Question

Find the fourier integral for f(x)= C |x|<=1 and f(x)= 0 |x|> 1 we here C is constant

Answer 1

Calculate the coefficients:

$A(\lambda)=\frac{1}{\pi}\int_{-\infty}^{+\infty}f(x)\cos(\lambda x)dx=\frac{c}{\pi}\int_{-1}^{+1}\cos(\lambda x)dx\\ =\frac{2c}{\lambda\pi}\sin(\lambda)$

Due to the fact that $f$ is an even function $B(\lambda)\equiv0$ . So, fourier integral:

$f(x)=\int_0^{+\infty}\frac{2c}{\lambda\pi}\sin(\lambda)\cos(\lambda x)d\lambda$