Davis: Present Status of Integral Equations 
9 
but perhaps the most important generalization for which they are 
responsible is the theory of the Fredholm equation of first kind. 
To Abel 198 belongs the credit for having given in 1823 the first 
solution of the equation now known by his name, 
/ x u(t) 
dt a < 1. 
c (x-t) a 
The inversion formula, 
sina- d /»x f(t) 
u(t). = — I dt, 
x dx ^ c (x -t)l “ a 
is one of the best known results of the early period in integral 
equations and has led to a number of generalizations and applica- 
tions. It is closely connected with Riemann’s operator for fractional 
differentiation and integration. 79 ’ 81 The equation had been 
previously obtained, tho not solved, by 8. Poisson* in the problem of 
the distribution of temperature in a conducting sphere, and J. 
Liouvillef in 1832 connected it with his theory of fractional deriva- 
tives. Three years later and independently of Abel he also gave a 
solution of the equation in the form in which it appeared in the work 
of Poisson. J A general treatment of the problem was given by R. 
Murphy in 1833, who studied it in connection with questions in 
electrostatics! . 
At about this same time both Liouville and Poisson were led to 
another problem which represented a real generalization of that of 
the inversion of an integral. In 1826 Poisson|| obtained an equation 
of the form 
4xk fx 
g(x) = u (x) - I f'.(x -t) ii (t) dt, 
3 " o 
which he succeeded in solving by expanding u(t) in powers of the para- 
meter k. The convergence of the resulting series was not established 
until later. 
Altho J. Gaque in 1864^f introduced integral equations into the 
discussion of a method for solving linear differential equations, and 
other investigators in applied fields incidentally found themselves 
confronted by the necessity of determining an unknown function 
under a sign of integration, it was apparently not recognized before 
1888 that these functional equations presented a new problem in 
analysis which was worth a systematic study of its own. In that 
"Journal de 1’ccole polytechnique, (1821, vol. 19), p. 299. 
tlbid. (1823, vol. 21), p. 9. 
tlbid. (1835, vol. 24), pp. 55-60. 
§Cambr. Phil. Trans, vol. 5 (1833), pp. 113-148, 315-394. 
IlMemoire sur la theorie du magnetisme en mouvement. Oeuvres, vol. 3, no. 5, pp. 41-72. 
‘iMethode nouvelle pour 1’ integration des equations cliff erentielles lineaires. Journal de math., 
vol. 29 (1864), pp. 185-222. 
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