Generalized Cauchy-Riemann Systems with a Singular Point

# Generalized Cauchy-Riemann Systems with a Singular Point

Introduction:

Generalized Cauchy-Riemann systems with a singular point are a fascinating topic in mathematics. These systems involve the study of functions that satisfy a set of partial differential equations, known as the Cauchy-Riemann equations, with a singular point. The presence of a singular point adds complexity to the analysis and leads to interesting mathematical properties.

## Main Title

### Definition

A generalized Cauchy-Riemann system with a singular point is a system of partial differential equations that describes the behavior of a function in the neighborhood of a singular point. The system is typically expressed in terms of the real and imaginary parts of the function and involves derivatives with respect to both the real and imaginary variables.

#### Properties

Generalized Cauchy-Riemann systems with a singular point exhibit several interesting properties:

1. Non-analytic behavior: Unlike regular Cauchy-Riemann systems, functions satisfying the generalized system may not be analytic in the neighborhood of the singular point.
2. Singularities: The presence of a singular point introduces singularities in the solutions of the system. These singularities can be classified and studied using techniques from singularity theory.
3. Applications: Generalized Cauchy-Riemann systems with a singular point find applications in various areas of mathematics, such as complex analysis, differential geometry, and mathematical physics.