Complex functions and integral transforms - Part I

This comprehensive textbook explores the theory and applications of complex functions in mathematics. Beginning with the foundations of complex numbers (z = x + iy), the book develops the geometric interpretation of the complex plane where these numbers reside. The core chapters examine mappings of complex numbers, covering linear transformations, Möbius transformations, and conformal mappings, with special emphasis on the Riemann mapping theorem that allows conformally mapping any simply connected domain onto the unit disk.The text progresses to analytic functions, characterized by the Cauchy-Riemann equations, setting the groundwork for complex integration. The integration chapters develop the theory methodically, explaining path integrals, Cauchy’s Integral Theorem (showing that integrals of analytic functions around closed contours equal zero), and path independence properties. Particular attention is given to function complex integration, covering contour deformation techniques, branch cuts for multi-valued functions, and integration on unbounded domains including Jordan’s lemma.