Holomorphic & Analytic Functions

Complex differentiability, Cauchy-Riemann, and the analytic equivalence.

The idea

A function $f \colon \mathbb{C} \to \mathbb{C}$ is holomorphic at $z_{0}$ when the limit $f'(z_{0}) = \lim_{h \to 0} \frac{f(z_{0} + h) - f(z_{0})}{h}$ exists, and holomorphic on an open set when it is holomorphic at every point of the set. The strength of the condition lies in the increment: $h$ is complex, it may approach $0$ along any path in the plane, and every path must give the same limit.

Write $f = u + iv$ with $z = x + iy$. Comparing the horizontal approach with the vertical one already forces the Cauchy-Riemann equations $u_{x} = v_{y}, \qquad u_{y} = -v_{x},$ and when the partial derivatives are continuous, these two equations are equivalent to holomorphy.

Holomorphy is far more restrictive than real differentiability. A function holomorphic on a disk is analytic there: it equals a convergent power series $\sum_{n \ge 0} c_{n}(z - z_{0})^{n}$ on the disk, with $c_{n} = f^{(n)}(z_{0})/n!$, and conversely every convergent power series is holomorphic where it converges. In particular, one complex derivative forces derivatives of every order.

Ways to work on it

• Walkthrough. Complex differentiability as a limit, the Cauchy-Riemann equations, and a function that fails them.
• Practice. Test the Cauchy-Riemann equations on a given function.
• Hardest. The holomorphic equals analytic equivalence: power-series coefficients and radius of convergence.

Not sure where to start? Take the ten-question placement test.