Analytic Within and On the Unit Circle

When discussing whether a function is analytic within or on the unit circle, we are referring to complex analysis, which deals with functions of a complex variable.

Analytic Within the Unit Circle

A function f(z) is said to be analytic within the unit circle if it is analytic (i.e., differentiable) at every point inside the unit circle. The unit circle is defined as the set of points z ∈ ℂ such that |z| = 1, and the region inside the unit circle is where |z| < 1.

  • Analytic Function: A function is analytic at a point if it is differentiable not only at that point but also in a neighborhood around that point. More formally, a function f(z) is analytic if it has a Taylor series expansion that converges to f(z) in some neighborhood around the point of interest.
  • Within the Unit Circle: The phrase “within the unit circle” means that the function is analytic at all points where |z| < 1. This includes all points strictly inside the circle but excludes the points on the boundary |z| = 1.

For example, the function f(z) = 1/(1 – z) is analytic within the unit circle because it can be expanded as a convergent power series:

f(z) = ∑n=0∞ zn   for   |z| < 1

This series converges for all z inside the unit circle, but the function becomes singular at z = 1 (on the unit circle), where it has a pole.

Analytic On the Unit Circle

A function f(z) is said to be analytic on the unit circle if it is analytic at every point on the unit circle, meaning |z| = 1. This is a stronger condition because the function must now be analytic at all points on the boundary of the disk defined by |z| = 1, not just within it.

  • To be analytic on the unit circle, the function must: