Note on singularities of power series

MA THEMA TICS: H. F. BOHNENBL UST

752

PRoc. N. A. S.

Yr) is 0, since yl, ..., Yr are distinct by the irreducibility of (1.2). Hence the set of r homogeneous equations just constructed has the unique set of solutions (3.1) c(y, - 1) = 0, ..., Cr(y'r - 1) = 0. If therefore none of cl, . .., r vanish, yi, ..., Yr are mth roots of unity. We shall show that the hypothesis that precisely h of cl, ..., cr vanish, where 0 < h < r, leads to a contradiction. For, if precisely h vanish, f(x) is a solution of an equation of type (1.1) of order r - h. By hypothesis the characteristic equation of this equation is irreducible and has rational coefficients. But it has r - h roots in common with (1.2), which is irreducible. Since r - h < r, we have a contradiction. Hence yi, ., Yr are mth roots of unity. It remains to be proved that yi, ..., Yr are primitive. Let ym - 1 = Po(y) ... PI(y) be the resolution of ym - 1 into factors irreducible in the rational domain, and let the roots of Po(y) = 0 be the q5 (m) primitive mth roots of unity. Then the roots of Pi(y) = 0, . . ., P,(y) = 0 are imprimitive mth roots of unity. But since m is by hypothesis a proper period, m is the least positive integer for which the equations (3.1) hold. Hence Yr are not nth roots of unity, n < m, and therefore they are the Yi, roots of Po(y) = 0. P

NOTE ON SINGULARITIES OF POWER SERIES By H. F. BOHNENBLUST DEPARTMENT OF MATHEMATICS, PRINCETON UNIVERSITY

Communicated October 3, 1930

THEOREM: If (1) the power series a,,z', >p,,z", and Zapnz' have radii of convertn=O n=O ,n=O gence equal to one; and (2) Eanz' has on its closed circle of convergence a single singularity at the " =O point z = 1; and (3) the coefficients pa are non-negative, p,, > 0; then the power series Eanptz' is necessarily singular at the point z = 1. " =O This theorem is on one hand a generalization of the theorem of Pringsheim on singularities of power series with non-negative coefficients. (Pring-sheim's theorem is obtained by putting all an = 1.) On the other

VOL. 16, 1930

MATHEMATICS: H. F. BOHNENBLUST

753

hand it is a contribution to the Hadamard theorem on singularities of the Hadamard product of two power series. Hadamard theorem applied to s2anza and Epez' states that the point z = 1 is a possible singular point of -n=O n=o Eanpnz', while our theorem states that this point is necessarily singular. n =o The proof is an immediate consequence of the theorems of Pringsheim and Hadamard. Considering the first condition of the theorem, we obtain that to any given positive number e, there exists a number N(e), such that co

co

co

Ia.I

< (1 + e)"T for all

pn

(1 + e)

<

n>N(e),

It

I

and a,| pA, > (1 - e)n for infinitely many indices. A simple calculation shows that, as soon as e < 1, we have

Ian, 2P, < (1 + 7e)n for all n > N(e),

I

and an 12p,> (1- 3e)n for infinitely many indices. These inequalities express that the radius of convergence of the power series E a. I2pnzn is is=o

I

equal to one. All the coefficients being positive we can apply the theorem of Pringsheim, which shows that the point z = I is singular for that series.

But the same series is the Hadamard product of annzn and Eaz'. no n=O The latter one has (condition 2) a single singularity at z = 1 on its closed circle of convergence. Assuming (against the theorem) Eanp to be n =O

regular at z = 1, it follows that E| an 12Pnz' should be regular at the same n =O point, contradicting the above result. This contradiction proves the theorem. It is essential to suppose that the power series has only one singun =0 larity on its closed circle of convergence; for, given any arc on the unit circle, there exists a function, singular at z = 1 and at a point zo(zo $ 1) of the given arc, which when "multiplied" by an adequate power series Ep,1z", satisfying conditions 10 and 30, gives a power series regular at n=0 z = 1. In fact, on any arc there exists a point zo, for which 4 = -1( being a positive integer). Let us then consider the -function 1 2 X,

EaXz"

1

z+ X+1

o

E anz

754

MA THEMA TICS: J. H. C. WHITEHEAD

PRoc. N. A. S.

where a. = 1 + 2-'. On the other hand let us take r

M=

1/3 when n

=

l.m,

m

even,

1 when n = l.m, m odd, 0 otherwise. co

The "product" is the function EC,,z", where n=O

r 1 when n = l.m, m even, n= - 1 when n = l.m, m odd, L 0 otherwise.

This function is equal to 1 + Z', and is therefore regular at z = 1.

Putting the coefficients p, equal to one or to zero, we obtain as a corollary to our theorem: "A power series having only one singular point z0 on its closed circle of convergence is such that all its sub-series having the same circle of convergence are singular at z0."

A METHOD OF OBTAINING NORMAL REPRESENTATIONS FOR A PROJECTIVE CONNECTION By J. H. C. WHITEHEAD DEPARTMENT oF MATHEMATICS, PRINCETON UNIVERSITY

Communicated October 3, 1930

Normal representations for a projective connection, H, are characterized by the equations (1.1) PakzJZk = 0, where Pig are the components of II in the normal representation z + z0; (Greek letters, used as indices, will take the values o, . .. n, and Roman letters the values 1, ... n). In a paper in a forthcoming issue of the Annals of Mathematics we give a construction for obtaining such representations. In the introduction to that paper we give a brief historical account of generalized projective geometry, in which references to the literature may be found. In this note we give an alternative method depending on solutions to the partial differential equations.