Double Series

an expression of the form

composed of the elements of an infinite matrix ǀǀu_mn 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:

u_mnis 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 S_m , 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 and 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 + n² ≤ N, and then consider limit . 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 u_mn . . 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.