La regla del trapecio para funciones complejas

Abstract

This monograph studies the application of the trapezoidal rule in the integration of complex functions over curves in the complex plane. Two fundamental approaches are presented: one based on the decomposition of the complex function into its real and imaginary parts, and another that directly uses the values of the function over a partition of the curve. Both methods allow the approximation of complex integrals through finite sums and provide a practical way to tackle calculations where analytical solutions are not possible. Additionally, the approximation error and its behavior are analyzed, showing that it decreases with the square of the number of partitions, provided the function involved is sufficiently smooth. The methods are illustrated with examples over closed contours, and the numerically obtained results are compared with exact values, demonstrating the effectiveness of the method even in complex settings. The theoretical foundation is supported by complex analysis and classical numerical analysis, which allows for a rigorous justification of all formulas used.