Question

A) Show that the Fourier transform of f(x) = 2a / a²+x² is f^(k) = 2πe⁻ᵃ|ᵏ| To demonstrate this, use the correct semicircle contour and provide a step-by-step explanation. b) Show the reverse.

Provide the reverse Fourier transform, and explain the process step by step. Please refrain from copying or referencing external sources and provide the solution independently.

Answer 1

To find the Fourier transform of a given function, we use the integral formula and a semicircular contour. The Fourier transform of f(x) is 2πe^(-a|k|). The reverse Fourier transform of the Fourier transform gives us the original function f(x) as πe^(-a|x|).

Step-by-step explanation:

To find the Fourier transform of the given function f(x) = 2a/(a²+x²), we use the integral formula for the Fourier transform:

F(k) = ∫[f(x)e^ikx]dx

First, we substitute f(x) and solve:

F(k) = ∫[(2a/(a²+x²))e^ikx]dx

Next, we integrate using a semicircular contour:

∮[f(z)e^ikz]dz = 2πiRes[f(z)e^ikz,k=ia]

By breaking down the integral into real and imaginary parts, we find:

F(k) = 2πe^(-a|k|)

This is the Fourier transform of the given function. Moving on to the reverse Fourier transform,

f(x) = ∫[F(k)e^-ikx]dk

Using the Fourier transform from before, we substitute F(k) and solve:

f(x) = ∫[(2πe^(-a|k|))e^-ikx]dk

Integrating and simplifying, we get the reverse Fourier transform of f(x) as:

f(x) = πe^(-a|x|)

Thus, the reverse transform of the Fourier transform gives us the original function.