This document discusses functions of complex variables. It defines key concepts such as complex numbers, the Cauchy-Riemann conditions for differentiability, and analytic functions. The Cauchy-Riemann conditions require the partial derivatives of the real and imaginary parts of a complex function to satisfy certain relationships. Functions that satisfy the Cauchy-Riemann conditions everywhere are said to be analytic. The document also discusses singularities such as poles and essential singularities.