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October 22nd, 2010, 10:49 PM   #1
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Multivalued functions and branch cuts

Two questions:

1) A branch point (BP) of a function is a point which has different values when you make a complete circuit around it. You solve this issue by creating a branch cut (BC) between the BPs. If you make a circuit which does not enclose any BP, then the function returns to its original value.

What happens if you make a circuit which does not enclose any BP, but does cross the BC? Are you even allowed to do that?

2) I don't quite get why a some functions are considered different values even though the value only differs by 2*Pi, and some functions are not considered different. For example,

w(z) = log(z)+i*(t+2*n*Pi), n is integer

Is considered multivalued. If z=x is real, positive, then make a circuit about z=0; initially, t=0, w(z) = w(x*e^{I*0}) = log(x). After the circuit, t=2*PI, w(z)=w(x*e^{i*2*Pi})=log(x)+2*pi*i

But w(z)=z is not considered multivalued. If z=x (x is real, positive), then t=0, w(z) = w(x*e^{i*0}) = x. After the circuit, t=2*Pi, z=x*e^{i*2*Pi}. Yet w(z)=z is not considered multivalued when w(z)=log(z) is. What's the difference?

Thanks for your help.
-J
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