find fourier transform of 1-|x| in -1<x<1 and 0 , elsewhere

The Fourier transform of a function f(x) is given by the integral from -∞ to ∞ of f(x) * e^(-2πixξ) dx.

The function f(x) = 1 - |x| is defined from -1 to 1, and is 0 elsewhere. Therefore, the integral becomes:

F(ξ) = ∫ from -1 to 1 of (1 - |x|) * e^(-2πixξ) dx.

This integral can be split into two parts, one for -1 < x < 0 and one for 0 < x < 1, because the absolute value function |x| behaves differently for negative and positive x.

So, F(ξ) = ∫ from -1 to 0 of (1 + x) * e^(-2πixξ) dx + ∫ from 0 to 1 of (1 - x) * e^(-2πixξ) dx.

Now, you can solve these two integrals separately.

For the first integral, use the substitution u = -2πixξ, du = -2πiξ dx