ON THE THEORY OF INTEGRAL EQUATIONS. 347 



was used by Rieniann | in some researches on the law of distribution of 

 prime numbers. He gave the inversion formula 



a + oa t 



*»-eJ 



t- r -'f(x)dx. 



This inversion formula has been studied very thoroughly by Mellin, 2 

 and is of considerable importance in the theory of the Gamma and 

 Hypergeometric functions. The solution of the integral equation (4) 

 when f(x) is given for positive integral values of x is known as the 

 Problem of the Moments and is of some importance in investigations on 

 the Law of Error. 3 The equation has been studied in these circum- 

 stances by Stieltjes, 4 in his famous researches on continued fractions. 

 He shows that the solution of the equation is not unique unless a 

 number of restrictions, indicated by the properties of an associated 

 divergent series, are imposed upon the functions / and rf>- The results 

 of Stieltjes are expounded and generalised by Borel in his book on 

 divergent series. 



Stieltjes is also led in the course of his researches to a solution of 

 the equation 



M = PS? 



) x + t 



Integral equations of the form (1) are of considerable importance in 

 the theory of divergent series which has been developed by Borel, 5 

 le Roy, 6 Barnes, 7 Hardy, 8 Cunningham, 9 and others. An excellent 

 account of the theory is given in Bromwich's ' Infinite Series.' A 

 divergent series 2 a n x n is associated with a function 



< j ,(xt) = 2a n X ± 



'-- 

 and finally with the function 



f(x) = [e'^{xt)dt. 



Many interesting properties of these integrals are given in papers by 

 the authors just mentioned. 



Lerch l0 has shown that if an integral of the type 







e- rt f(t)dt 



1 Werke (Weber), p. 140. 



J Acta Math., 1902, vol. xxv. ; Math. Ann., 1910, Bd. 68, Heft 3. 



* See for instance the papers by F. Y. Edgeworth, Comb. Phil. Irani., 1901, 

 vol. xx. 



* Annales de Toulouse, 1894, t. 8; 1895, t. 9. 



6 Leqons tur les series divergentes. Paris (1901). 

 6 Annaks de Toulouse (2), t. 2 (1902), p. 317. 



* Phil. Trans. A., vol. cxcix., pp. 411-500. 



" Quarterly Journ., vol. xxxv. (1903), p. 22 ; Cambr. Phil. Trans., vols, xix-xxi. 



* Proc. Lond. Math. Soc. ser. 2, vol. iii. (1904), p. H51. 

 ,0 Acta Math., 1903, p. 339. 



