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Complex analysis - the logarithmic function

  • Thread starter mariab89
  • Start date
11
0
1. 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.

2. Homework Equations



3. 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

Science Advisor
Homework Helper
26,258
618
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.
 
11
0
so to show that what i did was...

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

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

Dick

Science Advisor
Homework Helper
26,258
618
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).
 
11
0
oh ok i see now!
Thanks a lot!
 

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