Functions of Complex Variables is a branch of mathematics that studies functions defined on complex numbers. These functions can be expressed in terms of real and imaginary parts, allowing for a deeper understanding of their properties. Key concepts include analyticity, which indicates that a function is differentiable in a neighborhood of a point, and the Cauchy-Riemann equations, which provide conditions for a function to be analytic. This field has important applications in various areas, including engineering, physics, and applied mathematics. Techniques such as contour integration and residue theory are used to evaluate integrals and solve problems in fluid dynamics and electromagnetic theory.