Understanding Branch Cuts for Analytic Functions in the Complex Plane

[hoffmann]

I have a question with regards to branch cuts:

Say I have a function f(z) = log(z² - 1). Why is a simple branch cut connecting z = -1 and z = +1 not sufficient to define an analytic function? On the other hand, why is it sufficient for the function f(z) = (z^2 - 1)^(1/2) ?

This is in the complex plane where z = a + ib.

 

[olgranpappy]

I have a question with regards to branch cuts:

Say I have a function f(z) = log(z² - 1). Why is a simple branch cut connecting z = -1 and z = +1 not sufficient to define an analytic function? On the other hand, why is it sufficient for the function f(z) = (z^2 - 1)^(1/2) ?

This is in the complex plane where z = a + ib.

perhaps the point at infinity?

 

[hoffmann]

^^ what do you mean by this?

 

[olgranpappy]

^^ what do you mean by this?

Don't worry about.

How about just trying to write
$$z=1+r_1e^{i\theta_1}$$
and
$$z=-1+r_2e^{i\theta_2}$$
and investigate how the function
$$\log(z^2-1)=\log(z-1)+\log(z+1)=\log(r_1r_2)+i(\theta_1+\theta_2)$$
behaves as you go around the perspective "branch cut" that you mentioned... try it.