Unification of Holomorphy, Cauchy Theorem and Analyticity

Unification of holomorphy, Cauchy theorem, and analyticity handled independently in the three preceding chapters, Cauchy formula for derivatives, infinite differentiability of holomorphic functions, Morera theorem, Fundamental Theorem of Algebra, inverse function theorem and local structure of holomorphic functions, open mapping theorem, the number of zeros of holomorphic functions in a disk, Weierstrass theorems on analyticity of limits or sums of uniformly convergent sequences or series, Hurwitz theorem on the propagation of inexistence of zeros of the terms of sequences converging uniformly in compact sets to their limits, Hurwitz injection theorem on similar propagation of injectivity and of inclusion of ranges in a same set. Exercises, including analytic continuation by symmetry across line segments or circle arcs, Bloch theorem on the size of ranges of holomorphic functions and Bloch and Landau constants, location of zeros of polynomials including Sturm chains and the Routh criterion, and with applications to signal analysis and processing including reference to Fourier transform and to low-pass filters.