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### 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 ...
### To evaluate $\int_0^{+\infty} \frac{\;dx}{\sqrt[3]{x^3+a^3}\sqrt[3]{x^3+b^3}\sqrt[3]{x^3+c^3}}$
$$f(a,b)=\int_0^{+\infty} \frac{\;dx}{\sqrt{x^2+a^2}\sqrt{x^2+b^2}}$$ To use Landen's transformation $$f(a,b)=\int_0^{+\infty} \frac{\;dx}{\sqrt{x^2+(\frac{a+b}{2})^2}\sqrt{x^2+ab}}$$ ...
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 ... 1answer 439 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 ... 1answer 106 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: ... 2answers 231 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 ... 2answers 194 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 ...