u , PR. J. MERCEB: STDBM-UOUVOLK SERIES OP NORMAL 



M X tend, to - , wb. . is one of the end point, of (0, ,) ; let us suppose, in the 



L X that . : 0. From the results obtained in B 8, 9, we see that 



The right-hand member is 



_e^i_ rr e - > 



2 sinh /JTT L- 



where . is any positive number less than, or equal to, *. Proceeding as in 10, it 

 may he shown that 



C. 2 

 and that 



Pf P "**/()* ^^^ P e-*f(t)dt. 

 28inhp7rJo ** 



Hence we obtain the result 



From this it follows that 



and, in particular, that 



lim - 



^^.-oo 



whenever the right-hand member exists. 

 It may be shown in a similar way that 



/(ir-0) >:Tm7 -X (" K A (TT, t) f(t) dt > lim -X [' K A (ir, ) /() d 2: /(ir-0), 



A^.-o> Jo x >- * 



16. It will be observed that so far we have been concerned with the limit (32) for 

 a fixed value of ., and that, in consequence, the question of uniform convergence has 

 been left aside. We now proceed to prove that : 



//' the set of points at which f(s) is continuous includes a closed interval (y, 3) 

 lying wholly within (0, ir), then, as X tends to o>, 



converges uniformly to /(.<*) in this interval. 



* It is evident that the restriction a JT may be removed, if we adopt the convention of 10 in regard 

 to the definition of f(s) outside (0, *). 



