Complex variable function refers to a function with complex numbers as independent and dependent variables, 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 analytic functions in complex number field, so it is usually called analytic function theory.
The concept of complex number originated from finding the roots of equations. When finding the roots of quadratic and cubic algebraic equations, the square root of negative numbers appeared. For a long time, people couldn't understand this figure. However, with the development of mathematics, the importance of such numbers is increasingly apparent.
The theory of complex variable function came into being in18th century. 1774, Euler considered two equations derived from the integration of complex variables in one of his papers. Before him, French mathematician D'Alembert had obtained them in his paper on fluid mechanics.
Therefore, people later mentioned these two equations and called them "D'Alembert-Euler equations". In the19th century, when Cauchy and Riemann studied fluid mechanics, they studied the above two equations in more detail, so they were also called Cauchy-Riemann conditions.
Properties of complex variable function
Complex functions have many interesting properties, the most important of which is that they are analytic. This means that they are infinitely differentiable on the complex plane and satisfy cauchy-riemann equations, that is, the real part of pure virtual function and the imaginary part of real function are both continuously differentiable's.
In addition, due to the existence of polar angles and modules, complex variable functions also have periodicity, parity and analytical continuation. Some important concepts in complex variable function include: pole, zero, residue and so on.