Analyzing Multivalued Functions in Complex Domains

[indigojoker]

We are asked to find a branch where the multi-valued function is analytic in the given domain.

The function: $(4+z^2)^{1/2}$ in the complex plane slit along the imaginary axis from -2i to 2i.

The principal branch is $\exp(\frac{1}{2} Log(4+z^2))$ and so we want analyticity from -2i to 2i. It seems to me that this is obeyed and the cuts appear when |z|>2i. When |z|>2i we see that the term inside Log become negative, when |z|<2i,the term inside the Log is positive. Does this mean the principal branch is a branch that is analytic in the given domain? I referred to the back of the book and it gave a different branch, namely $z \exp(\frac{1}{2} Log(1+\frac{4}{z^2})})$, what went wrong?

The function is: $(z^4-1)^{1/2}$ in |z|>1.

The principal branch is $\exp(\frac{1}{2}Log(z^4-1))$ and so we want analyticity when |z|>1. However, we see that this is already the case. Because within the unit circle, z^4 is positive, and subtracting 1 makes z^4-1 negative. We want analyticity when |z|>1 which is the case since z^4 is now positive and larger than 1, making z^4-1 positive when |z|>1. Again, I feel that the principal branch works, however, the back of the book gives a different result $z^2 \exp{\frac{1}{2} Log(1-\frac{1}{z^4})}$

The function in d is: $(z^3-1)^{1/2}$ in |z|>1.

I could make the same argument as above, and say the principal branch of $\exp(\frac{1}{2} Log(z^3-1))$ works. However, again, this disagrees with the result in the back of the text: $z\exp(\frac{1}{3} Log(1-\frac{1}{z^3}))$

 

[mighty2000]

Hello,

Thank you for your post. In order to determine the correct branch for a multi-valued function, we need to consider the behavior of the function on the entire complex plane, not just within a specific domain. The principal branch of a function is chosen so that it is single-valued and analytic on the entire complex plane, except for some isolated points where the function is not defined.

In the first example, the principal branch of (4+z^2)^{1/2} is \exp(\frac{1}{2}Log(4+z^2)), which is analytic on the entire complex plane except for the branch cut along the imaginary axis from -2i to 2i. This means that the function is not defined at any point on this branch cut, but is otherwise single-valued and analytic. This is the correct branch for the given function in the given domain.

In the second and third examples, the principal branch of (z^4-1)^{1/2} and (z^3-1)^{1/2} is \exp(\frac{1}{2}Log(z^4-1)) and \exp(\frac{1}{2}Log(z^3-1)), respectively. These branches are chosen so that they are analytic on the entire complex plane, except for the points where the function is not defined. In these cases, the points where the function is not defined are the roots of the polynomial in the argument of the logarithm, which are z=1 and z=1, respectively.

The branches given in the back of the book, z \exp(\frac{1}{2}Log(1-\frac{1}{z^4})) and z\exp(\frac{1}{3}Log(1-\frac{1}{z^3})), are not the principal branches for these functions. They may work in some specific domains, but they are not analytic on the entire complex plane. The principal branch is always chosen to be analytic on the entire complex plane, except for isolated points where the function is not defined.

I hope this helps clarify the concept of choosing the correct branch for a multi-valued function. If you have any further questions, please don't hesitate to ask.