# Complex: Argument and log

M. next
Hello!

So I have questions on this equivalence:

Imlog[(1+x)/(1-x)] = arg [(1+x)/(1-x)] where x: complex number

How is this true? Is it always applicable no matter what form of complex function is under calculation?

Thank you.

M. next
Yes, indeed. Excuse me for the late reply.

Homework Helper
Note, a complex number z can always be written in the form z=a+ib: a,b are real.

M. next
Yes?

Staff Emeritus
Gold Member
If I ask you for the imaginary part of log(z) can you tell me what it is?

From there it should be fairly obvious what the imaginary part of log(f(z)) is in terms of f(z).

Homework Helper
I don't see that the "(1+ x)/(1- x)" is really relevant. If z is any complex number, $z= re^{i\theta}$ where "$\theta$" is the "argument" of z. Then $log(z)= log(re^{ix\theta})= log(r)+ i \theta$. That is, $Im(log(z))= \theta$, the argument of z.

Homework Helper
Yes?