All Questions
Tagged with integration special-functions
1,101
questions
0
votes
0
answers
37
views
Writing $\ln(x+1)e^{-ax}$ in terms of Meijer-G function
Is there any way to write $f(x)=\ln(x+1)e^{-ax}$ in terms of Meijer-G function? I tried calculating Mellin transform of $f(x)$ to no avail. Frustrated, I used Mathematica to get the following answer
$$...
- 3,263
4
votes
2
answers
133
views
Reference for $\int_{-\infty}^{\infty}e^{a x^4+b x^3+cx^2}dx\;$?
In my research I encounter an integral of the form
$$
\int_{-\infty}^{\infty}{\rm e}^{\large ax^{4}\ +\ bx^{3}\ +\ cx^{2}}\,{\rm d}x\qquad a < 0,\quad b, c \in \mathbb{R}
$$
So the integral is ...
- 41
2
votes
0
answers
24
views
Orthogonality of Whittaker functions
Is there a known orthogonality property of Whittaker functions $W_{\kappa,\mu}(iz)$ with respect to the first index as an integral over the argument? I am particularly interested in the case $\mu=0$ ...
- 433
1
vote
2
answers
109
views
Definite integral $\int_b^\infty \frac{e^{-ax}}{\sqrt{\cosh{x}-\cosh{b}}}dx$? [closed]
For positive $a$ and $b$, consider the integral
$$ I(a,b)=\int_b^\infty \frac{e^{-ax}}{\sqrt{\cosh{x}-\cosh{b}}}dx. $$
Does $I(a,b)$ have a closed-form expression (far-fetched hope)? If not, does it ...
- 21
3
votes
0
answers
84
views
What is the origin of the non-analytic behavior of the integral $\int_0^\infty \frac{t^3}{t^4+1} J_2(x t) dt$?
I came across integrals of the type $I(x) = \int_0^\infty \frac{t^3}{t^4+1} J_2(x t) dt$
From numerical integration, it seems that when $|x| \rightarrow 0$, the integral is reaching a limiting value ...
- 2,424
4
votes
0
answers
182
views
Definite integral involving K Bessel function and a square root
I have recently been trying to evaluate some integrals involving the modified Bessel function $K_0(x)$. The specific integrals are
$$L(x,u) = \int_0^{1} K_0\left( 2x \sqrt{r(1-r)} \right) \exp(2ixur) ...
- 51
1
vote
0
answers
34
views
Mellin transform of confluent Lauricella hypergeometric function
The $F_D^{(n)} $ Lauricella's hypergeometric function can be defined as follow
$$F_D\left(a,b_1,\cdots,b_n;c;x_1,\cdots,x_n\right) = \sum_{m_1=0,\cdots,m_n=0}^{\infty}\frac{\left(a\right)_{m_1+\cdots+...
- 3,263
4
votes
1
answer
98
views
Integral of $\exp(\langle x, a+ib\rangle)$ over hypersphere
I am looking to compute the following integral over $S_{n-1}$ the unit hypersphere in $\mathbb{R}^n$
\begin{equation}
I(a,b) = \frac{1}{|S_{n-1}|}\int_{S_{n-1}}e^{\langle x,a+ib \rangle}dx
\end{...
- 43
2
votes
1
answer
127
views
Inverse Laplace transform of product of three exponentials
Consider the Laplace Transform of the form
$$
\operatorname{F}\left(s\right) =
\frac{1}{\left(s - s_{1}\right)^{\large a_{1}}
\left(s - s_{2}\right)^{\large a_{2}}
\...
- 3,263
1
vote
0
answers
70
views
Laplace transform of special function
The Confluent hypergeometric function of first kind (aka Kummer's function) is defined as
$${\mathbf{M}}\left(a,b,z\right)=\frac{1}{\Gamma\left(a\right)\Gamma\left(b-a%
\right)}\int_{0}^{1}e^{zt}t^{a-...
- 3,263
5
votes
1
answer
206
views
How to prove the result of the following integral? [duplicate]
How to prove that
$$
\int _0^{\infty }\frac{K\left(\frac{1}{2}-\frac{1}{2 \sqrt{x+1}}\right)}{\sqrt[4]{x+1}}e^{-x}{\rm d}x =
\frac{1}{2} \sqrt{e \pi } K_{1/4}\left(\frac{1}{2}\right)
$$
where $K(x)$ ...
- 239
0
votes
1
answer
44
views
Another general form of beta function evaluation
Is there a general formula for this integral, $$\int_0^1 \frac{x^a (1-x)^b}{(c+hx)^{a+b+2}}dx$$
I encountered an integral which broke into two smaller integrals and both of them were of the above ...
1
vote
0
answers
123
views
How to evaluate the integral $\int_0^\pi e^{i(a\sin x + b\cos x)} dx$
We know that (from e.g. here)
$$
\int_0^{2\pi}{\rm e}^{{\rm i}\left[a\sin\left(x\right) + b\cos\left(x\right)\right]}{\rm d}x =
2\pi\operatorname{J}_{0}\left(\sqrt{a^{2} + b^{2}}\right)
$$
where $\...
- 123
1
vote
0
answers
21
views
PDF of the difference of two Beta Prime distributions
I am struggling to find the PDF of the difference of two Beta Prime distribution.
Definition
A random variable is said to have a Beta Prime distribution $\text{B}'(\alpha, \beta)$ with $\alpha, \beta&...
- 506
2
votes
2
answers
154
views
Solving an "impossible" integral, $\int \frac{dx}{a^{x^2}+b^{x^2}} $
First a saw this problem
$$\int \frac{dx}{a^{x^2}+b^{x^2}} $$
where a,b are natural numbers
This problem would be easier if $a=b$
then error function appears here
I have tried with Wolfram alpha but ...
- 29