Series expansions

Series expansions are mathematical representations of functions as infinite sums of terms, where each term is calculated from the function's derivatives at a specific point. These expansions allow functions to be approximated or expressed in a more convenient form for analysis, computation, or theoretical work. There are several types of series expansions, but the most common ones include: 1. **Taylor Series**: This representation expands a function \( f(x) \) around a point \( a \) using derivatives at that point.

A Dirichlet series is a type of infinite series of the form: \[ D(s) = \sum_{n=1}^\infty \frac{a_n}{n^s} \] where \( s \) is a complex variable, \( a_n \) are complex coefficients, and \( n \) ranges over the positive integers. The series converges for certain values of the complex variable \( s \) depending on the properties of the coefficients \( a_n \).

The Fox–Wright function is a special function that generalizes several well-known functions, including the hypergeometric function. It is defined through a series representation involving parameters that can take various values, leading to a wide range of applications in mathematics and physics.

The Kapteyn series, named after the Dutch astronomer Jakob Kapteyn, is a conceptual framework used in astronomy to describe the distribution of stars in the Milky Way galaxy. However, it is primarily known for its application in statistical analysis concerning stellar populations and the assessment of the spatial distribution of stars. In particular, the Kapteyn series can refer to different forms of distributions within the context of stellar distribution models.

The Laurent series is a representation of a complex function as a series, which can include both positive and negative powers of the variable. It is particularly useful for analyzing functions that have singularities (points at which they are not defined or fail to be analytic).

The Madhava series refers to a series of mathematical expansions developed by the Indian mathematician Madhava of Sangamagrama in the 14th century. Madhava is credited with creating early developments in calculus, particularly in the context of infinite series and trigonometric functions. One of the most notable contributions of the Madhava series is the expansion for calculating the value of \(\pi\) and other trigonometric functions.

Schlömilch's series is a series associated with a particular type of mathematical expansion, often related to functions in mathematical analysis, particularly in the study of special functions and series expansions. Specifically, it typically refers to a series that arises in the context of approximating certain kinds of functions or in the solution of differential equations. One notable example of a Schlömilch series is related to the expansion of the logarithm function or other functions in terms of powers of certain variables.