Stone-Weierstrass Theorem (1885) states the approximation of continuous functions by polynomials, which is both theoretically and numerically useful.

In complex analysis, we can see a more general approximation: rational approximation. Recall that a function is meromorphic on the extended complex plane (Riemann sphere) if and only if it is rational function. i.e. rational function is the only kind of meromorphic functions that also meromorphic on the infinity. In the similar sense, we can see polynomials as meromorphic functions which have the only singularity (pole) on infinity. So the following are 2 very important rational approximation in complex analysis.

Runge’s Theorem (1885) reveals a kind of approximation of holomorphic functions by rational functions. This approach is under the geometry that function is holomorphic on the neighborhood of a compact set. The proof makes use of this condition. Surely this condition is a little bit strong, as in more intuitive sense, we would say, function is holomorphic on the interior of that compact set and continuous on the boundary. This is the spirit of Cauchy integral theorem/formula, the fundamental of complex analysis. We need to find a way to keep this spirit, and Mergelyan is the one who did this.

Mergelyan’s Theorem (1951) indicates the approximation of holomorphic functions by polynomials, which only requires holomorphic on the interior and continuous on the boundary for that function. The only additional requirement is “finite connected components for complement of K“. This theorem still has only one constructive proof provided by Mergelyan so far, and it is easily extend to a broader version: approximated by rational functions. That’s maybe the reason why it costs a long time to prove between this theorem and the other 2.

An open problem in Complex Analysis directly derived from Mergelyan’s theorem is checking the possibility to extend the geometric condition of Mergelyan’s theorem (complex plane) to the general higher dimensional complex spaces. We have several partial results so far, but no one prove it in general, so that’s still an open problem in complex analysis.