Compute the exponential of a complex number

Question

I have a pretty basic question about complex numbers.

Let $z=x+yi$ be a complex number, I want to compute the real and imaginary parts of the number $w=e^{e^z}$.

Thanks in advance for any help.

Answer 1

As DJC and Fredrik Meyer suggest, you need a repeated application of

$$e^z=e^{x+iy}=e^x(\cos y+i\sin y)= e^x\cos y+ie^x\sin y$$

to get something like

$$e^{e^z}=e^{e^x\cos y}\cos (e^x\sin y)+ie^{e^x\cos y}\sin (e^x\sin y).$$