I have written my final exam in complex analysis last month but I am not quite satisfied with how my professor graded my exam.
I am in the process of petitioning for regrade. I would really appreciate it if you could help me take a look why I am getting the mark I got OR if you think I should be receiving a higher grade.
Question: Consider the function $$f(z) = \frac{z^3 + 2z^2}{z^4 -1}$$. Define the extension of function f so that it becomes a holomorphic function $f: C(hat) -> C(hat)$.
My solution: Note that $$ z^4 - 1 = (z^2 + 1)(z^2 - 1) => z = +/- i, +/- 1 => z = i,-i,1,-1
See that f: C{i,-i,1,-1} -> C is holomorphic => f: C -> C is meromorphic with poles at z = i, -i, 1, -1
So we an extend the function by defining f: C(hat) -> C(hat) such that f(i) = infinity, f(-i) = infinity, f(1) = infinity, f(-1) = infinity.
Professor then added f(infinity) = ? But I thought from we only need to perform treatments on the poles? Why is x = infinity a concern here?
Thank you very much!!
I'm afraid your professor was right, there. As long as we only consider finite complex values, the function is meromorphic, has a pole at $z=1$. To remedy that, we can add the point $\infty$, making the complex plane a sphere (Riemann sphere). But that's not only about adding that point to the set of values of a function, it's also about adding it to the set of arguments. Obviously, we'd have $f(\infty)=0$, but you'd still have to show it's analytic, there. Meaning you can express it as a power series in $1/z$, and that's elementary. But you didn't show that.