Determine the Fourier Transform and Fourier Series of the function

Question

$$ f(t)=\frac{\sin(at)}{t} $$

Since the term is parameterized, it's easy to see that if I take the first derivative with respect to 'a', then the function becomes considerably easier. I do this to the Fourier Transform and obtain: $$ \frac{\partial }{\partial a}\Im (f(t))=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{\infty }\cos(at))e^{itx}dt $$

However, this is an integral of an even function times an odd function, which equals 0 and raises my suspicion. I've tried implementing Euler's cosine form and got nowhere.

Also I'm using the imaginary symbol as the Fourier transform. Why? It looks cool.

Is the complex exponential an odd function ? – Damien L Mar 22 '13 at 09:40 — Damien L, Mar 22 '13 at 09:40

score 2 · Answer 1 · edited Apr 13 '17 at 12:21

2

The differentiating inside the integral trick requires several conditions be checked first. If you notice, the integral on the right is not even defined.

Check out http://ocw.mit.edu/courses/mathematics/18-304-undergraduate-seminar-in-discrete-mathematics-spring-2006/projects/integratnfeynman.pdf

and The Integral that Stumped Feynman?

Also you have to be a bit careful with how you're defining everything. You should call the Fourier transform $\hat{f}(x)$ rather than $f(t)$ since it is a different function in the variable $x$.

edited Apr 13 '17 at 12:21

Community

1

answered Mar 20 '13 at 08:09

muzzlator

7,325

Edited for definition – Gerg Mar 20 '13 at 08:22
I'll take a look at the link – Gerg Mar 20 '13 at 08:22
1

That first link also has a version of the integral you are interested in. They choose a different parameter to differentiate with respect to so that the function is dominated by $|e^{-bt}|$ – muzzlator Mar 20 '13 at 08:31
I see it now. Took the derivative to the wrong one. Sorry, for the late reply, I'm working on about 10 other things for students tomorrow. – Gerg Mar 20 '13 at 09:46
2

@muzzlator : the integral on the right is very well defined in the best integration theory. – Damien L Mar 20 '13 at 10:39

Determine the Fourier Transform and Fourier Series of the function

1 Answers1