From the analytic view, all the derivatives of this function in the definition domain are 0. Let the Taylor expansion of this function be f (z) = f (z0)+f' (z0) * z+f' (z0)/2 * z2+= f (z0), where z0 is a point in this domain.

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 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.

The theory of complex variable function involves a wide range of applications, and many complex calculations are solved by it. For example, there are many different stable plane fields in physics. The so-called field is a region, each point corresponds to a physical quantity, and their calculation is solved by complex variable function.

For example, Russian Rukovski used the theory of complex variable function to solve the structural problems of aircraft wings when designing aircraft, and he also made contributions to solving the problems of fluid mechanics and aviation mechanics with the theory of complex variable function.

The theory of complex variable function is widely used not only in other disciplines, but also in many branches of mathematics. It has been deeply involved in differential equation, integral equation, probability theory and number theory, and has had a great influence on their development.