Theorem LM.

Theorem 5.2.1   Let $C$ be a path of length $L$ and $f$ be a function, continuous at every point of $C$ . Moreover suppose that there exists a positive real number $M$ such that $\forall z \in C,\vert f(z)\vert \leq M$ . Then:

$$\left\vert \int_C f(z) \; dz \right\vert \leq LM$$

The proof is based on the fact that

$$\left\vert \int_C f(z) \; dz \right\vert \leq \int_C \vert f(z)\vert \; dz,$$

which can be proven using Riemann sums.

Example 5.2.2   Let $C$ be the upper half of the unit-circle; then $L=\pi$ . Consider the function $f(z)=e^z$ ; we have: $\forall z \in C, f(z)=e^{\cos t + i \sin t} = e^{\cos t} \cdot e^{i \sin t}$ , therefore $\forall z \in C, \vert f(z)\vert \leq e$ . It follows that $\left\vert \int_C e^z \; dz \right\vert \leq e \pi$ .