404 REPORTS ON THE STATE OP SCIENCE. 



portions of the curve between z x and z 2 , z 2 and z 3 . . . . z m _ 1 and z m . If 

 f n (z), g„( z ) denote the values of f(z) and g(z) outside C, 

 M z )>9i( z ) >. » „ inside C, 



then the conditions to be satisfied are that 



fo =/n g = g\ on the portion z m z { of C 



7 3 ]\>0 i y f \J n l on the portion * A+1 



/.-«! /l +'2 01 (A = 1,2,... TO). 



Plemelj ' has shown that this problem of Eiemann may be solved in a 

 much simpler way by using Cauchy's integral formula in place of 

 Green's functions. 



Hilbert also gives sufficient conditions that a given complex 

 expression 



it(s) + iv(s) 



may be the boundary value on C of a function of a complex variable 

 which is regular inside or outside C. 



21. The Solution of Linear Differential Equations by means of 



Definite Integrals. 2 



Laplace's method of solving linear differential equations by means 

 of definite integrals has been extended by Heine, Pincherle, Jordan, 

 Pochhammer, Schlesinger, and many other writers. The general method 

 of solving an equation h f (u) = by means of a definite integral of the 



typG f(x) = i(x,t)f(t)dt 



depends upon the formation of an identity of the form 



where w{x,t) may or may not be identical with k. If M,(v) = is the 

 differential equation adjoint to M,(w) = we have in general 



hrif) = [h^)f(t)dt =f M,(nO*(0<W = [J-B + \w{x,t)M t ^)dt 



where B is the bilinear concomitant. The first integral can usually be 

 made to vanish by a suitable choice of the limits, and the second integral 

 vanishes if f(t) is a solution of the equation 



M t (v) = 



which is called the transformed equation relative to the kernel /.-. In 

 the case when k(&,£) = c 1 ' we may take w(x,t) = e xi , and then if 



1 Sitzungslerichte, Vienna, May 10, 1906 ; Monatshefte fur Matltematilt und Physik' 

 1908 ; Jahresbericht der deutsche?i Math. Vercin.. 1909, p. 15. 



2 A report and bibliography on the theory of linear differential equations from 

 1805-1907 is given by Schlesinger, Jah'-csbericht der devtschen Math. Verein., 

 1909, pp. 133-266. 



