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General Identities

General Mathematical Identities for Analytic Functions












General Identities





Transformations


Transformations and argument simplifications

Argument involving related functions (compositions)

This formula shows how to generate the series expansion of a general composition at the point , using the known series expansions for each of the functions and at .

This formula shows how to generate series expansion of the general composition at the point , using known series expansions for each function and at zero.

Products, sums, and powers of the direct function

Products of the direct function

This formula shows how to multiply power series of two functions and at the point .

This formula shows how to multiply power series of two functions and at the point .

This formula shows how to multiply power series of three functions , and at the point .

Ratios of the direct function

This formula represents the reciprocal of a power series for function at the point .

This formula represents the ratio of a power series for functions and at the point .

This formula represents the ratio of a power series for functions and at the point .

This formula represents the ratio of a power series for functions , and , at the point .

Sums of the direct function

This formula represents the summation property for the power series of the functions and at the point .

Powers of the direct function

This formula represents the series squared property of the functions at the point .

This formula represents the integer power of the series for the functions at the point .

This formula represents the arbitrary power of the series for the functions at the point .

This formula represents the generic arbitrary power of the series for the functions at the point .

Related transformations

Inversion

This formula represents the series expansion of the inverse function through the known power series of the direct function at the point .

This formula represents the series expansion of the inverse function through the known power series of direct function at the point .

Re-expansions in different points

This formula shows how to generate the series expansion of the function at the point , if this function is presented through the power series at the point .

This formula shows how to generate the series expansion of the function at the point , if this function is presented through the power series at the point .

This formula shows how to generate the series expansion of the function at the point , if this function is presented through the power series at the point .

Expansions through classical orthogonal polynomials

This formula shows how to generate the series expansion of the function at the point , if this function is presented through the power series at the point .

This formula shows how to generate the series expansion of the function at the point , if this function is presented through the power series at the point .

This formula shows how to generate the series expansion of the function at the point , if this function is presented through the power series at the point .

Determinants

This determinant is called the Vandermonde determinant.





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