FUNCTIONS IN TIIK THEORY OF INTEGRAL EQUATIONS. 187 



corresponding to f(s) will converge at s = 0, and have lim 5, (0) for their common 



- 



Hum, provided that this limit exists and is finite. 



Let us suppose that /(O + O) exists and is finite. Then it is easily seen that 



5 1 



8n 



It follows from this, by a proof similar to that employed above, that the series 

 mentioned will all converge to /(O + O), if, as ft diminishes indefinitely, the integral 



sin 



converges uniformly to zero, for positive integral values of n. The reader will be 

 able to show that this condition is satisfied when f(s) has limited total fluctuation in 

 an arbitrarily small neighbourhood to the right of s = ; and also when 



possesses a Lebesgue integral in an interval (0, a). 



As corresponding remarks apply at the end point s = ir, we have the theorems : 

 If f(s) has limited total fluctuation in an arbitrarily small neighbourhood to the 



\ ft / \ > t nen those canonical Sturm- Liouville series corresponding to f(s), 

 ivhose normal functions do not satisfy the boundary condition u = at , 



converge and have the sum *.\ _< at this point. 

 If /.) ( exists as a finite number, and 



J \ I 



has a Lebesgue 



t 



integral with respect to t in an interval (0, a), then those canonical Sturm- Liouville 

 series corresponding to f(s), whose normal functions do not satisfy the boundary 



condition u = at , converge and have the sum -y > _< at this point. 



20. We saw in 17 that all canonical Sturm- Liouville series corresponding to f(s) 

 will converge uniformly in (y, 8), if 



converges uniformly to zero in this interval. After what was said in 18, the reader 



2 B 2 



