General Identities General Mathematical Identities for Analytic Functions

Operations

Limit operation

This formula reflects the definition of a limiting value for a function at the point , when approaches in any direction (the so-called epsilon‐delta definition): the formula means that the function has a limit value if and only if for all , there exists such that whenever . A limiting value does not always exist, and (if it exists) it does not always coincide with the value of the function at the point (the last value also may not exist). But in the "best situation", when all the values exist and coincide: , and the function is called continuous at the point .

This formula reflects the definition of a limiting value for a function at the infinite point : the formula means that the function has a limit value if and only if for all , there exists such that whenever . A limiting value may not always exist, and (if it exists) may not always coincide with the value of the function at the point (the last value also may not exist). But in the "best situation", when all the values exist and coincide: , and the function is called continuous at the point .

This limit shows that analytic functions are continuous functions.

This limit defines the derivative of function at the point , if it exists. For analytic functions this limit exists.

This limit defines the derivative of a function with an argument .

L’Hospital’s rule appeared in the first textbook on differential calculus, Treatise of L’Hospital, in 1696. It allows you to evaluate the limit of the ratio of two functions through the limit of the ratio of their derivatives in the cases when and are equal to zero or .

This formula reflects the definition of the convergent integral at the argument .

This formula reflects the commutativity of the two operations definite integration and limit.

This formula reflects the reordering of the two operations infinite summation and limit.

This formula reflects the reordering of the two operations infinite product and limit.

The Riemann–Lebesgue lemma shows that sine coefficients of the trigonometric Fourier series tend to zero as .

The Riemann–Lebesgue lemma shows that cosine coefficients of the trigonometric Fourier series tend to zero as .