Morera's theorem

Theorem:

Suppose that f : D → C is continuous in the domain D, and that

∫_T f(z)dz = 0

for any triangle T contained in D. Then f is analytic in D.

Proof:

For any z ∈ D we can find r > 0 such that the disc E = {w ∈ C : |w - z| < r} is contained in D. Define F : E → C by

F(w) = ∫_{[z, w]} f(z)dz

where [a, b] denotes the directed line segment joining a, b ∈ C. For h with |h| < r - |w| the line segments [z, w], [w, w+h], [w+h, z] form a triangle contained in D. Thus

h^-1(F(w+h) - F(w)) = h^-1∫_{[w, w+h]} f(z)dz → f(w)

as h → 0 by continuity of f. Thus dF/dz = f in E, so F is analytic in E. The derivative of an analytic function is analytic, so it follows that f too is analytic in E.

Hence f is differentiable at all z ∈ D, so f is analytic in D.

Corollary:

If f_n : D → C is a sequence of analytic function converging uniformly to f on D then f is analytic, and f_n' converges to f' on D.

Proof:

For any triangle T in D

∫_T f_n(z)dz = 0

by Cauchy's theorem. So for any n

|∫_T f(z)dz| = |∫_T (f(z) - f_n(z)) dz| ≤ L(T)*sup{|f(z) - f_n(z)| : z ∈ D}

where L(T) denotes the length of the circumference of T. RHS → 0 as n → ∞, so LHS must be 0 since it is independent of n. It follows from Morera's theorem that f is analytic.

For any z ∈ D there is some r > 0 such that there is a circle C of radius r centred at z contained in D. By the Cauchy integral formula

|f'(z) - f_n'(z)| = |∫_C (f(w) - f_n(w))/(w-z)² dw|/2π ≤ sup{|f(z) - f_n(z)| : z ∈ D}/r → 0 as n → ∞