How to transform from exponential to trigonometric

Question

I'm having trouble transforming an expression with exponentials to one with sines or cosines only.

$$u = - b a e^{-kz} e^{i(kz-wt)}$$

I know $e^{i(kz-wt)}$ is $\cos(kx-wt)$, but I can't figure out how to go from $e^{-kz}$ to a trig expression.

Could any enlightened soul please help me out?

Answer 1

In general, $e^{\pm ix} = cos(x) \pm isin(x)$. I think you forgot to mention what $z$ is, but I assume (reading that "you know that..." ) that $z = x + iy$ an arbitrary complex number, then: \begin{equation} e^{-kz} = e^{-k(x+iy)} = e^{-kx}e^{-iky} = e^{-kx}(cos(ky)-isin(ky)) \end{equation}