Complex Analysis of Real Functions VII: A Simple Extension of the Cauchy-Goursat Theorem
arXiv:1808.07026
Abstract
In the context of the complex-analytic structure within the open unit disk, that was established in a previous paper, here we establish a simple generalization of the Cauchy-Goursat theorem of complex analytic functions. We do this first for the case of inner analytic functions, and then generalize the result to all analytic functions. We thus show that the Cauchy-Goursat theorem holds even if the complex function has isolated singularities located on the integration contour, so long as these are all integrable ones.
arXiv comment17 pages, 6 figures. Small formatting corrections and bibliography update
References in corpus (12)
- Fourier Theory on the Complex Plane II: Weak Convergence, Classification and Factorization of Singularities
- Fourier Theory on the Complex Plane I: Conjugate Pairs of Fourier Series and Inner Analytic Functions
- Fourier Theory on the Complex Plane V: Arbitrary-Parity Real Functions, Singular Generalized Functions and Locally Non-Integrable Functions
- Fourier Theory on the Complex Plane IV: Representability of Real Functions by their Fourier Coefficients
- Complex Analysis of Real Functions I: Complex-Analytic Structure and Integrable Real Functions
- Complex Analysis of Real Functions II: Singular Schwartz Distributions
- Complex Analysis of Real Functions III: Extended Fourier Theory
- The Riemann Mapping Problem
- Fourier Theory on the Complex Plane III: Low-Pass Filters, Singularity Splitting and Infinite-Order Filters
- Complex Analysis of Real Functions IV: Non-Integrable Real Functions
- Complex Analysis of Real Functions V: The Dirichlet Problem on the Plane
- Complex Analysis of Real Functions VI: On the Convergence of Fourier Series