Two problems from complex analysis

I sit in this class on mathematical methods in physics and engineering that’s pretty good. (I was telling Aashish the other day that I’m actually surprised at how much ground we’ve covered since starting at ‘A vector is…’, two months ago). The class is doing complex analysis right now, and something called the maximum principle came up.

The maximum principle is ironically named! It states that an analytic function of a complex variable cannot have a maximum in its domain of analyticity. To see why, one could use Cauchy’s formula to show that the value of the function at a point is the average of the values on a circle around it, or indeed the average of values on a two-dimensional ball around it.

I, being me, had to point out that this can be shown using only the Cauchy-Riemann conditions, or that the components of a complex function are harmonic. I was told to show how. Um, let’s see. It is obvious for the one-dimensional case, i.e. if the second derivative of a function of one variable is zero, the function value at a point is the average of function values at two points ‘around’ it.

For the two-dimensional case, this can be shown by contradiction. Assume that there is a maximum, Taylor-expand your function along two axes, use the fact that first derivatives are zero, and that you can choose your steps along the two axes to be the same, and you’ll end up with a contradiction.

That you can resolve your function into two directions for which the Taylor expansion is of the kind shown above is a result of the Laplacian being rotation-invariant. I have a feeling there must be a more elegant proof that I’m not able to think of.

Here’s a second problem. Say there’s a series in a complex variable. It can be shown that the series converges uniformly within a certain radius of convergence of a point, if we assume that the series is convergent on the boundary of the domain (you argue that if the worst case is convergent, every other case is uniformly convergent). That is, if the first sum converges, the second sum converges uniformly in the region indicated. (Note: The z^j are powers, not components!)

Now, is the series also convergent outside the domain? It seems to me that one should be able to analytically continue the series outside the domain and it should converge. It might be a problem that the sums are powers of absolute values and not analytic powers. I’m not sure.


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