REFERENCE SHEET

Holomorphic functions & power series

Complex differentiability in a neighborhood is far more rigid than real differentiability.

Before this Gradients, Hessians & change of variables, Series & uniform convergence

uₓ=vᵧ, uᵧ=−vₓ for f=u+iv

Definitions

Complex derivative

f′(z)=limₕ→₀ [f(z+h)−f(z)]/h, with complex h approaching zero from every direction.

Holomorphic

Complex differentiable at every point of an open set. Entire means holomorphic on all of ℂ.

Analytic

Locally representable by a convergent power series. For complex functions, analyticity and holomorphicity are equivalent.

Key theorems

Cauchy–Riemann criterion

If u,v have continuous first partial derivatives on an open set, the displayed equations hold throughout that set exactly when f=u+iv is holomorphic there.

Identity theorem

Holomorphic functions on a connected open set that agree on a set with an accumulation point inside the domain agree everywhere.

Liouville theorem

A bounded entire function is constant.

Notable applications

Rigidity and continuation

Knowing a holomorphic function on a small accumulating set can determine it on a connected domain.

Conformal maps

A holomorphic map with nonzero derivative preserves oriented angles locally.

Work it out

Try before revealing

01 / Check

Where is f(z)=the complex conjugate of z complex differentiable?

Hint

Write u=x and v=−y.

Solution

Nowhere. uₓ=1 but vᵧ=−1, so the necessary Cauchy–Riemann equation fails at every point.

02 / Apply

An entire f satisfies |f(z)|≤7 for all z. What can you conclude?

Hint

Use Liouville.