MHB How Do You Calculate the Real and Imaginary Parts of \( e^{e^z} \)?

[Deanmark]

Let f(z) = $e^{e^{z}}$ . Find Re(f) and Im(f).

I don't know how to deal with the exponential within an exponential. Does anybody know how to deal with this?

 

[Euge]

Yes. Recall Euler’s formula: $e^{iy} = \cos y + i\sin y$ for all real $y$. For $z = x + yi$, we would then have $e^z = e^{x + yi} = e^x e^{yi} = e^x(\cos y + i\sin y)$. If $w = e^z$, then $$e^w = e^{e^x\cos y + i(e^x\sin y)} = e^{e^x\cos y}\, e^{i(e^x\sin y)} = \cdots$$ Take it from here.