Monthly Archives: June 2013

Cosine transform of sinc cubed

Some dude asked to evaluate the following integral: $$\int_0^{\infty} dt \, \frac{\sin^3{\pi t}}{(\pi t)^3} \cos{u t}$$ I propose to perform a direct computation using Cauchy’s theorem, i.e., extension into the complex plane. I will then verify the solution using the convolution theorem. DIRECT EVALUATION Rewrite the integral as $$\frac12 \int_{-\infty}^{\infty} dt \, \frac{\sin^3{\pi t}}{(\pi t)^3} […]

An integral that is much easier than it looks

This is one of those cases where some poor, inexperienced user puts up a tough-looking integral. In Math.SE, there is a large contingent that jumps on these and gets quite upset if the consensus is that it can’t be done. Thus, when such a user posted this integral for evaluation: $$I(a,b)=\int_0^1 dt \, t^{-3/2}(1-t)^{-1/2}\exp\left(-\frac{a^2}{t}-\frac{b^2}{1-t} \right)$$ […]

Integral with hyperbolic functions

The problem here is to evaluate the following integral: $$\displaystyle \int_{0}^{\infty} dx \frac{\cosh (ax) \cosh (bx)}{\cosh (\pi x)} $$ such that $|a|+|b| < \pi$. This one can be done a number of ways, including complex analysis. Atypical for me, I chose a different way, one which led to a surprising result. It should be noted […]