I'm the teaching assistant for an undergraduate complex analysis course at my current university. I was reviewing some old prep materials to try and find some good practice problems to give to my students, and happened on the following:
Consider $\Bbb C^2$ with coordinates $z,w$. Can the function $f=e^{-1/zw}$ be written as a difference of two functions $g,h$, where $g$ is holomorphic away from $z=0$ and $h$ is holomorphic away from $w=0$? Explain your answer.
This was at the end of some final exam prep and was marked with a (*) to indicate it was supposed to be challenging. (This was also set by a professor who has since been hired away from my institution and was known for making some, uh, difficult exams.) I'm interested in finding a good solution.
The best I've been able to do so far is that if $g$ and $h$ were globally represented by power series, allowing for negative exponents for $z$ in $g$ and negative exponents in $h$, then by looking at negative-degree terms in $z$ and $w$ we ought to reach a contradiction: $\frac{1}{zw}$ cannot be represented as a sum of elements of the form $\frac{p(z,w)}{z^n}$ and $\frac{q(z,w)}{w^n}$, which should lead to what we need. But this argument feels untrustworthy – I know that if a function of one variable is entire on $\Bbb C$, I can find a global power series for it, but I'm deeply unsure about generalizing that to something like $\Bbb C\times\Bbb C^\times$, and it feels like I'm not really taking full advantage of how bad the singularities of $e^{-1/zw}$ are near $zw=0$. Can you help?
Best Answer
What you could do is to consider doing $\int_{z \in C} \int_{w \in C} u(z,w) \, dw \, dz$ to everything, where $C$ is the curve going counterclockwise one time around the origin. If you plug in $u(z,w) = e^{-1/zw}$ you get $4\pi$, and if you plug in for $u(z,w)$ a function that is either holomorphic for $z \ne 0$ or holomorphic for $w \ne 0$, you get $0$.