Answer to Question #224180 in Differential Equations for mansoor1290

Question

Find the Fourier Sine and Cosine transformations of the following function f (x) =2 if x< 0<a and f(x)=0 if x>a

Answer 1

Solution,

For the Fourier sine transform,

By definition;

$f_s[{f(x)}]=\hat f_s(w)=\sqrt{\frac2π}\int_0^\infin f(x)sin(wx)dx$

Hence;

$\hat f_s(w)=\sqrt{\frac2π}[\int_0^a2sin(wx)dx+0]$

$\hat f_s(w)=\sqrt\frac2π[\frac{-2cos(wx)}{w}|_0^a]$

$\hat f_s(w)=-\frac2w\sqrt\frac2π(cos(wa)-cos0)$

$\hat f_s(w)=-\frac2w\sqrt\frac2π(cos(wa)-1)$

For cosine Fourier transform,

By definition;

$\hat f_c(w)=\sqrt\frac2π\int_0^\infin f(x)cos(wx)dx$

$\hat f_c(w)=\sqrt\frac2π[\int_0^a2cos(wx)dx+0]$

$\hat f_c(w)=\sqrt\frac2π(\frac{2sin(wx)}{w}|_0^a)$

$\hat f_c(w)=\frac2w\sqrt\frac2π(sin(wa))$