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:

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

The proof is based on the fact that

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

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$ .

Noah Dana-Picard 2007-12-24