Finding analyticity of a complex function involving ln(iz)

[tixi]

Homework Statement

The function ln (principal value of ln) is known to be analytical in complex plane with removed negative real axis. What is the largest region in which f(z)=ln(iz)-i pi/2 is analytical? Evaluate f′(z).

Relevant Equations

Principal value of the logarithm: ln(z) = ln(r) + iArg(z)
Chain rule for complex functions
Inverse functions differentiation rule

Hey everyone! I got stuck with one of my homework questions. I don't 100% understand the question, let alone how I should get started with the problem.

The picture shows the whole problem, but I think I managed doing the a and b parts, just got stuck with c. How do I find the largest region in which f(z) is analytical and how do I get started trying to differentiate it? Do the differentiation rules for the real ln translate to the complex one?

Thanks in advance <3

 

[benorin]

Start here: The function ln (principal value of ln) is known to be analytical in complex plane with removed negative real axis, let this region be denoted by $D$.
The function $f(z)=\bar{\ln} (iz)-i\tfrac{\pi}{2}$ has a domain of analyticity obtained from $D$ by a rotation (this rotation is induced by the argument $iz$). Think you can figure the rest of that out?
My complex is rusty, but if I recall correctly $\tfrac{d}{dz}\bar{\ln }(h(z))=\tfrac{1}{h(z)}\cdot h^\prime (z)$ by the chain rule. That should get you going!

 

[tixi]

Thank you so much! I never got a notification that my thread was answered, but this still helped! The exercise was explained eventually but it was very abstract and they just gave a very simple answer, so the motivation you provided was still helpful! Thanks