# Compute the exponential of a complex number

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.

• Just use that $e^{a+ib} = e^a e^{ib} = e^a (\cos b + i \sin b)$ – JavaMan Oct 23 '11 at 19:49
• HINT: $e^z=e^{x+iy}=e^x(\cos y+i\sin y)$. – Fredrik Meyer Oct 23 '11 at 19:52

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