Assignment:
(a) The odd function f(x) of period L may be expanded as
Give expressions for kn and An. Given that f(x) = 1 for 0 < x < L/2, obtain an expression for An. Hence, or otherwise, show that
(b) The Fourier transform f(k) of a non-periodic function f(x) is defined to be
write down the inverse transform. The function f(t, x) satisfies the partial differential equation
Show that the Fourier transform
satisfies
Given the initial condition
show that
Provide complete and step by step solution for the question and show calculations and use formulas.