I'm trying to brush up with Fourier series with Apostol's Mathematical Analysis. I was looking through the Fourier chapter and its Fourier integral theorem. I'm slightly confused on how to approach it with improper integrals and how to determine if the integral is either odd or even. Especially with problems such as
$$\int_{0}^{\infty} \frac{x \sin ax}{1+x^{2}}\mathrm dx = \frac{a}{\left | a \right |} \frac{\pi }{2}e^{-\left | a \right |}$$
$$\int_{0}^{\infty} \frac{\cos ax}{b^{2}+x^{2}}\mathrm dx= \frac{\pi }{2b} e^{-{\left | a \right |b}}$$
Any help is much appreciated!
Best Answer
$$ \int _0^\infty \frac{\cos (ax)}{b^2+x^2}dx=\frac{1}{2}\int _{\mathbb{R}}\frac{\cos (ax)}{b^2+x^2}dx=\frac{1}{2}\Re \left[ \int _{\mathbb{R}}\frac{1}{b^2+x^2}e^{iax}dx\right] =\frac{\sqrt{2\pi}}{2}\Re \left[ \left[ \mathcal{F}^{-1}(f)\right] (a)\right] , $$
where $f(x)=\frac{1}{x^2+b^2}$ and $\mathcal{F}$ denotes the Fourier Transform. The Fourier Transform of $e^{-b|a|}$ is $\sqrt{\frac{2}{\pi}}\frac{b}{x^2+b^2}=b\sqrt{\frac{2}{\pi}}f$, so that $\left[ \mathcal{F}^{-1}(f)\right] (a)=\sqrt{\frac{\pi}{2}}e^{-b|a|}/b$. Thus,
$$ \int _0^\infty \frac{\cos (ax)}{b^2+x^2}dx=\frac{\pi}{2b}e^{-b|a|}. $$
The other one can be done similarly. Hope this helps!