13. Some applications

Let $U \subseteq C$ be an open subset, and $f : U \to C$ be a holomorphic function. With our latest version of Cauchy's Theorem, we have two nice applications.

Application 1:
Let $α, β : [t_{0}, t_{1}] \to U$ be two non-intersecting $C^{1}$ paths. Suppose in addition, 1;;that the line segment between $α (t)$ and $β (t)$ for $t \in [t_{0}, t_{1}]$ is contained in $U$ . Then we can define a linear homotopy $ϕ : [t_{0}, t_{1}] \times [0, 1] \to U$ , which is a $C^{1}$ map. It is namely the map $ϕ (t, s) = (1 - s) α (t) + s β (t)$

Now notice that the boundary of the rectangle can be given by the following curves: $α, ϕ (t_{1}, s), - β, - ϕ (t_{0}, s)$ . We choose let $γ$ run along these curves, and we apply Cauchy's theorem to get

\oint_{α} f (z) d z - \oint_{β} f (z) d z = \oint_{ϕ (t_{0}, -)} f (z) d z - \oint_{ϕ (t_{1}, s)} f (z) d z

Application 2:
Suppose that $α (t_{0}) = β (t_{0})$ and $α (t_{1}) = β (t_{1})$ . Then using the above integrals, we are left with 1;; $\oint_{α} f (z) d z = \oint_{β} f (z) d z$

Application 3: (Annulus)

If $α (t_{0}) = α (t_{1})$ and $β (t_{0}) = β (t_{1})$ , then we have Cauchy's theorem for an annulus. That is, 1;; $\oint_{α} f (z) d z = \oint_{b} f (z) d z$ . This statement says that under reasonable conditions, it doesn't matter what ring around a point you integrate.