in the Tlieory of Integral Equations 



179 



Thus, to derive (7) from (5) let 



f{x) = coth-i [P{oc)], (/> (0 = coth-i [(^1 {t)\ : 



then (5) becomes 



0-' [coth-' [F{x)] + coth-i {(^1 {t) W 



Now let (^"^ (^) = u, 



then ir = <^(m) = coth"' 1^1 («)}, 



whence ^i {u) = coth 0, 



and u = 4>r^ (coth 2) ; 



that is (/)-' (^) = 01-1 (coth ^), 



and therefore (5) reduces to 



_^\ F{x)4>,(t)+l ] 

 F{x) + (ji,(t) 



0r 



which is of the form (7 ). 



As an example of the use of (2) and (3) in the determination 

 of relations between integrals, let 



/i (•'^) = sin X, /, {x) = cos X, 



and, using the form (6) for 0, let 



0{x, «) = e»'-'o.'^', 



K (x) = X. 



bi = b.2 = a, Ui = «y = 0, 



and 



Then, putting 



we have from (2) yfr^ (t) = sin x . e^iog'' 



J 



dx 



(log t . sin a — cos a) e^^^st 4. 1 



^ l + (logO' ~ 



and 



yfr..it)=\ cos.'r.e-'"'°s'rf.« 



Jo 



(log t . cos a + sin a) e"^^^ — log ^ 



l+(logO-^ 



Substituting these values in (3), and then putting log^ = l/r 

 for brevity, we obtain 



'*" [x sin {x — a) + 7' cos {a; — a)} e'**'"' , 





r' + «' 



' -' it' sin X -\- r cos x 



i ■-' X SI 

 •'0 



y.2 _j_ ,^2 



f/./ 



= 0: 



13—: 



