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I have a pretty basic question about complex numbers.

If $z=x+yi$, a complex number, i want to compute the real and the imaginary part of the number $w=e^{e^z}$.

Thanks in advance for any help.

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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
up vote 2 down vote accepted

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

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

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