Chapter 7.12, Problem 24P

(a) Find the exponential Fourier transform of $f(x)=e^{-\left|x\right|}$ and write the inverse transform. You should find ${\displaystyle {\int }_0^{\infty }\frac{\cos \alpha x}{{\alpha }^2+1}}d\alpha =\frac{\pi }{2}{e}^{-\left|x\right|}$.

(b) Obtain the result in (a) by using the Fourier cosine transform equations (12.15).

(c) Find the Fourier cosine transform of $f(x)=1/\left(1+x^2\right).$ Hint: Write your result in (b) with $x$ and $\alpha$ interchanged.