I need an way to solve this Fourier transform problem. $$ f(t)= \begin{cases} \cosh(t) & \text{ For } |t|<1\\ 0 & \text{ For }|t|>1 \end{cases} $$ The given answer for the problem is given as
This is what I have got so far. I can't find a way to progress any further toward that answer.
I realize I can create a $\sinh$ or a $\sin/\cos$ function from what I have. I have tried, but it does not get the given answer.
I would appreciate any help with this. I have a few Fourier Transform sums for my assignment and this is supposedly the easiest :(
You are almost there
$$F(\omega) = \frac{1}{2\sqrt{2 \pi}} \int_{-1}^1 (e^t+ e^{-t}) e^{-j \omega t}dt$$
But
$$ \frac{1}{1-j\omega}(e e^{-j\omega}-e^{-1} e^{j\omega})+\frac{1}{1+j\omega}(e e^{j\omega}-e^{-1} e^{-j\omega})=$$
$$= \frac{1}{e}\frac{1}{1+\omega^2}[(1+j\omega) (e^2 e^{-j\omega} - e^{j\omega})+(1-j\omega) (e^2 e^{j\omega} - e^{-j\omega})]=$$
$$=\frac{1}{e}\frac{1}{1+\omega^2}[ (e^2-1)2 \cos(\omega) + (e^2+1)2 \omega\sin(\omega)] $$
(there might be some error with the signs, this needs revising, but you get the idea)