Theorem 6.1.  Suppose that is a continuous complex-valued function defined on a set containing the contour C.  Let be any parametrization of C for  .  Then  

            .  

Proof.

    We omit the proof of Theorem 6.1 because it involves ideas (e.g., as the theory of the Riemann--Stieltjes integral) that are beyond the scope of this book. A more rigorous development of the contour integral based on Riemann sums is presented in advanced texts such as Lars Valerian Ahlfors, Complex Analysis, 3rd ed. (New York: McGraw-Hill, 1979).  

Complex Analysis for Mathematics and Engineering