Show that
is single-valued and analytic in . Calculate .
I don't even know where to begin with this one.
I don't understand why this is listed as a hybrid function, since anyway... So I'm going to assume we can just write the function as for all .
Now convert it back to Cartesians and use the Cauchy Riemann Equations.
i.e. For , the function is analytic if and
Well, we don't know exactly what you are allowed to use - can you use the fact that the cosine function is entire and square root is analytic in ?
Also, it is not defined at 0 since, as you said, is defined as , and has a branch point at 0.
I don't think it is 1:1, though -- . Regarding the calculation of the derivative - you should probably do it by definition.
One-to-one might be the wrong wording here. You're supposed to show that it's not multivalued. You're allowed to use that cos is entire, but like I wrote previously, I think you need to show that f and the partial derivatives are continuous. Anyways, this is just an exercise, but I would really like to know how to solve it to further my knowledge.
The Cauchy-Riemann equations is only a necessary condition in order for the function to be analytic. In order to prove that the function is analytic you also need to show that the partial derivatives are continuous. I don't know how to get the partial derivatives for the function though since it's a split function. I would think that the derivative of the constant part is 0, but the answer is