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_{\alpha}f(z)dz-\oint_{\beta}f(z)dz=\oint_{\phi(t_{0},-)}f(z)dz-\oint_{\phi(t_{1},s)}f(z)dz

A p p l i c a t i o n 2 : S u p p o s e t h a t $ α (t_{0}) = β (t_{0}) $ a n d $ α (t_{1}) = β (t_{1}) $ . T h e n u s i n g t h e a b o v e i n t e g r a l s, w e a r e l e f t w i t h == 1;; $ \oint_{α} f (z) d z = \oint_{β} f (z) d z $ == A p p l i c a t i o n 3 : (A n n u l u s) I f $ α (t_{0}) = α (t_{1}) $ a n d $ β (t_{0}) = β (t_{1}) $, t h e n w e h a v e C a u c h y^{'} s t h e o r e m f o r a n a n n u l u s . T h a t i s, == 1;; $ \oint_{α} f (z) d z = \oint_{b} f (z) d z $ == . T h i s s t a t e m e n t s a y s t h a t u n d e r r e a s o n a b l e c o n d i t i o n s, i t d o e s n^{'} t m a t t e r w h a t r i n g a r o u n d a p o i n t y o u i n t e g r a t e .