Complex Analysis 9 | Power Series [dark version]

Power series in complex numbers behave in a structured way: they converge inside a disk and diverge outside it, with the boundary left as the only uncertain zone. That “disk of convergence” is the central takeaway because it turns an infinite-sum definition into a geometric object in the complex plane—making it far easier to predict where a complex power series defines a valid function.

The discussion starts by extending the real-variable definition of a power series to complex coefficients. Given complex numbers a0, a1, a2, …, a power series defines a function by summing a_k times (z − z0)^k over k from 0 to infinity. The key subtlety is the domain: the series only defines a function at those z for which the infinite sum converges. In other words, the domain is precisely the set of complex numbers where the series converges.

A concrete example anchors the theory: the exponential function exp(z) is written as the power series Σ_{k=0}^∞ z^k / k!. The usual “infinite sum” interpretation is emphasized as a limit of complex numbers, and the same convergence ideas from real analysis apply once distance in the complex plane is used to measure limits.

To build intuition, the geometric series Σ_{k=0}^∞ z^k is analyzed. It equals 1/(1−z) when |z|<1, and it diverges when |z|≥1. This yields a clear picture: convergence occurs inside the open disk centered at 0 with radius 1, while divergence occurs outside it. The transcript then generalizes this pattern: for any complex power series, there exists a maximal radius R such that the series converges for all z with |z−z0|<R, and diverges for all z with |z−z0|>R. The only place that can vary from one power series to another is the boundary where |z−z0|=R—some series may converge there, others may diverge, or mix behaviors.

The “maximal radius” R is formalized as the radius of convergence. The domain can be described using open balls (for guaranteed convergence) and closed balls (to capture divergence outside the boundary). The transcript notes that a standard proof can be built using the root criterion combined with the geometric series.

Finally, an explicit way to compute R is given via the Cauchy–Hadamard formula: 1/R equals limsup_{k→∞} (|a_k|)^{1/k}, with the limsup taking values in [0,∞]. This lets one determine the convergence disk without guessing. The exponential function is suggested as an exercise to compute its radius of convergence, and the next topic is set to move toward uniform convergence.