# Manual Fourier Transform of exp(-t^2 /tau)

1. Feb 20, 2014

### amin5236

Hello all,
I have got into a confusion. In the function below, I need to numerically integrate the function with respect to (q) after the Fourier transform.
$$f(t)=e^{-Dq^2 t}$$
I use two approaches. Approach-1 is working, but Approach-2 does not give the same result, and I wonder what could be the mathematical explanation, or maybe it's because of the Matlab algorithms that I use.

1. First approach, to analytically solve the FT with Mathematica:
$$2 \int_0^{\infty } e^{-D q^2 t} \cos (t \omega ) \, dt = \text{ConditionalExpression}\left[\frac{2 D q^2}{D^2 q^4+\omega ^2},\left| \Im(\omega )\right| \leq \Re\left(D q^2\right)\land \Re\left(D q^2\right)>0\right]$$
take the result function and numerically integrate over (q) in Matlab using 'quadl' function:

Code (Text):

q1=1e6;     %integ. low limit
q2=3e10;    %integ. hi limit
D=8e-12;    % a constant

omega = 2*pi*logspace(-2,10,50); %ang.frequency

for i = 1: length(omega)
end

where 'Distr' contains the function 2Dq^2 / (q^4 + omega^2)

finally plot it as a function of ω:
https://dl.dropboxusercontent.com/u/12535433/PhysicsForum/01.jpg [Broken]

2. In the second approach, I take the f(t) function (better to say f(t,q) )and try to numerically solve the double integral in Matlab (dblquad function):
Code (Text):

t1 = 0;
t2 = 1E6;

for ii = 1: length(omega)
J2_w(ii) = dblquad(@(q,t) exp(-D .* q.^2 .* t.^a) .* cos(omega(ii).*t), q1,q2,t1,t2);
end

and plot it as a function of ω:
(This double integral takes too long)
loglog plot
https://dl.dropboxusercontent.com/u/12535433/PhysicsForum/Graph2.BMP [Broken]

semilogx plot

https://dl.dropboxusercontent.com/u/12535433/PhysicsForum/Graph1.BMP [Broken]

Black=Approach-1