I cannot really understand how the reflection principle works. I have found various forms of the theorem, but the notes i use are not so clear on how we actually extend functions and what is the formula used each time. So i gathered two examples i need help with. Hopefully their solutions will make things clearer for me. Here it goes:
1) Let $D$ be the unit disc, $I=\left \{ e^{it}:t\in(0,1) \right \}$ an open arc on its the boundary and a function $f$ continuous on $D\cup I$ and holomorphic on $D$. If $f(z)=e^z$ for $z\in I$, show that $f$ has an expansion which is entire.
[Related: if $f$ is instead identically zero on that arc, how can we extend-reflect in order to make $I$ be a set whose points are interior points of a new domain and analytically continuate so that we get that $f$ is identically zero everywhere?]
2) Let $H$ be the upper half-plane, $I=[0,1]$ and let $f:H\cup I\rightarrow \mathbb{C}$ be a continuous function, holomorphic on $H$, satistfying $f(x)=\frac{1}{x+i}$ for every $x\in [0,1]$.Show that $f$ has a extension F on $D=\left \{ z:|z+i|>\frac{1}{3} \right \}$.
All help will be welcome and appreciated.