I'm trying to reduce the following complex exponential to an expression having a single complex exponential:
$\frac{e^{jwM_1}-e^{-jw(M_2+1)}}{1-e^{-jw}}$
Can anybody help me how to reduce this expression to another expression that has only one complex exponential? I tried to use Euler rule but I couldn't figure it out. Any helps would be appreciated.
$$ \frac{e^A-e^{-B}}{1-e^{-C}} = e^{\frac{A-B+C}{2}}\cdot\frac{sinh(\frac{A+B}{2})}{sinh(\frac{C}{2})} $$
Then:
$$\frac{e^{jwM_1}-e^{-jw(M_2+1)}}{1-e^{-jw}} =$$
$$ e^{\frac{jw}{2}\cdot(M_1-M_2)}\cdot\frac{sinh(\frac{jw}{2}\cdot(M_1+M_2+1))}{sinh(\frac{jw}{2})}$$