Davis: Present Status of Integral Equations 21 
present themselves. In his work on the potential problem where 
corner points are admitted in the boundary of the region, 0. D. 
Kellogg 443 found that the Fredholm series failed to converge and the 
singular equation had to be examined by a different method. Inte- 
grals in which Cauchy’s principal value was used were studied in 
this connection by Kellogg and Hilbert. A general theory of 
inversion formulas connected with integrals of which the principal 
value must be taken has been given by G. H. Hardy. 302 H. Villat, 298 
in a memoir which appeared in 1916, studied the Fredholm equation 
with singular kernel from this point of view. 
The theory of integral equation has been greatly broadened by 
work which has been done in connection with the analytic integral 
equation , 
r P(x,t) 
u(x) = f(x) + I n.(t) dt, (7) 
J s N (x,t) 
where P(x,t) and N(x,t) are functions analytic in two complex 
variables in the regions R and H' , f(x) is a function analytic in R 
except for a finite number of poles, and S is some path of integration 
between two fixed points a and (3 in Rb In particular a and p may 
coincide and the path of integration be a circuit as in Cauchy’s 
formula, or the path may be composed of one or more loop circuits. 
1 
It is interesting to note that if the kernel is and S is a 
1 t-x 
circuit about x, then X = — is a characteristic value for the 
2xi 
homogeneous equation, corresponding to which any analytic function 
of x is a solution. The analytic case of the Fredholm equation is 
clearly included in equation (7) when «' and p are real and the path of 
integration is the real axis. 
Important investigations have been made for the analytic 
equation by E. Picard, 258 and D. Pompeiu. 297 In a recent article 
G. Julia 294 has applied ideas of Hermite to obtain further insight 
into the character of its solutions. 
The numerical solution of integral equations has not been 
neglected altho, with increasing use, there is need for greater 
development along this line. F. L. Hitchcock and E. T. Whittaker 
(see section XXIII of the Bibliography) have contributed methods of 
attack. Hitchcock’s method applies to a Fredholm equation with 
inter yal (0,1), while Whittaker considers Volterra equations of 
first and second kinds for kernels of the form K(x— t). In the 
n 
equation of first kind he considers the case where K(x) = x _p 2 a n x n , 
n=l 
0 < p < 1, and for the equation of second kind he develops the 
4-36790 
