Assignment:
(a) The odd function f(x) of period L may be expanded as
![697_odd.JPG](https://secure.tutorsglobe.com/CMSImages/697_odd.JPG)
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
![1622_Exp.JPG](https://secure.tutorsglobe.com/CMSImages/1622_Exp.JPG)
(b) The Fourier transform f(k) of a non-periodic function f(x) is defined to be
![240_Transform.JPG](https://secure.tutorsglobe.com/CMSImages/240_Transform.JPG)
write down the inverse transform. The function f(t, x) satisfies the partial differential equation
![2103_Partial.JPG](https://secure.tutorsglobe.com/CMSImages/2103_Partial.JPG)
Show that the Fourier transform
![938_1.JPG](https://secure.tutorsglobe.com/CMSImages/938_1.JPG)
satisfies
![2434_2.JPG](https://secure.tutorsglobe.com/CMSImages/2434_2.JPG)
Given the initial condition
![2071_Condition.JPG](https://secure.tutorsglobe.com/CMSImages/2071_Condition.JPG)
show that
![1236_Show.JPG](https://secure.tutorsglobe.com/CMSImages/1236_Show.JPG)
Provide complete and step by step solution for the question and show calculations and use formulas.