Past exam of the mathematics course of the University of Cambridge / 2004 / ib / Paper 2 / 16A / Solution

Define a candidate antiderivative by integration along the straight segment from zero:For $z$ in the open unit disc and sufficiently small $h$, the triangle with vertices $0,z,z+h$ lies within the disc, since the disc is convex. By the Cauchy integral theorem, its boundary contour integral vanishes. ConsequentlyFor $h\ne 0$, subtract $f(z)$ after division by $h$. The absolute value of the remainder is at most ${\sup }_{0\le t\le 1}|f(z+th)-f(z)|$, which tends to zero by continuity of the holomorphic function $f$ at $z$. Hence the complex derivative exists and $g^{\prime }(z)=f(z)$ for every point of the disc. This proves that $g$ is a holomorphic function and supplies the requested primitive directly from the Cauchy integral theorem.