# Cool proofs again

In between marking exams, marking essays, external examining, getting my two PhD students to submit their theses, and, well you get the idea, I’ve been trying to come up with a cool proof of a theorem in complex analysis:

Let $f(z)=\sum_{n=0}^\infty a_n z^n$ be a power series with radius of convergence $R>0$. Then, $f'(z)=\sum_{n=1}^\infty na_n z^{n-1}$.

I.e., the derivative of $f$ is what you expect it to be, the termwise differentiation of the series.

I’ve not been able to find a good proof of this. My plan of attack has been to show first via the root test that the formal derivative has radius of convergence $R$. The main sticking point is then to show that this formal derivative really is equal to $f'(z)$. The clearest proof I found was in James Pierpoint‘s book Functions of a Complex Variable which you can find online here. Like many old books (this is from 1914) I’ve found that, if you allow for slightly odd notation and nomenclature, it contains excellent material that is currently passed over. (Does anyone still teach Raabe’s test?)

Anyhow, the proof is on pages 172-173 and relies on pages 84-85. For me the proof is very clear because of equation 3 on page 84: the expansion of $f(z+h)$ via rows can be turned into a power series in $h$ by summing the columns. My problem is that when I try to turn everything into a rigorous proof this becomes obscure. Also, we have to prove that all the formal derivatives of $f$ have radius of convergence $R$. This in itself is not difficult but it means we have to set up the notion of all formal derivatives rather than just the first one. For me this may be too much extraneous clutter for students.

The hunt continues…