Let $$J_1=\int_0^1\frac{1}{\sqrt{1-t_2}}dt_2,$$ $$J_2=\int_0^1 \int_0^{t_2}\frac{1}{\sqrt{1-t_2}}(\frac{1}{\sqrt{1-t_3}}+\frac{1}{\sqrt{t_2-t_3}})dt_3dt_2,$$ $J_3=\int_0^1 \int_0^{ …

Put $\eta = r+it, r>0$ and $K_{\eta}(z, w, \lambda)= \frac {\mid \lambda\mid }{2\sinh(\eta \mid \lambda \mid) } e^{-\frac{\mid\lambda\mid \coth(\eta | \lambda |) \mid z-w \mid ^{ …

given an integral equation $$ g(s)= \int_{0}^{\infty}dx \ f(x)K(xs)dx $$ if we name the function $ M(n+1)= \int_{0}^{\infty}K(t)t^{n}dt $ and $ g(s) $ has an expansion $ g(s)= \ …

I need to find the expectation of $\ln (x-\epsilon) $ with respect to a probability distribution $\mathbb{P}(x)$. A direct evaluation seems very difficult as the expression for $ \ …

I have the following double integral: $\int\limits_0^x {\int\limits_0^y {{e^{ - {K_1}(u + v)}}{I_0}\left( {2{K_1}\sqrt {uv} } \right)dudv} }$ where $K_1$ is a constant. Do you have …

How do I solve the Integral $$ \frac{1}{2\pi j} \oint \frac{b^{ - s} \Gamma[2 + i - s] \Gamma[s] \Gamma[-1 - i + s]}{ (2 + i - s) \Gamma[3 + i - s]} \:\mathrm{d}s$$ This integral …

A smooth function $f(x)$ of variable $x=(x_1,\ldots,x_n)>0$ is called completely monotone if for any multiindex $\alpha \in \mathbb{N}^n_0$ the equality holds: $$ (-1)^{|\alpha …

Dear all, please help me solve the following integral. I need to solve this integral for one of my problems. $$(\frac{1}{2\pi})^2\int_0^\infty\int_{-\infty}^\infty \frac{J_0(\rho …

Let $\Phi(x,y,\theta)$ be a phase function defined on $X \times X \times (\mathbb R^n-0)$ where $X$ is some domain in $\mathbb R^n$, let $A(x,y,\theta)$ be an amplitude function. A …

Problem. To give a concrete example of a geodesically reversible Finsler metric on the two-sphere that is not just the sum of a reversible Finsler metric and an exact $1$-form. So …

I wish to solve the following integral equation, preferably analytically, and find $\int_{-l}^l f(x) dx$. If analytical solution is too complicated, any suggestion for the computat …

Hi, I'm trying to employ Mercer's theorem on the kernel $k(x,y)=\min(x,y)$. It is known (and easy to verify) that this is a nonnegative-definite kernel over $[0,T]$ for any $T>0$. …

Assume that $f:[0,2\pi]\to [0,2\pi]$ is a continuous function such that $f(0)=f(2\pi)$ and define the function $$g(s)=\int_{-\pi}^\pi \frac{\sin f(t+s)-\sin f(s)}{\sin t/2} dt.$$ I …

Theorem (Michel). A $1$-form on the projective plane is exact if and only if its integral over any projective line is equal to zero. Is there a simple proof of this result due, I …

Hello! It may be a stupid question, i'm trying to find a closed form for an integral similar to a Fourier transform on $S^{n}$ but i'm stuck... Let $\alpha>0$, the integral i can' …