Past exam of the mathematics course of the University of Cambridge / 2016 / ib / Paper 3 / 13G (Complex Analysis)

Let $T$ be a closed triangle contained, together with a neighbourhood, in the domain of a holomorphic function $f$. We prove Cauchy theorem for a triangle without assuming continuity of $f^{\prime }$.

Join the edge midpoints to subdivide $T$ into four triangles, each with half its diameter and perimeter. Their positively oriented contour integrals sum to the integral around $T$, because the internal edges cancel. At least one subtriangle has integral of modulus at least one quarter of the original. Repeat that choice to obtain nested closed triangles $T_n$ satisfyingTheir intersection is a single point $z_{\ast }$, by compactness and shrinking diameter. complex differentiability at a point at $z_{\ast }$ givesThe first two terms have polynomial primitives, hence integrate to zero around each triangle. For sufficiently large $n$, $|r|<\epsilon$ on $T_n$, so the complex line integral estimate givesCombining the bounds gives $|{\int }_{\partial T}f(z)dz|\le \epsilon d_0{P}_0$. Since $\epsilon$ is arbitrary,The orientation can be reversed with no change to the conclusion.

For a closed piecewise continuously differentiable curve avoiding $a$, its winding number isEquivalently, for a parametrization $\gamma :[u,v]\to \mathbb{C}$, it is $(2\pi i{)}^{-1}{\int }_u^v{\gamma }^{\prime }(t)/(\gamma (t)-a)dt$, summed over the smooth pieces. A continuous choice of the argument of $\gamma (t)-a$ along the parameter interval changes by an integer multiple of $2\pi$ between the endpoints; that integer is the winding number.

For every closed piecewise smooth curve $\gamma$ in $U$, the winding number of $f\circ \gamma$ about zero is the integerIf $f=g_k^k$ for a holomorphic $k$th root, $g_k$ is nowhere zero, andThus $m_{\gamma }$ is divisible by every $k$ for which a root exists. An infinite subset of the positive integers is unbounded, so a fixed integer divisible by all those $k$ must be zero. Hence all periods of the holomorphic function $f^{\prime }/f$ vanish.

Fix $z_0\in U$. Since a domain is connected and open, its points can be joined by piecewise smooth paths. Vanishing periods makeindependent of path. It is a holomorphic primitive with $H^{\prime }=f^{\prime }/f$. Differentiating gives $(f{e}^{-H}{)}^{\prime }=0$, so $f{e}^{-H}=f(z_0)$. Choose a complex number $c$ with $e^c=f(z_0)$. ThenThis constructs a global holomorphic logarithm without requiring $U$ to be simply connected.