Functions with complex numbers as independent variables are called complex variable functions, and the related theory is complex variable function theory. Analytic function is a kind of analytic function in complex variable function. Complex variable function theory mainly studies the analytic function in complex number field, so it is usually called analytic function theory.

The mathematical form describing many natural phenomena can be partial differential equations, especially the mathematical descriptions of many important basic laws of physical mechanics and engineering processes are partial differential equations, such as the basic laws of fluid mechanics and electromagnetism. These partial differential equations reflecting the laws of physical and engineering processes are the so-called mathematical physical equations.