The Present Status of Integral Equations 
By Harold Thayer Davis, Assistant Professor of Mathematics in 
Indiana University. 
1. Introduction. The development of the branch of mathe- 
matics called “analysis’ ’ has centered in large measure around 
problems which have arisen in connection with the solution of 
functional equations. The most important class of such equations 
and the first to be studied was very naturally that in which deriva- 
tives of the unknown function appeared. But it must remain one 
of the curiosities of the evolution of mathematical science that more 
than two hundred years elapsed after the discovery of integral 
calculus before functional equations involving the unknown function 
under signs of integration were generally recognized as adding a 
new chapter to analysis and a powerful tool to the hand of the 
investigator in applied science. 
Altho little more than a quarter of a century has passed since 
the work of Volterra and Fredholm first called general attention to 
integral equations, the development of the subject has been so rapid 
and complete that these equations are already challenging the place 
of differential equations in the work of applied science. It is 
hazardous to predict, yet one cannot help but observe that in the 
discontinuous behaviour of atomic radiation there is a serious threat 
to those who must state all laws of physics by means of differential 
equations. Thus Poincare, in discussing the theory of quanta 
radiation, says :f “It is useless to remark how far removed this 
concept is from our customary ideas, since the laws of physics will 
no longer be susceptible of being expressed by means of differential 
equations.” 
The modern theory of integration is admirably adapted to the 
treatment of the discontinuous behaviour of functions, and if differ- 
ential equations shall ultimately prove unable to cope with the 
modern theory of radiation, it is highly improbable that these laws 
cannot be ex pressed by means of integral equations. 
*Presented at the meeting of the Indiana section of the Mathematical Association of America, 
May, 1925, at Bloomington. 
fComptes Rendus, vol. 153 (1911) p. 1103. 
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