All Questions
15 questions
1
vote
1
answer
79
views
PDF of the difference of two Beta Prime distribution
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&...
- 393
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 ...
3
votes
1
answer
499
views
Does the integral $\int_0^{\infty}e^{cx^2+dx}dx/(a+bx)$ have a closed form?
The integral is
$$\DeclareMathOperator{\dm}{d\!}
\int_0^{\infty}\frac{e^{-cx^2+dx}}{a+bx}\dm x.
$$
Here I assume that $a,b,c,d$ are chosen to make this integral convergent. Rewritting the rational ...
- 375
2
votes
0
answers
252
views
Power series of the modified Bessel function of the second kind
I am looking for a power series representation of
$$ \frac{1}{K_{\nu}(x)}, $$
where $K_{\nu}$ denotes the modified Bessel function of the second kind and $\nu>-1/2$ is not an integer.
I know that ...
- 83
8
votes
0
answers
296
views
Is there a real-analytic approach to evaluate a definite integral (with an elementary integrand) whose value involves Lambert $W$?
I have never seen a real-analytic approach to evaluate integrals of the form below
$$\int_a^b\text{elementary function}(x)\,dx=\text{constant non-trivially involving}\,W(\cdot)\tag1$$ The elementary ...
- 1,459
1
vote
0
answers
35
views
How to relate this integration with the integral expansion of special functions?
I encounter the following integral when trying to find the inverse Fourier transform of the characteristic function of a certain sum of random variables. Here, $p\ge0$, $q\ge0$ are real, and $n,a,b$ ...
- 11
1
vote
1
answer
276
views
Exponential decay bound on integral
I have an integral of the form
$$ \int_R^{\infty} e^{-x} x^n \vert L_m^{\alpha}(x) \vert^2 \ dx,$$
where $L_m^{\alpha}$ is the generalized Laguerre polynomial and $n \ge 0.$
I would to get a nice ...
- 73
2
votes
1
answer
194
views
Generalized Selberg integral
I am interested in expressing the following generalization of the Selberg integral in terms of Gamma functions
$$ \int_0^1 \ldots \int_0^1 \prod_{i=1}^d u_i^{\frac{k_i-1}{2}} \prod_{m=1}^d (1-u_m)^{\...
- 83
2
votes
0
answers
249
views
Calculating $\int_1^{\infty}\frac{\operatorname{ali}(x)}{x^3}dx$, where $\operatorname{ali}(x)$ is the inverse function of the logarithmic integral
It is well-known that we can compute the closed-form of the integrals $$\int_1^{\infty}\frac{\log x}{x^2}dx$$
and $$\int_1^{\infty}\frac{\operatorname{li} (x)}{x^3}dx,$$
where $\operatorname{li} (x)$ ...
4
votes
1
answer
351
views
Asymptotic behaviour of function using Fox $H$-function representation
In equation (9) of this paper, it is claimed that the limiting behaviour
$$
\int_0^\infty \frac{1-\cos(kx_0)}{s+Dk^\alpha}dk
\sim
\frac{\Gamma(2-\alpha)\sin(\pi(2-\alpha)/2)x_0^{\alpha-1}}{(\alpha-1)D}...
- 85
-1
votes
1
answer
149
views
How to choose compactly supported smooth $h$ so $h^2(x)+ h^2(x-1)=1$ for all $x\in [0,1],$ and $\int_{-3/4}^{3/4} |h(x)|^2 dx =3/2$? [closed]
It is known that we may choose smooth $f:\mathbb R \to [0,1]$ such that $f(x)=1$ if $x\geq \frac{3}{4} $ and $f(x)=0$ if $x\leq -\frac{3}{4}+1.$
Define $h(x)= \sin (\frac{\pi}{2} f(x+1))$ if $x\...
- 305
3
votes
1
answer
473
views
A generalized logarithmic function
Consider the function
$$f_\epsilon(x):=\int_0^1 \frac{(1-z)^x-1}{\log(1-\epsilon z)}\, dz,$$
defined for a parameter $\epsilon \in (0,1]$ and $x \geq 0$. When $\epsilon=1$, this is $\log(1+x)$. One ...
- 1,324
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
votes
1
answer
227
views
Solving the integral identity $ \int_{a}^{b} f(x)dx = \int_{a}^{b} f(x)g(x)dx. $ [closed]
We know that 0 is the additive identity and 1 is the multiplicative identity. In the same spirit let us define the integral identity as follows.
Definition: Let $f(x)$ be integrable in $(a,b)$. If ...
- 5,470
4
votes
1
answer
461
views
How to get an expression for this integral (Numerically/Analytically)
I have the following problem:
I need to evaluate the integral $$\int_{\cos(\alpha)}^1 P_l(t)P_{l'}(t) \, dt $$ for $\alpha \in [0,\pi]$ and each combination of $l$ and $l'$, where $P_l$ is the $l$-th ...
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