Re: complex fourier transform

Juryu wrote:
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).

I haven't heard of "contour bending" either, I suspect they mean
starting with a contour and then expanding it to infinity or some such.
Your method of completing the square doesn't quite work with a complex
w because after you change the variables you end up with an integral of
exp(-u^2/2) where u is not in the real line.

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?

There is no "i" in the exponent after you take the absolute value so
the previous result doesn't apply AFAICT.

--
Jan Bielawski

.

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