variate noncentral Beta distribution (similar to this observation, but I open a new question because yet I'm not sure it is ...
- 2,560
5
votes
3
answers
1k
views
Perform an integration involving the product of two hypergeometric functions
I've encountered the following product,
\begin{equation}
\, _2F_1\left(3 d+2,3 d+2;6 d+4;1-\frac{1}{t^2}\right) \,
_3F_2\left(-\frac{d}{2},\frac{d}{2},d;\frac{d}{2}+1,\frac{3 d}{2}+1;t^2\right)
\...
- 723
4
votes
1
answer
211
views
Perform an integration over the unit interval of a two-parameter expression involving a Gauss hypergeometric function
In a quantum-information-theoretic context, I've encountered the problem
of integrating over $r \in [0,1]$, the function
\begin{equation}
r^{2 d-1} \, _2F_1\left(-\frac{d}{2},\frac{d}{2};\frac{d+2}{2}...
- 723
5
votes
1
answer
1k
views
Integral involving Laguerre, Gaussian and modified Bessel function
I am trying to prove that the integral
\begin{align}
\int_{0}^{\infty } e^{-\frac{r^2}{2B}} r^{l-n}
L_n^{l-n}\left(\frac{r^2}{C}\right) I_{l-n}\left(\rho r \right) r dr
\end{align}
has ...
- 53
2
votes
2
answers
855
views
Integral with Bessel function and hypergeometric function ${}_2F_2$: explicit expression for these polynomials?
This question follows this one, where the general problem has apparently no simpler form than the integral one. I focus now on the limit case:
\begin{align}
\int_0^T e^{-x}\frac{nI_n(x)}{x}dx=\int_0^T ...
- 634
5
votes
1
answer
522
views
Closed form for $\int_0^T e^{-x}\frac{I_n(\alpha x)}{x}dx$
EDIT: Some additional details and corrections, I would appreciate any information about the highlighted expression.
I try to solve $\int_0^T e^{-x}\frac{I_n(\alpha x)}{x}dx$ where $I_n(x)$ is the ...
- 634
1
vote
0
answers
121
views
On the Integration and Manipulation of Expressions Involving Hypergeometric Functions
I would like to ask the following two:
For the integral:
\begin{equation}
\int_{0}^{t}\left( 1-s^p \right)^{\frac{1-p}{p}}ds
\end{equation}
I know that it is reduced to the following product ...
- 111
6
votes
1
answer
468
views
Definite integral involving inverse regularized incomplete beta functions
In my research I encountered the following integral
$$J = \int_0^1 I_t^{-1}(a_1,b_1) \: I_t^{-1}(a_2,b_2) \: \mathrm{d}t$$
which I would like to evaluate as a closed form expression, that is, as a ...
- 1,538
4
votes
0
answers
238
views
Generalize $\frac1{48^{1/4}\,K(k_3)}\,\int_0^1 \frac{dx}{\sqrt{1-x}\,\sqrt[3]{x^2+27x^3}}=\,_2F_1\big(\tfrac13,\tfrac13;\tfrac56;-27\big)=\frac47\,$?
These involve separate cases $a=\tfrac13,\,a=\tfrac14,\,a=\tfrac16$ of,
$${_2F_1\left(a ,a ;a +\tfrac12;-u\right)}=2^{a}\frac{\Gamma\big(a+\tfrac12\big)}{\sqrt\pi\,\Gamma(a)}\int_0^\infty\frac{dx}{(1+...
- 12.6k
4
votes
1
answer
1k
views
Indefinite integral of squared hypergeometric function
I am trying to compute the indefinite integral
$$
\int_0^u {}_2F_1\left(\frac{1}{4},\frac{5}{4},2,1-v^2\right)^2 dv
$$
for $0<u<1$. Using Clausen's formula for the square of the hypergeometric ...
- 165
1
vote
0
answers
302
views
Integration involving modified bessel function, exponential and power
I need to find the following integration.
$$
\int_0^a e^{-(N-1)x}(\sqrt{4x}K_{1}(\sqrt{4x}))^N
$$
where
$$
a>0, \quad N \geq 1
$$
Any help will be much appreciated.
BR
Frank
- 119