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.

  • 1
    $\begingroup$ Just use that $e^{a+ib} = e^a e^{ib} = e^a (\cos b + i \sin b)$ $\endgroup$ – JavaMan Oct 23 '11 at 19:49
  • 1
    $\begingroup$ HINT: $e^z=e^{x+iy}=e^x(\cos y+i\sin y)$. $\endgroup$ – Fredrik Meyer Oct 23 '11 at 19:52

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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