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Let , where pk are complex numbers, have 0 < ρ ≤ ∞ for radius of convergence and assume that P(x) ≠ 0 for α ≤ x < ρ (α < ρ is some real constant). Assuming that is convergent for all (x ∈ [0, ρ), we define the P-limit of the sequence s = {sk} byThis, so called, power method of limitation (see (3), Definition 9 and (1) Definition 6) will be denoted by P. The best known power methods are Abel's (P(x) = 1/(1 – x), α = 0, ρ = 1) and Borel's (P(x) = ex, α = 0, ρ = ∞). By Cp we denote the set of all sequences, P-limitable to a finite limit and by the set of all sequences, P-limitable to zero.

A curve Γ has been defined as a normal multiple of a curve C if it contains an involution I n , of order n , which has the properties: (1) the sets of I n are in (1–1) correspondence with the points of C ; (2) each set I n consists of n points which are distinct in the birational sense; (3) I n is generated by an Abelian group G of order n of birational transformations of Γ into itself. We shall denote the field of complex numbers by k , the function field of C by k ( C ), and the function field of Γ by k (Γ). Conditions (1) and (3) imply that k (Γ) is a commutative normal (i.e. Galois) extension of k ( C ). The object of this note is to show how, given the curve C and the Abelian group G of order n , we can construct a curve Γ with the properties (1), (2), (3).

For the group of real numbers R, an exponential monomial is defined as a function of the form xr(−ixz), for some non-negative integer r and some complex number z. Similarly, an exponential polynomial is a function P(x) exp (−ixz), for a polynomial P. In a now famous paper ((15)), Schwartz proved that every closed translation invariant subspace (variety) of the space of continuous functions on R is determined by the exponential monomials it contains. His techniques do not generalize to groups other than R as they use the theory of functions of one complex variable. A shorter proof of this result, using the Carleman transform of a function, was given by Kahane in his thesis ((9)). Ehrenpreis ((5)) proved results similar to those of Schwartz for certain varieties in the space of analytic functions of n complex variables, and Malgrange ((13)) proved the related result that any solution in ℰ(Rn) (for the notation see (16)) of the homogeneous convolution equation μ*f = 0, for some μ∈ℰ′, belongs to the closure of the exponential polynomial solutions of the equation.

Methods are developed for the computation of the complex zeros of (½z)−νJν(z) when the index ν is an arbitrary complex number. These methods, which do not require an explicit knowledge of the Jv(z), are susceptible to rapid numerical evaluation on a computer. Beyond the interest in the zeros in their own right, these methods now make feasible the use of the infinite product representation of Jν(z) for the rapid computation of Bessel functions of complex order.

LetZ, Q, Cdenote respectively the ring of rational integers, the field of rational numbers and the field of complex numbers. Minkowski (4) solved the problem of minimizingforx, y ∈ Z(i)orZ(ρ), wherea, b, c, d ∈ Chave fixed determinant Δ ≠ 0. Here ρ = exp 2/3πi, andZ(i)andZ(p)are the rings of integers inQ(i)andQ(ρ)respectively. In fact he found the best possible resultsforZ(i), andforZ(ρ), wherewhile Buchner (1) used Minkowski's method to show thatforZ(i√2). Hlawka(3) has also proved (1·2), and Cassels, Ledermann and Mahler (2) have proved both (1·2) and (1·3). In a paper being prepared jointly by H. P. F. Swinnerton-Dyer and the author, general problems of the geometry of numbers in complex space are discussed and a systematic method given for solving the above problem for all complex quadratic fields Q(ϑ). Here, ϑ is a non-real number satisfying. an irreduc7ible quadratic equation with rational coefficients. The above problem is solved in detail forQ(i√5), for whichand the ‘critical forms’ can be reduced to

The only really useful practical method for solving numerical algebraic equations of higher orders, possessing complex roots, is that devised by C. H. Graeffe early in the nineteenth century. When an equation with real coefficients has only one or two pairs of complex roots, the Graeffe process leads to the evaluation of these roots without great labour. If, however, the equation has a number of pairs of complex roots there is considerable difficulty in completing the solution: the moduli of the roots are found easily, but the evaluation of the arguments often leads to long and wearisome calculations. The best method that has yet been suggested for overcoming this difficulty is that by C. Runge (Praxis der Gleichungen, Sammlung Schubert). It consists in making a change in the origin of the Argand diagram by shifting it to some other point on the real axis of the original Argand plane. The new moduli and the old moduli of the complex roots can then be used as bipolar coordinates for deducing the complex roots completely: this also checks the real roots.

1. It is trivial thatfor all integers n ≥ 0 implies sn = 0 (n = 0, 1, 2, …). This conclusion is no longer true if it is only known that (1·1) is true for an infinity of n. But we shall show that the truth of (1·1) for, roughly speaking, one-half of all positive integers, together with an order condition on the magnitude of sn, ensures sn = 0 for all n. This follows fromTheorem 1. Let a1 < a2 < … be positive integers, n(R) the number of a's not exceeding R, sn a sequence of complex numbers. If(ii) sn = ο(nK)as n tends to infinitythen sn = P(n), where P(x) is a polynomial of degree less than K.

This problem was originally attacked by Stokes, and the formula obtained by Stokes for the case of steady motion (given at the end of § 1 below) is now generally used in experimental work. However, this formula throws no light on the way in which the velocity of the sphere approaches its steady value: and the only known discussion of this problem was given by Boggio (his results will be found in § 2 below). Unfortunately, when the specific gravity of the sphere exceeds ⅝ (or the constant σ, defined later, is less than 4) the solution so obtained cannot be converted into numerical results, because the constants α, β become complex numbers; and even when α, β are real, the formula is not very convenient for large values of t (as the velocity approaches the steady value).

If {λ; n }, { b n } are sequences of complex numbers, and we consider the series ∑ b n exp (−λ n x ), given as convergent in (0, 1) (i.e. the open invertal (0,1)) to f(x) ∈ L , then, writing (if λ n = 0 the corresponding term is ½ b n x 2 ) where the series is supposed is to be uniformly convergent in (0, 1), we have for 0 h h(x) .If we know that the second member of (2) tends to f(x) as h → +0, it will follow that F(x) is a repeated integral of f(x) ((1), 671). If there is a sequence { φ v (x) } of integrable functions with the property that then, on multiplying (1) by φ v (x) and integrating over (0,1), we obtain a formula for b v in terms of F(x) . On integrating by parts twice, b v will be expressed in terms of f(x) , and this will constitute a uniqueness theorem for the series ∑ b n exp (− λ n x ).

Let A = (ank) be an infinite matrix of complex numbers ank (n, k = 1, 2,…) and X, Y two subsets of the space s of complex sequences. We say that the matrix A defines a (matrix) transformation from X into Y, and we denote it by writing A: X → Y, if for every sequence x = (xk)∈X the sequence Ax = (An(x)) is in Y, where An(x) = Σankxk and the sum without limits is always taken from k = 1 to k = ∞. The sequence Ax is called the transformation of x by the matrix A. By (X, Y) we denote the class of matrices A such that A: X → Y.

A square matrix A, of order n, having complex coefficients can be inverted without the aid of any operations involving complex numbers. This can be done if the coefficients ars + iαrs are replaced by their matrix equivalents and the resulting 2n × 2n real matrix A1 is inverted. The inverse will be a 2n × 2n matrix of similar form in which the subsidiary 2 × 2 matrices can be replaced by the equivalent complex numbers, thus yielding the inverse of A. It is the purpose of this note to show that a similar technique can be employed to evaluate det A.