PURE MATHEMATICS 521 



{Jahresher. d. Math. Ver., 31, 1922, 125-38), who show that if 

 P be symmetrical to the real axis and we construct the circles 

 on the joins of symmetrical points, then every non-real root of 

 a T-polynomial, with real coefficients, lies inside or on one of 

 these circles, an extension similar to that of Jensen to Gauss's 

 theorem on the zeros of the derivative of a polynomial. 



If a polynomial have only real zeros, all its derivatives have 

 only real zeros. If an analytical function is the limit of poly- 

 nomials with only real zeros, it belongs to a well-defined class 

 of integral functions and none of its derivatives have unreal 

 zeros. What other functions have this property ? 



G. Polya {Math. Zs., 12, 1922, 36-60) gives a partial answer 

 to this question, confining himself to (i) rational functions and 

 (2) integral functions of finite genus which have only a finite 

 number of zeros. He proves the following theorems : 



I. If R(0) is a rational fraction, the number of unreal zeros 

 of R"(2) tends to infinity with n unless either (i) R has only 

 one finite pole (of any order) or (2) R has only two finite poles 

 which are symmetrical with regard to the real axis. In case 

 (i) the total number of zeros of R is bounded, and in (2), if 

 R takes real values for real values of z, the number of unreal 

 zeros of R (0) is the same from a definite value of n on. 



II. If P(s) and Q,{z) are polynomials and Q (o) =0, and we 

 put G{z) = P{z) e^'^', then the number of unreal zeros of G" {z) 

 tends to infinity with n unless either (i) Q{z) is of degree one, 

 or (2) Q(s) is of degree two and equals bz — cz^, h being real and 

 c positive. In case (i) the total number of zeros is bounded, 

 and in case (2), if G takes real values for real z, the number of 

 unreal zeros never increases with n. 



These theorems are particular cases of more general ones : 



III. If F(s) is a meromorphic function, we form the enumer- 

 able aggregate of zeros of F(s), . . . F"(0) . . . The number 

 of limit-points depends only on the position of the poles of Y{z) 

 and not on their multiplicity. A point 2; is a limit-point only 

 if the two nearest poles are at equal distances. 



We can get a geometrical description of the limit-points. 

 If a is a pole of F(s) we call the " region of action " of a the 

 points z which are nearer to a than to any other pole of Y{z). 

 If a,h are two poles, the common points of their two regions 

 lie in the straight line bisecting ab at right angles. The region 

 of a is the inside of a convex polygon, which may extend to 

 infinity and in this case may have an infinite number of sides. 

 For/> iz) the regions form fundamental regions and are centrally 

 symmetrical hexagons with three pairs of equal and parallel 

 sides. In III the point set is identical with the set of points 

 of the boundaries of the regions of the poles of Y{z). 



In the region of a take a point z and develop Y{z) as a power 



