Question

The inverse Fourier cosine transform is​

Answer 1

Answer:

In the process of solution of a PDE via Fourier cosine transform the author assumes at one step

F−1cexp(−tkw2)=12kt−−−√exp(−x24kt)

where Fc^{-1} is fourier cosine inverse operator. I know this is derived from formula of inverse fourier cosine transform which is

F−1c[Fc(w)]=f(x)=2π−−√∫∞0cos(wx)Fc(w)dw

It means I have to prove

2π−−√∫∞0cos(wx)exp(−tkw2)dw=12kt−−−√exp(−x24kt)

There are similar questions like inverse fourier transform but I am unable to comprehend analogy between them and my question.