Articles of improper integrals

The Laplace transform of $\exp(t^2)$

A naive attempt to calculate the Laplace transform of the function $f(t)=e^{t^2}$ results in integrals of the form $$\int_0^\infty e^{t^2-st}dt,$$ which obviously don’t exist as the integrand grows too large. However, Wolfram Alpha/Mathematica gives the following result $$F\left(\frac{s}{2}\right)-\frac{1}{2} i \sqrt{\pi } e^{-\frac{s^2}{4}} $$ where $F$ is the Dawson function. Is this really the Laplace transform […]

Show that $\int_0^\infty\frac{1}{x ((\ln x)^2+1)^p}dx$ converges for any $p\geq 1$ and find its value.

Suppose that $p\geq 1.$ In this question, Robert answered that the following integral $$\int_0^\infty \frac{1}{x ((\ln x)^2+1)^p} dx$$ converges for any $p\geq 1.$ However, I am not able to show Robert’s claim. Below is my attempt: Use the substitution $u = \ln (x).$ Then we have $$\int_0^\infty \frac{1}{x ((\ln x)^2+1)^p} dx = \int_{-\infty}^\infty \frac{1}{(u^2+1)^p} du.$$ […]

Finding values for integral $\iint_A \frac{dxdy}{|x|^p+|y|^q}$ converges

Given the following integral $$\iint_A \frac{dxdy}{|x|^p+|y|^q}$$ where $A=|x|+|y|>1$. How can one find for which $p$ and $q$ values the integral converges? Since the function is non-negative it is sufficient to show convergence/divergence on any Jordan exhaustion of $A$ in order to show convergence/divergence on $A$. I tried to use polar coordinates here, but I don’t […]

Why does $\left(\int_{-\infty}^{\infty}e^{-t^2} dt \right)^2= \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(x^2 + y^2)}dx\,dy$?

Why does $$\left(\int_{-\infty}^{\infty}e^{-t^2}dt\right)^2 = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(x^2 + y^2)}dx\,dy ?$$ This came up while studying Fourier analysis. What’s the underlying theorem?

Is there any close form solution possible for the following integral?

I need to solve the following integral $$\int_{a}^{\infty}\Gamma\left(b,\frac{d}{x}\right)x^{k-1}e^{-cx}dx$$ where $a>0$, $b>0$ and $c>0$. I checked in wolfram alpha but it does not provide any answer. Further, I checked the book of Gradeshteyn (Eq. 6.456) but it can be seen that there are slight differences in my problem Eq. 6.456. Thanks in advance. My Attempt: In […]

Convergence testing of the improper integral $\int_{0}^{\infty}\frac{\ln x}{\sqrt{x}(x^2-1)}\ dx$

I’ve tried to test this integral for convergence for a couple of hours, actually I know that $$\int_{2}^{\infty}\frac{\ln x}{\sqrt{x}(x^2-1)}\ dx$$ converges with no problem with the help of Dirichlet test for convergence. But the problematic part is: $$\int_{0}^{2}\frac{\ln x}{\sqrt{x}(x^2-1)}\ dx$$ and I don’t really know how to prove the convergence there. The problematic points are […]

Any good approximation for this integral?

I am interested in the following integral $$ \mathcal{I}=\int_{-\infty}^\infty\mathop{dz}\left[\frac{1}{\sqrt{a+b}(z^2)^{n/4}}-\frac{1}{\sqrt{a+b\cos^2\theta}(R^2+z^2)^{n/4}}\right], $$ where $R\ll 1$, $n<2$ and $$\cos\theta=\frac{z}{\sqrt{R^2+z^2}}.$$ I was thinking of Taylor Expanding the integrand for $z<R$, and integrating the result from $-R$ to $R$, but the answer seems not good when compared numerically. Any good approximation if not the exact answer would work. Any ideas? […]

A closed form for the integral $\int_0^1\frac{1}{\sqrt{y^3(1-y)}}\exp\left(\frac{i A}{y}+\frac{i B}{1-y}\right)dy$

Yesterday, during reviewing my old lecture notes on advanced quantum mechanics, i stumbeled over the following integral identity, which seems, on a first glance, too nice to be true $$ I_{A,B}=\int_0^1\frac{1}{\sqrt{y^3(1-y)}}\exp\left(\frac{i A}{y}+\frac{i B}{1-y}\right)dy=\sqrt{\frac{i\pi}{B }}e^{i(\sqrt{A}+\sqrt{B})^2} $$ with $A,B>0$ After working on it for a few hours i came up with a solution, which i think is […]

Compute $\int_0^{\infty}\frac{\cos(\pi t/2)}{1-t^2}dt$

Compute $$\int_0^{\infty}\frac{\cos(\pi t/2)}{1-t^2}dt$$ The answer is $\pi/2$. The discontinuities at $\pm1$ are removable since the limit exists at those points.

Multiple self-convolution of rectangular function – integral evaluation

I am trying to find an $n$-multiple convolution of a rectangular function with itself. I have a function $f(x) = 1$ for $0<x<1$, 0 otherwise. I define $$ g_2 (y) = \int_{-\infty}^{\infty} f(y+x) f(x) \mathrm{d} x $$ and recursively $$ g_n (y) = \int_{-\infty}^{\infty} g_{n-1}(y+x) f(x) \mathrm{d} x \, . $$ To make it easier […]