PURE MATHEMATICS 519 



has integer coefficients, i.e. in is a factor of T", This condition 

 is clearly not sufficient ; for example, the series 



2 + s^ + 2" + . . . + ^'•i + . . . 



does not represent an algebraic function, the circumference of 

 the unit circle being a natural boundary for the function. 

 But there are classes of functions for which the condition is 

 sufficient ; such are (i) hypergeometric functions (Errera), 



(2) functions whose derivatives are rational functions (Polya), 



(3) meromorphic functions (Borel), or more generally single- 

 valued functions with only isolated singularities (Polya). In 

 cases (2) and (3) the functions satisfying the conditions will be 

 rational functions. Of recent years a good deal of work has 

 been done, notably by Borel, Polya, and Carlson, on functions 

 which can be represented by power series with integer co- 

 efficients without being rational functions. Suppose we have 

 a simply connected region D in the complex 0-plane, containing 

 only inner points and containing the point z = o, can we find 

 an analytic function f{z), which is not a rational function, 

 expressible by a power series with integer coefficients, which is 

 uniform and has only isolated singularities in D ? The answer to 

 this question has been given by G. Polya {Proc. L.M.S., 21, 1922, 

 22-38 ; see also a summary in Jahresber. d. Math. Ver., 31, 1922, 

 107-55). The domain D can be conformly represented upon a 

 circle in the zti-plane with the point z = o corresponding to the 

 centre of the circle. If we add the condition that the magnification 



dw 



-J- is unity for this point, then the radius of the circle p will 



be uniquely determined. Polya's result is that \i p -^ i such a 

 non-rational function can be found, but that \ip> i , the function 

 must be rational. As a corollary it follows that if /?= i, then 

 the circle \z\ < I belongs to D, a result that had been proved by 

 Szego. Also, if C is a curve with no double points containing 

 2 = 0, and the interior and exterior be conformally represented 

 respectively by \w\ < pi and \w\ > p2, with zd = o and 00 corre- 

 sponding to = o and 00 , and the magnifications unity at these 

 points, then p^ ^ p^. A special case is that if a power series with 

 integer coefficients converges in unit circle it is either a rational 

 function or else its region of existence is limited by the circle 

 of convergence ; a theorem conjectured in 191 6 by Polya and 

 proved bj^ Carlson (Math. Zs., 9, 192 1, 1-13; compare also 

 Szego, Math. Ann., 87, 1922, 103-7). Polya's proof is interest- 

 ing as it employs theories which at first sight seem wholly dis- 

 connected. The first is Kronecker's conditions (1881) for a 

 power series Go + a^z + a^* + . . . to represent a rational 

 function, namely that only a finite number of the determinants 



