Spreading of a pulse as it propagates in a dispersive medium
Hello everyone!!
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 1dim pulse can be written as: u(x,t) = 1/2*1/√2∏* ∫A(k)*exp(ikxiw(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) (kko)^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!! 
Re: Spreading of a pulse as it propagates in a dispersive medium
that is a trick.you have to write cos(k_{0}x) as Re(e^{ik0x}),you will get only exponentials then you will have to complete the square in powers of exponentials and use of a simple gaussian integral.
∫_{0}^{∞} e(x^{2})dx=√∏/2 
Re: Spreading of a pulse as it propagates in a dispersive medium
Thank you so much! I could solve it!! It wasnt too hard after all :) thanks again.

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