Complex line integrals and the ML estimate for $\int_{|z|=R}\frac{\log z}{z^2}\,dz$

1. \(f : D \to \mathbb { C }\) be a continuous function, where \(D \subset \mathbb { C }\) is a domain.Let \(\alpha : [ a , b ] \to D\) be a smooth curve. Give a precise definition of the complex line integral

\[ \int _ { \alpha } f . \]

2. Assume that there exists a constant M such that \(| f ( \tau ) | \le M\) for all \(\tau \in { \mathrm { I m a g e } } ( \alpha )\) . Prove that

\[ \big | \int _ { \alpha } f \big | \leq M \times \mathrm { l e n g t h } ( \alpha ) . \]

3. Let \(C _ { R }\) be the circle \(| z | = R ,\) , described in the counterclockwise direction, where \(R > 1\) Provide an upper bound for \(| \int _ { C _ { R } } \frac { \log { ( z ) } } { z ^ { 2 } } |\) , which depends only on R and other constants.