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

Thank Yao Liang Chung for so detailed explanation of yours.

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

Dear Mirjana Vukovic,

thank a lot for your kindness.

Feng Qi (F. Qi)

f0g= exp(Ln(f0g) )

Yes, you are right. Thanks.