complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...

, a branch of

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the Mittag-Leffler star of a complex-analytic function is a set in the

complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...

obtained by attempting to extend that function along

rays Ray may refer to: Fish * Ray (fish), any cartilaginous fish of the superorder Batoidea * Ray (fish fin anatomy), a bony or horny spine on a fin Science and mathematics * Ray (geometry), half of a line proceeding from an initial point * Ray (gra ...

emanating from a given point. This concept is named after

Gösta Mittag-Leffler Magnus Gustaf "Gösta" Mittag-Leffler (16 March 1846 – 7 July 1927) was a Swedish mathematician. His mathematical contributions are connected chiefly with the theory of functions, which today is called complex analysis. Biography Mittag-Leffle ...

Definition and elementary properties

Formally, the Mittag-Leffler star of a complex-analytic function ''ƒ'' defined on an

open disk In geometry, a disk (also spelled disc). is the region in a plane bounded by a circle. A disk is said to be ''closed'' if it contains the circle that constitutes its boundary, and ''open'' if it does not. For a radius, r, an open disk is usua ...

''U'' in the

centered at a point ''a'' is the set of all points ''z'' in the complex plane such that ''ƒ'' can be continued analytically along the

line segment In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...

joining ''a'' and ''z'' (see

analytic continuation along a curve In complex analysis, the monodromy theorem is an important result about analytic continuation of a complex-analytic function to a larger set. The idea is that one can extend a complex-analytic function (from here on called simply ''analytic func ...

). It follows from the definition that the Mittag-Leffler star is an open

star-convex set In geometry, a set S in the Euclidean space \R^n is called a star domain (or star-convex set, star-shaped set or radially convex set) if there exists an s_0 \in S such that for all s \in S, the line segment from s_0 to s lies in S. This defini ...

(with respect to the point ''a'') and that it contains the disk ''U''. Moreover, ''ƒ'' admits a single-valued

analytic continuation In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new ...

to the Mittag-Leffler star.

Examples

* The Mittag-Leffler star of the complex

exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...

defined in a neighborhood of ''a'' = 0 is the entire complex plane. * The Mittag-Leffler star of the

complex logarithm In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

defined in the neighborhood of point ''a'' = 1 is the entire complex plane without the origin and the negative real axis. In general, given the complex logarithm defined in the neighborhood of a point ''a'' ≠ 0 in the complex plane, this function can be extended all the way to infinity on any ray starting at ''a'', except on the ray which goes from ''a'' to the origin, one cannot extend the complex logarithm beyond the origin along that ray. * Any open star-convex set is the Mittag-Leffler star of some complex-analytic function, since any open set in the complex plane is a

domain of holomorphy In mathematics, in the theory of functions of Function of several complex variables, several complex variables, a domain of holomorphy is a domain which is maximal in the sense that there exists a holomorphic function on this domain which cannot be ...

Uses

Any complex-analytic function ''ƒ'' defined around a point ''a'' in the complex plane can be expanded in a

series Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used i ...

polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...

s which is convergent in the entire Mittag-Leffler star of ''ƒ'' at ''a''. Each polynomial in this series is a linear combination of the first several terms in the

Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...

expansion of ''ƒ'' around ''a''. Such a series expansion of ''ƒ'', called the Mittag-Leffler expansion, is convergent in a larger set than the Taylor series expansion of ''ƒ'' at ''a''. Indeed, the largest open set on which the latter series is convergent is a disk centered at ''a'' and contained within the Mittag-Leffler star of ''ƒ'' at ''a''

References

* *

External links

* {{DEFAULTSORT:Mittag-Leffler Star Analytic functions