Because the analysis must first satisfy the differentiability, we should first consider whether the above functions are differentiable.

Because when △y and △x converge at different speeds, the limit of △f/△z is different (for example, △y=k△x, and the ratio of the above formula can be related to k). Therefore, the latter is not differentiable everywhere on the whole complex plane, so it is not analytical.

Extended data:

Complex numbers are functions of independent variables and dependent variables, and the related theory is the theory of complex variable functions. 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.

Settings? (z) is a complex variable function in the plane open set D. For z∈D, if the limit exists and is finite, it is called? (z) is derivable at z, and this limit value is called? (z) The derivative at z, written as? (z). This is a generalization of the concept of derivative of real variable function, but the existence of derivative of complex variable function contains rich contents.

This is because z+h is any point in the two-dimensional neighborhood of z, and the existence condition of limit is much stronger than that of one-dimensional real number. If a complex function has derivatives everywhere in a neighborhood of z, then it must have higher derivatives at z and can be expanded into a convergent power series.

Baidu Encyclopedia-Complex Variable Function