I am trying to calculate the Fourier transform of the following function:
$$f(x)=\begin{cases}e^{-x}&|x|<1\\0&\text{otherwise}\end{cases}$$
What I have so far is:
$$\hat{f}(\omega)=\int\_{-\infty}^\infty f(x)e^{-j\omega x}dx=\int\_{-1}^1e^{-x}e^{-j\omega x}dx=\int\_{-1}^1e^{-(1+j\omega)x}dx$$
which I got as:
$$(1-j\omega)\frac{e^{(1+j\omega)}-e^{-(1+j\omega)}}{1+\omega^2}$$
which I have tried to simplify to the following:
$$2(1-j\omega)\frac{\sinh(1)\cos(\omega)+j\cosh(1)\sin(\omega)}{1+\omega^2}$$
which if I separate the real and imaginary parts left me with:
$$\frac{2\left[\sinh1\cos\omega+\cosh1\,\omega\sin\omega\right]+2j\left[\cosh1\sin\omega-\sinh1\,\omega\cos\omega\right]}{1+\omega^2}$$
My question is am I missing an identity that makes this a lot easier or is this as simple as I can get it? Thanks