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72
Thus
G. H. HARDY:
T (s) sin i sn (i - 2~1-s) £ (i + s).
Writing s instead of i -j- s, we obtain the ordinary form of
the functional equation, viz.
cos i sn T
(3)
2. It remains to justity the inversion of summation and
integration implied in (2). The series (i) is boundedly
convergent ; that is to say, there is a constant K such that
V7 si
\]x
2m
for all values of x and n\ and xs~l> where o< J< I, is
inte-grable over any finite range (ö, X). We have therefore
(4)
Also
= Urn
/e^’
\
J
lim
= lim
(5)
provided only that the last series is convergent and
Hm
= ö.
(6)
But
, - ~
(2m -\-
––– \ ^s- i sjn /2^ -I-
2m + i JY v
Ys-1cos(2m+i)Y+(s~i)\
j
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