Question 3

Question

Complex exponential and logarithm.

Answer

Complex Exponential

The complex exponential $\exp :\mathbb{C}\to \mathbb{C}$ is defined by
$\exp (z):=e^x(\cos (y)+i\sin (y))$
which is also
$\exp (z)=e^x\times e^{iy}$

Complex Logarithm

In principle it is the inverse of $\exp (z)$

But it is a function $\mathrm{Log}:\mathbb{C}\setminus \{0\}\to \mathbb{C}$
$\mathrm{Log}z=\ln |z|+i\mathrm{Arg}z.$

For $z=x>0$

We get $\mathrm{Log}x=\ln x$.