ON THE THEORY OF INTEGRAL EQUATIONS. 353 



by Liouville, 1 who appears to have obtained the solution quite inde- 

 pendently of Abel. 



Liouville connected the equation with many interesting problems in 

 geometry and analysis, and gave practically a rigorous proof of the 

 inversion formula 



z 



u(z) = ainXv ± [ fW dx ... (3) 

 W tt dz J (z - xf- x ■ Ki 



which holds if f(x) is continuous in an interval a 1 x 5 b if /(a) = 0, and 

 if the integral in the last equation possesses a continuous derivative, so 

 that the function u(z) given by the formula is continuous. It can be 

 shown that there is only one continuous solution of the equation. The 

 transition from formula (3) to (2) can be effected when f(x) possesses 

 a finite derivative which has only a finite number of discontinuities in 

 the interval (a < x < b). 



The case in which f(a) =J= has been considered by Goursat. 2 If 

 f(x) satisfies the foregoing conditions, the solution is given by 



u(z) = sin X;r _ /fa) _ J. s ' mXlT f f(x )dx 

 w tt \z - af~ x tt J (z - x) 1 -*' 



o 



Careful derivations of these results are given in Bocher's tract on 

 integral equations. 



Liouville first studied the integral equation in 1833 in connec'tion 

 with the theory of fractional differentiations and integrations. He was 

 thus led to some analogous forms of the equation in which A is negative, 

 and gave the formula 



CO 



j" x"- 1 <f>{x+a)dx = (- 1)" I» (AY n <p(a), 







which, however, is only valid when a large number of conditions are 

 satisfied. 



Boole 3 extended Liouville's results and gave the following general 

 formula for the solution of the equation : — 



a 



^(a) = [ {a—x) n - l 4>{x)dx, 







where n is a positive fraction. 



i i 



0(a) = ft'7* — v (?)[ v~ l (l-v) 1 -"- 1 Mav)dv, 

 r(«)r(» — n) \daj) v ' rv ; 



i being the integer next above n and i — 1 being less than the first 

 exponent in the ascending development of ^(a). This formula is 



1 Journal de VEcole Polytrchnique, 1835. s Acta Math., 1903. 



3 Camb. Math. Joitrn., 1815, vol. iv., pp. 82-87. 



