BOUNDARY VALUE OF INFINITE SERIES 197 



f^ / 2c \sm{2n + l)t v , 



as is well kno'wi]. 



The transformation z = tt — t makes the second integral of (19) be- 

 come 



Jo •'V 2m+1 J smt 



which becomes, when {2n + l)i = y, equal to 



y 



Cfi . 2(c-j/) \ 2n+l sin» 

 /sm~--^ 



on applyuig the first law of the mean this becomes 



'1 ,- / 2 (c — i')\ r° sin w , r> ^ ^ ^ 



r 

 . f 



2n+l 



When n = oo the value of this is 



'sin t 



dt. 

 t 



Hence the value of the Fourier series at a; = tt — is »Soo equal to 

 i|/(+,r-0)+/(-x+0)| + {/( + ^-0)-/(-,r-0)|ip-^-rfi. (21) 



We find approximatelj^ 



f^sin t , ir.nr> "" 



-^f/«= 1.1/96 -• 



Jo f ^ 



Therefore at a; = -r — the function S has a shear whose lower end is 

 half the smn, and whose upper end is this half sum plus half of 1.1796 times 

 the difference of/ (x) and/ (— tt). 



In the same way we find the value of the series when x converges to 

 — TT + to be )S„ equal to 



