(b) Discuss the branches and branch points of thefunction log z.

1.	(b) Discuss the branches and branch points of thefunction log z.
Answer» A branch point is a point such that if you go in a loop around it, you end elsewhere then where you started. A branch cut is what you use to make sense of this fact. This is best illustrated with an example, so let us consider the complex logarithm. We have a definition of the logarithm as the inverse of the exponential functionexexfor the real numbers. But just as we can extend the exponential function to the complex numbers by: ex+iy=exeiy=ex(cos(y)+isin(y))ex+iy=exeiy=ex(cos⁡(y)+isin⁡(y)) we would like to be able to extend the logarithm as well. Using the fact that we can express any complex number in the formreiθreiθ, let us naively define the logarithm as: This will be fine for every point except00, becauselogloghas a singularity at that point. But that shouldn't worry us too much. What should concern us more is the following. Considering going around in a loop around00. The unit circle will do nicely.

Answer»

A branch point is a point such that if you go in a loop around it, you end elsewhere then where you started. A branch cut is what you use to make sense of this fact.

This is best illustrated with an example, so let us consider the complex logarithm. We have a definition of the logarithm as the inverse of the exponential functionexexfor the real numbers. But just as we can extend the exponential function to the complex numbers by:

ex+iy=exeiy=ex(cos(y)+isin(y))ex+iy=exeiy=ex(cos⁡(y)+isin⁡(y))

we would like to be able to extend the logarithm as well. Using the fact that we can express any complex number in the formreiθreiθ, let us naively define the logarithm as:

This will be fine for every point except00, becauselogloghas a singularity at that point. But that shouldn't worry us too much. What should concern us more is the following. Considering going around in a loop around00. The unit circle will do nicely.

(b) Discuss the branches and branch points of thefunction log z.

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