Complex Analysis: Solving Logarithmic Equations and Limits

[Dassinia]

Hello,
I'm solving the problems given in previous exams, and there's this question:

Homework Statement

a/ Give the value of ln(i), ln(-i) and i^i
b/ If zo=-1-i , what is the value of

lim [ ln(zo+e)-ln(zo+i*e) ] when e-> 0
Same question with zo=1+i

Homework Equations

The Attempt at a Solution

a. ln(i)=ln|i|+i*arg(i)=i*pi/2
ln(-i)=-i*pi/2
i^i=exp(-pi/2)
b. I found that when zo=-1-i and zo=1+i the limit is 0, I don't know if I'm missing something, this question is so easy that it seems suspect

Thanks !

 

[jambaugh]

There is a cut line where the limits will differ. Since exp is a many to one function in the complex plane we have to make a restriction on it's domain when we invert. But that cut is typically made on the negative real ray so only there do you have discontinuity in the Ln function. But away from there it is a continuous function so limit=value.

Easy $\ne$ Simple, sometimes the easy answer is easy because you understand some very non-simple concepts. Sometimes a simple answer isn't easy too.