3
votes
0answers
45 views

Analytic Continuation of a nowhere existing Mellin Transform

I'm trying to give sense to the (self-made) statement: The analytic continuation of $\int_0^\infty e^t\;t^{s-1}\;dt$ is holomorphic in $s=0$. At first sight this could seem completely ...
36
votes
1answer
983 views

To evaluate $\int_0^{+\infty} \frac{\;\mathrm dx}{\sqrt[3]{x^3+a^3}\sqrt[3]{x^3+b^3}\sqrt[3]{x^3+c^3}}$

$$f(a,b)=\int_0^{+\infty} \frac{\;\mathrm dx}{\sqrt{x^2+a^2}\sqrt{x^2+b^2}}$$ To use Landen's transformation $$f(a,b)=\int_0^{+\infty} \frac{\;\mathrm ...
0
votes
1answer
151 views

Problem evaluating an improper integral $\int_0^{\infty} \frac{(\sin{2x}-2x\cos{2x})^2}{x^6}$ using fourier transform

This is a question from one of the past papers of my university which I am unable to do. I am not being able to do question 2 from below. Let $f(x)= a^2-x^2 \,\,\,\,\, |x|<a ...
2
votes
1answer
682 views

Evaluating improper integrals using laplace transform

I want to calculate the following improper integral using Laplace and transforms (and laplace transforms only). $$\int_0^{\infty} x e^{-3x} \sin{x}\, dx$$ I propose the following method. I plan to ...
0
votes
1answer
120 views

Question about L2 Inner Product and Integrals

Does exists $f\in L_2(\mathbb{R}^d)$ such that for all $g\in L_2(\mathbb{R}^d)$ which is not identically zero: ...
4
votes
2answers
260 views

Mellin Transform of $\sin$

Can one show that the following integral converges on $-1<\Re s < 1$ and define holomorphic function of $s$? $$\int_0^\infty \sin(y) y^{s-1} dy$$ I've googled for a while, but I could not ...
1
vote
2answers
210 views

calculate $\int_{-\infty}^{+\infty} \cos(at) e^{-bt^2} dt$

Could someone please help me to calculate the integral of: $$\int_{-\infty}^{+\infty} \cos (at) e^{-bt^2} dt.$$ a and b both real, b>0. I have tried integration by parts, but I can't seem to ...