RECENT ADVANCES IN SCIENCE 197 



Die klassichen Probleme der Analysis des Unendlichen (Leipzig, 

 1910). 



A. Pringsheim (Sitzungsber . der k. b. Akad. der Wiss. zu 

 Munchen, 191 5, 387-400) gives an elementary proof of Weier- 

 strass's product theorem for whole transcendental functions, 

 and shows how Weierstrass's method of making his products 

 convergent can also serve to derive a criterion for conditional 

 convergence of infinite products and for determining the 

 variation of value of such a product when the order of the 

 factors is changed. 



G. Valiron {Bull, de la Soc. Math, de France, 191 6, 44, 45-64) 

 has a paper on the rate of growth (croissance) of the maximum 

 modulus of series having the character of a whole function, in 

 which he uses a method given by him in 191 3. 



Dunham Jackson (Annals of Math. 1916, 17, 172-9) shows 

 that in certain cases a function of several complex variables 

 analytic except for certain non-essential singularities can be 

 expressed as the quotient of two functions analytic in all the 

 variables . 



T. H. Gronwall (ibid. 1916, 18, 70-73) establishes two of 

 the most important properties of the function log (1 -\-z) from 

 the point of view of function-theory by actually performing by 

 simple methods the analytic continuation of the power series 

 used to define this function. It may be noted that Gronwall 

 (ibid. 65-9) deals with a problem in geometry connected with 

 the analytic continuation of a power series. Gronwall gave in 

 1916 a translation, with notes, of J. W. L. V. Jensen's ele- 

 mentary exposition of the theory of the Gamma function (ibid. 

 17, 124-66), and later (ibid. 1916, 18, 74-8) he extended to the 

 boundary of the region of convergence the discussion given by 

 Jensen of Binet's factorial series for logT (s) and -*/r (s). 



P. Appell (Acta Math. 40, 291-309) collects his investiga- 

 tions on Theta functions of the fourth degree. 



W. L. Hart (Annals of Math. 191 6, 18, 99-104) treats trigo- 

 nometric series in which the arguments of the sines and cosines 

 in the various terms are 2ira ]i t, where the numbers a k are not 

 necessarily commensurable, so that such series are generalisa- 

 tions of Fourier series. When the numbers last mentioned are 

 known as well as the function f(t) represented, as they are 

 by observation or possibly theory in problems in applied 

 mathematics, it is shown, under certain hypotheses as to the 



