## Complex Power Series

**Point of Post: **In this post we discuss the basic ideas behind when a complex power series converges and discuss the holomorphicity of functions representable by power series.

**Motivation**

We have now discussed the notion of holomorphic functions but beside’s some extremely trivial ones (like polynomials) we don’t have any good class of examples of such functions. In this post we describe what is the most informative example of holomorphic functions–power series. Why are power series the most informative example? Well we shall eventually see that the holomorphic functions are precisely those that are locally just power series.

I assume anyone reading this is fairly well-acquainted with power-series and the related notions (the Weierstrass M-test, etc.) , for I will gloss over some stuff.

**Power Series**

Let be a sequence of complex numbers. We say that the series *converges *if the sequence converges in the usual sense. Since is complete we know that a sequence converges if and only if it’s Cauchy, and thus we see that converges if and only if for every there exists such that implies that .

Recall the notation that if with then , the *infinity norm, *is defined to be equal to . Clearly needn’t be finite. Recall then that we say that a sequence of functions mapping *converges pointwise *to a function if for each one has that . We say that converges *uniformly *to if for each there exists some such that implies that . This says roughly that “the same” for each . We say that is *uniformly Cauchy *if for all sufficiently large one has that . It’s a fundamental fact that since is complete that converges uniformly (to some function) if and only if it is uniformly Cauchy. It is a common fact that if is a sequence of uniformly Cauchy continuous functions then converges uniformly to some continuous function .

If is compact then defines a norm on the space of all complex valued continuous functions on .

When we speak of the (uniform) convergence of a *power series * we really are discussing the (uniform) convergence of the partial sum polynomials .

Suppose we are given a power series . We define the *radius of convergence *of the series to be

(here means the limit supremum) where if and if .

The fundamental fact about radii of convergence is the following:

**Theorem: ***Let be a series with radius of convergence . Then, and **converge uniformly on (here denotes the ball of radius centered at and the overline means closure) for each . Moreover, diverges for .*

Note that nothing is said about points on (here means boundary)–this is because nothing really can be said. It’s totally possible that converges nowhere, everywhere, or just some places on .

Now onto our first indication that power series are important for the discussion of holmorphic functions. Let . Say that is *locally representable by power series *if for every there exists a ball and a power series such that on . The big theorem then is the following:

**Theorem: ***Let be open. Then, if is locally representable by power series one has that , and moreover if is locally at then is locally at .*

I’ll leave you to look up the theorem (it’s in [3] for example)–it’s just an -chase. The important thing to note though is that since is locally representable by power series it is, again, holmorphic! Thus, we actually find the following:

**Theorem: ***If is locally representable by power series then is infinitely differentiable on , and if is locally then is locally . In particular, .*

In particular, if two series are equal, say on some open set then we know that . In particular:

**Theorem: ***Let be open. If on then for all $latex n.*

**References:**

[1] Greene, Robert Everist, and Steven George Krantz. *Function Theory of One Complex Variable*. Providence: American Mathematical Society, 2006. Print.

[2] Conway, John B. *Functions of One Complex Variable I*. New York: Springer-Verlag, 1978. Print.

[3] Rudin, W. *Real and Complex Analysis.* New York,NY: McGraw-Hill, 1988. Print.

[4] Ahlfors, Lars V. *Complex Analysis; an Introduction to the Theory of Analytic Functions of One Complex Variable*. New York: McGraw-Hill, 1966. Print.

[…] time we proved that every function on an open subset of that is locally representable by power series is […]

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