Understanding a step in a proof of 2.1.12 from Ash and Novinger

Question

I have trouble understanding one particular step in the proof of "Extended Cauchy theorem for triangles" 2.1.12 from Complex Variables by Ash and Novinger. The goal is to prove that given function $f$ continuous on an open set $\Omega$ and analytic on $\Omega\setminus \{z_{0}\}$ and given a triangle $T$ such that its convex hull is a subset of $\Omega$ we have that $$\int_{T}f(z) dz =0$$ The part of the proof I have problems with is the following. In the case when $z_{0}$ is one of the vertices of $T$ and when $T$ is nondegenrate we choose some arbitrary points $a$ and $b$ on the sides $[z_{0},z_{1}]$ and $[z_{0},z_{2}]$ of $T$ respectively. After that we state that: $$\int_{T}f(z)dz = \int_{[z_{0},a]}f(z) dz + \int_{[z_{0},b]}f(z) dz $$ As I understand, we prove it using the fact that $$\int_{[a,b,z_{2},z_{1},a]}f(z)dz =0$$ Now I see two ways proving this equality. The first one is to dissect the quadrilateral into two triangles and use the Cauchy theorem for triangles which was already proven. But now the book references theorem 2.1.9. From this I infer that the authors meant to prove that integral is zero using the fact that analytic functon on star-like region has a primitive. But I don't see a clear way to find a region $\Omega_{1}\subset \Omega\setminus \{z_{0}\}$ which will be starlike and contain $[a,b,z_{2},z_{1},a]$. So my question is this, can you give a hint on how to hind such a region or show how to do it ?

Answer 1

That isn't what the author says on page 8 of https://faculty.math.illinois.edu/~r-ash/CV/CV2.pdf. Using your notation, he says that $$\int_{T}f\,dz = \int_{[z_0,a]}f\,dz + \int_{[a,b]}f\,dz + \int_{[b,z_0]}f\,dz.$$ To prove this, it is good to draw a picture. You will see that the right hand side is a line integral over a triangle contained in $T$ and that it differs from $\int_{T}f\,dz$ by a line integral of $f$ over the boundary of a trapezoidal region contained in $T$, disjoint from $z_0$. Since $f$ is analytic on a neighborhood of the trapezoid, the regular Cauchy theorem implies this integral is $0$.