analytic continuation
Suppose you only know a holomorphic function on a small patch of the plane — say a power series sum a_n (z - z_0)^n that converges inside a single disk. The disk is its first home, but it may not be its only possible home. Analytic continuation asks: can we extend this function to a bigger region so that it is still holomorphic everywhere, and so that on the original patch it agrees with what we started with? When such an extension exists, the new function is called an analytic continuation of the old one. The everyday picture is jigsaw puzzle pieces: each disk is one piece, and we try to fit more pieces around the edges, each agreeing with its neighbours on the overlap.
Concretely, here is how one step works. Pick a point z_1 inside the original disk of convergence, near its boundary. Compute the value and all the derivatives of the function at z_1, and form a new Taylor series sum b_n (z - z_1)^n centred there. That new series has its own disk of convergence, and a basic fact (the radius reaches the nearest singularity) often pushes that disk partly OUTSIDE the original one. On the lens-shaped overlap the two series describe the same function — they must, because they share all derivatives at z_1 — so we have honestly enlarged the domain. Repeat: re-centre at a fresh point near the new boundary, and keep going. A classic example: the geometric series sum z^n equals 1/(1 - z) only inside |z| < 1, but the formula 1/(1 - z) is holomorphic on the whole plane except z = 1, so it is THE analytic continuation, valid far beyond the original disk.
Why it matters: it lets a function defined by some restricted recipe — a power series, an integral, a sum that only converges in part of the plane — claim a much larger, canonical domain. The Riemann zeta function lives this way: its defining sum sum 1/n^s only converges for the real part of s greater than 1, yet analytic continuation extends it to the whole plane minus s = 1, and the famous questions about its zeros live in the continued region. The deep payoff (next entries) is that the extension is essentially forced — there is almost no freedom in how to continue.
The series sum z^n converges only for |z| < 1, where it equals 1/(1 - z). Re-centre at z_1 = -1/2: the new Taylor series of 1/(1 - z) there has radius of convergence 3/2 (the distance from -1/2 to the singularity at 1), so its disk reaches out to z = 1 on the right and to z = -2 on the left, covering points the original series never did. Patching such disks all over the plane recovers 1/(1 - z) everywhere except z = 1.
A power series knows the whole function — re-expanding at a new centre reveals values outside the first disk.
Continuation is about extending, not redefining: on the original patch the continued function must agree exactly with the one you started with. Also, an extension need not exist — some functions are walled in by a natural boundary they can never cross.