# Determine the Fourier Transform and Fourier Series of the function

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

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 ? Commented Mar 22, 2013 at 9:40

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

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 for definition
– Gerg
Commented Mar 20, 2013 at 8:22
• I'll take a look at the link
– Gerg
Commented Mar 20, 2013 at 8:22
• 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 $|e^{-bt}|$ Commented Mar 20, 2013 at 8:31
• 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
Commented Mar 20, 2013 at 9:46
• @muzzlator : the integral on the right is very well defined in the best integration theory. Commented Mar 20, 2013 at 10:39