How to define and compute the power-exponential function of a complex variable? For example, how to define and compute the complex function $(\ln z)^{\sin z}$? where $z=x+iy$ is a complex variable. Thank you for your help.
The exponential function of a complex variable is defined via its power series
$e^{z} = \sum_{n=0}^\infty \frac{z^n}{n!}$
where $z= x +iy, x, y \in \mathbb{R}.$
Also, $e^z = e^{x+iy} = e^{x}e^{iy} = e^x (\cos y + i \sin y).$
Some interesting properties of $e^z$: (a) $e^z \neq 0$ and (b) $|e^z| = e^{\Re z}.$
The complex logarithm $\ln z$ is defined as 'inverse' of $e^z.$
Note that $\ln z$ is a multiple-valued function because $e^z$ is a periodic function with period $2 \pi.$
The complex power function is defined as
$z^a = e^{a \ln z}$
where $z \neq 0 \in \mathbb{C}$ and $a \in \mathbb{C}.$
So, $(\ln z)^{\sin z} = e^{\sin z \ln(\ln z) }.$
Hope it helps.
Thanks.
Regards
Yao Liang Chung
Hi Prof Feng Qi
You're welcome. Just share what I know in the field of Complex Analysis.
Best regards
Yao Liang Chung
Hi colleague Feng Qi
I just had the intention to help you, but the Professor Yao Liang Chung was faster than me.
Best regards and wishes,
Mirjana Vukovic
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