Definite Integral to Infinity of Exponential of -a x^2 by Cosine of b x

Theorem

 * $\displaystyle \int_0^\infty e^{-a x^2} \cos b x \rd x = \frac 1 2 \sqrt {\frac \pi a} \map \exp {-\frac {b^2} {4 a} }$

where $a$ is a strictly positive real number.

Proof
Fix $a$ and define:


 * $\displaystyle \map I b = \int_0^\infty e^{-a x^2} \cos b x \rd x$

for all $b \in \R$.

Then, we have:

Note that:

So:

We then have:


 * $\displaystyle \frac {\map {I'} b} {\map I b} = -\frac b {2 a}$

Integrating, by Primitive of Function under its Derivative and Primitive of Constant:


 * $\displaystyle \ln \size {\map I b} = -\frac {b^2} {4 a} + C$

for some $C \in \R$.

So:


 * $\displaystyle \map I b = A \map \exp {-\frac {b^2} {4 a} }$

for some $A \in \R$.

We have:

on the other hand we have:

So we have:

$\displaystyle \map I b = \int_0^\infty e^{-a x^2} \cos b x \rd x = \frac 1 2 \sqrt {\frac \pi a} \map \exp {-\frac {b^2} {4 a} }$

for all $b \in \R$ as required.