Complex analysis - the logarithmic function

[mariab89]

Homework Statement

Show that the function Log(-z) + i(pi) is a branch of logz analytic in the domain D* consisting of all points in the plane except those on the nonnegative real axis.

Homework Equations

The Attempt at a Solution

I know that log z: = Log |z| + iArgz + i2k(pi)
I'm not sure where to start with this question, any help would be greatly appreciated!
thanks :)

 

[Dick]

You know Log(z) is analytic except for a branch cut on the negative real axis. That means Log(-z) is analytic except for a branch cut on the positive real axis. So is Log(-z)+i*pi. Now you just have to show that it is a branch of log(z) by showing exp(Log(-z)+i*pi)=z.

 

[mariab89]

so to show that what i did was...

exp(Log(-z) + i*pi) = exp(Log(-z))exp(i*pi) = (-z) (-1) = z

but.. I am still unclear how this shows that Log(-z) + i*pi is a branch of log z.

 

[Dick]

If g(z) satisfies exp(g(z))=z then it's a branch of log(z). That's what defines log(z). It's an inverse function of exp(z).

 

[mariab89]

oh ok i see now!
Thanks a lot!