Find the Fourier Cosine Transform of the function $f(x)$ given by $$f(x) = \begin{cases} 1, & 0<x<a \\ 0, & x>a \end{cases}$$
Hence find the FCT (Fourier Cosine Transform) of $\frac{\sin(ax)}{x}$
FCT: $$\mathcal F_c(f(x))=\sqrt \frac{2}{\pi} \int_0^{\infty}f(x)\cos(sx)dx=F_c(s)$$
I need to find the FCT using the piecewise material. Any help appreciated
The Fourier Cosine transform of $f(x)$ is $\mathcal F_c\{f(x)\}=F_c(s)=\sqrt{\frac{2}{\pi}}\int_0^{\infty} f(x)\cos(sx)\mathrm d x$ that is $$ F_c(s)=\sqrt{\frac{2}{\pi}}\int_0^{a} \cos(sx)\mathrm d x=\sqrt{\frac{2}{\pi}}\left.\frac{\sin(sx)}{s}\right|_0^a=\sqrt{\frac{2}{\pi}}\frac{\sin(as)}{s} $$ So you have the pair $$ f(x) \longleftrightarrow \sqrt{\frac{2}{\pi}}\frac{\sin(as)}{s} $$ and then the function $g(x)=\frac{\sin(ax)}{x}$ has Fourier Cosine transform $$\mathcal F_c\{g(x)\}=G_c(s)=\sqrt{\frac{\pi}{2}}f(s)$$