For log z = ln|z| + i θ where pi/4 < θ < 9pi/4 find the domain of analyticity (in terms of angle theta) in complex plane.

Question

State the forbidden and allowed paths. I understand that log function is not differentiable along negative axis and 0 but how do I found the domain angle? And what is meant by forbidden and allowed paths?

Answer 1

Denote by $\Omega$ the complex plane with the $45^\circ$ ray $\{t(1+i)\,|\,t\geq0\}$ removed. The function $$\theta(z):={\rm arg}(z)\>\cap\ ]\pi/4,\>9\pi/4[\>$$ is a well defined and continuous real-valued function on $\Omega$. It follows that $$f(z):=\log|z|+i\theta(z)\ .$$ is a well defined and continuous complex-valued function on $\Omega$. Checking the CR equations shows that $f$ is in fact analytic in $\Omega$, and one finds that $$f'(z)={1\over z}\qquad(z\in\Omega)\ .$$ What is the relation between this $f$ and the principal value of the logarithm, denoted by ${\rm Log}\>$? The latter is defined in the domain $\Omega_*\>$, obtained by making a slit along the negative real axis. Since both $f$ and ${\rm Log}$ have the same derivative $z\mapsto{1\over z}$ the difference between these two functions is locally constant. But: The constant may be different in different components of $\Omega\cap\Omega_*$. I suggest you draw a figure and find out what exactly is happening in this example.

There is no such thing as "forbidden" or "allowed" path, unless further context is established.