The necessary and sufficient condition for a complex function to be differentiable is that the real part and imaginary part u(x, y)v(x, y) are fully differentiated at (x, y), Ux=Vy, u =-VX, so that its derivative can be derived: f'(z)=Ux(x, y)+iVx(x, y) is also true.

Complex variable function theory is a basic branch of mathematics, and its research object is complex variable function. The theory of complex variable function has a long history, rich content and perfect theory. It is widely used in many branches of mathematics, mechanics and engineering science. Complex numbers originate from finding the roots of algebraic equations.

Origin of complex variable function

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.