Double Series

double series

[¦dəb·əl ′sir‚ēz]
(mathematics)
A two-dimensional array of numbers whose sum is the limit of Sm, n, the sum of the terms in the rectangular array formed by the first n terms in each of the first m rows, as m and n increase.

Double Series

 

an expression of the form

composed of the elements of an infinite matrix ǀǀumn ǀǀ (m, n = 1,2,...). These elements may be numbers (then the double series is called a double series of numbers) or functions of one or several variables (double series of functions). An abbreviated notation is used for the double series:

umnis called the general term of the double series. The finite sums

are called partial sums of the double series. If the limit

exists when m and n tend to infinity independently from each other, then this limit is said to be the sum of the double series and the double series is said to be convergent. The theory of the convergence of double series is considerably more complex than the corresponding theory for simple series; for example, in contrast to the simple series, the convergence of a double series does not imply that its partial sums are bounded.

The expression

is called a repeated series. Here we are required to sum first the series

composed of the sums Sm , If the repeated series (1) is convergent and has theis convergent and has the sum S, then it is called the row sum of the double series. The column sum S’ of the double series is defined in an analogous manner. The convergence of the double series does not imply the convergence of the series Double Series and Double Series so that the row sums and column sums may not even exist. Conversely, if the double series diverges, it may turn out that the sums over the row sums and column sums exist and that S ≠ S. However, if the double series converges and has the sum S and if the row sum and column sums exist, then each of these sums is equal to 5. This fact is always used in the actual calculation of the sum of a double series.

The most important classes of the double series are the double power series, the double Fourier series, and the quadratic forms with an infinite number of variables. For the double Fourier series

one of the standard concepts concerning the sum of such series is the following: we form the circular (or spherical) partial sums

where the summation is over all pairs of integers (m,n) for which m2 + n2N, and then consider limit Double Series. This limit is called the spherical sum of the double Fourier series (2). Many important functions are represented with the aid of double series, for example, Weierstrass’ elliptic function.

A multiple series (more precisely, an s-multiple series) is an expression of the form

constructed from the members of the table ǀǀ umn . . p ǀǀ. Each member of this table has s indexes m, n, . . . , p, and these indexes run independently through all the natural numbers. The theory of multiple series is completely analogous to the theory of double series.

REFERENCES

Fikhtengol’ts, G. M. Kurs differentsial’nogo i integral’ nogo ischislenia, 6th ed., vol. 2. Moscow, 1966.

S. B. STECHKIN

References in periodicals archive ?
The Indian women's team are fresh from their victory against South Africa, where they scripted history by clinching a double series win in a single tour for the first time ever.
The SGP this year will see Great Britain's double series winner, Tai Woffinden, taking on reigning World Champion, Greg Hancock as well as 2012 World Champion Chris Holder, who has been hit by a series of injuries.
Compared with [26, 27, 30], the random response of this paper for the stochastic van der Pol system is directly expressed as the double series form of deterministic response using the known orthogonal polynomial functions by means of the sequential orthogonal decomposition.
It details the elementary theory of infinite series; the basic properties of Taylor and Fourier series, series of functions, and the applications of uniform convergence; double series, changes in the order of summation, and summability; power series and real analytic functions; and additional topics in Fourier series, such as summability, Parseval's equality, and the convolution theorem.
Victory in the opening round of the six-stage series sets the double series winners up well for a hat-trick this season, where the top four teams qualify for rugby's Olympic debut in 2016.
This yields a double series [summation over (m,n)] as for [I.
Since Pringsheim introduced the notion of convergence of a numerical double series in terms of the convergence of the double sequence of its rectangular partial sums in [9], several authors have contributed to this topic during the last century.
For giving the double series representation we shall use the Bailey pair method.
The double series is 110cm wide with an 18 tons per hour capacity.
But we haven't even discussed the amazing double series that was, and is, Spaced.
It would be unfair to deny the interest of a number of individual shows--among my official favorites were Hans van der Meer's double series of soccer matches, "Dutch Fields," 1995-97, and "Football on Stage in Provence," 2004, and Raphael Dallaporta's "Antipersonnel," 2004, chillingly aestheticized photos of landmines from around the world (both in the "Contemporaries" category).