complex fourier transform

Here I am with another problem studying for my complex variables
qualifying exam.

Problem:

***

The complex fourier transform

Ff(w) = int_[from -infinity to infinity] exp(-2*pi*i*t*w) f(t)dt

is defined, at least for real w, for any rapidly vanishing complex
valued function f on the real line. Determine the Fourier transform of
f(t) = exp (-t^2/2), using contour bending and the well-known fact that


int_[from -infinity to infinity] exp(-t^2/2) dt = sqrt(2*pi).

Determine the set of w in the complex plane for which the integral is
absolutely convergent.

***

First of all, I don't know what they mean by "contour bending", but I'm
able to solve the first part by completing the square on the
expontential, the answer is exp(-2*pi^2*w^2)*sqrt(2*pi).

For the second part, if I want it to be absolutely convergent, call w =
x+iy and do the same for int_[from -infinity to infinity]
abolute_value_of( exp(-2*pi*i*t*w) * exp(-t^2/2) ) dt, and we get
answer exp(2*pi^2*y^2)*sqrt(2*pi) < infinity, so it is absolutely
convergent for ALL values of w?? Doesn't seem like an answer... Am I
doing something wrong?

.

Relevant Pages

  • Re: complex fourier transform
    ... The complex fourier transform ... is defined, at least for real w, for any rapidly vanishing complex ... valued function f on the real line. ... For the second part, if I want it to be absolutely convergent, call w = ...
    (sci.math)
  • Re: complex fourier transform
    ... The complex fourier transform ... absolutely convergent. ... First of all, I don't know what they mean by "contour bending", but I'm ... Your method of completing the square doesn't quite work with a complex ...
    (sci.math)
Loading