Spreading of a pulse as it propagates in a dispersive medium
Im studying the spreading of a pulse as it propagates in a dispersive medium, from a well known book. My problem arise when i have to solve an expression.
Firstly i begin considering that a 1-dim pulse can be written as:
u(x,t) = 1/2*1/√2∏* ∫A(k)*exp(ikx-iw(k)t) dk + cc (complex conjugate)
and then i showed that A(k) can be express in terms of the initial values of the problem, taking into account that w(k)=w(-k) (isotropic medium):
A(k) = 1/√2∏ ∫ exp(-ikx) * (u(x,0) + i/w(k) * du/dt (x,0)) dx
I considered du/dt(x,0)=0 wich means that the problems involves 2 pulses with the same amplitud and velocity but oposite directions.
So A(k) takes the form:
A(k) = 1/√2∏ ∫ exp(-ikx) * u(x,0)
Now i take a Gaussian modulated oscilattion as the initial shape of the pulse:
u(x,0) = exp(-x^2/2L^2) cos(ko x)
Then the book says that we can easily reach to the expression:
A(k) = 1/√2∏ ∫ exp(-ikx) exp(-x^2/2L^2) cos (ko x) dx
= L/2 (exp(-(L^2/2) (k-ko)^2) + exp(-(L^2/2) (k+ko)^2)
How did he reach to this?? How can i solve this last integral???
Then, with the expression of A(k) into u(x,t) arise other problem. How can i solve this other integral.
Thank you very much for helping me!!