RECENT ADVANCES IN SCIENCE 9 



of a general theory of series of two forms of which the first 

 is 2c»g(x + n). In this sequel he determines the asymptotic 

 character of a function defined by a series of that first form. 



G. Polya (Math, es term. ert. Budapest, 1916, 34, 754-8) 

 proves a generalisation of the theorem of Eisenstein on the 

 representation of an algebraic function by a power series with 

 rational coefficients. 



M. Riesz (Arkiv for Mat. 1916, 11, No. 12) gives new proofs 

 and a generalisation of two theorems of Fatou on the co- 

 efficients of power series. 



S. Wigert (ibid. No. 5) proves theorems on the convergence 

 of series associated with the coefficients of a power series. 

 Wigert also (ibid. No. 21) gives a theorem on whole functions, 

 and M. Alander (ibid. No. 15) continues his work of 1914 on 

 the zeros of the derivatives of whole real functions. 



F. R. Berwald (ibid. No. 23) generalises certain identities 

 due to Euler. 



The main theorems on factorisation of analytic functions 

 of several variables were given by Weierstrass (cf. his Abhand- 

 lungen ilber Functionentheorie, 1886, p. 107), but his presenta- 

 tion does not make clear what is essential, and so W. F. Osgood 

 (Annals of Math. 191 7, 19, 77-95) gives a systematic presenta- 

 tion of the theory as a whole. Dunham Jackson (ibid. 142-51) 

 points out some of the cases in which theorems about the be- 

 haviour of analytic functions of several complex variables in 

 the neighbourhood of a point have analogous theorems corre- 

 sponding to them for the case of a function which is analytic 

 with respect to a particular one of its arguments, but merely 

 continuous with regard to the whole set of variables ; Bocher 

 (ibid. 1910-1 1, 12, 18-26) called such functions " semi-analytic." 

 Jackson also points out a simple case in which the above 

 analogy is not preserved unimpaired. His method of treat- 

 ment is closely related to that of an earlier paper of his (ibid. 

 191 5-16, 17, 172-9). 



Paul du Bois-Reymond (1876) gave the first example of a 

 continuous function whose Fourier's development diverges at 

 one or more points ; Haar (1909) showed how to construct a 

 continuous function whose Sturm-Liouville development 

 diverges and one whose development in Legendre's functions 

 is divergent, and Gronwall (1914) gave a function whose deve- 

 lopment in Legendre's functions is not summable at a certain 



