A treatise on the theory of Bessel functions, by G. N. Watson.

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INTEGRAL REPRESENTATIONS

163

Hence, for all * values of v,
(l1+, -1-) V-
V~ eizt (t2 - 1)- dt= 2i cos v7. r(v+)C (m+ ( + t +1)
JA m=o (2m)! r(v-j m +1)
= 2v+ i r () P (v + 4) cos ^r. J ().
Therefore, if v + I is not a positive integer,
(1) JI~ (z)- v (-)(~)/ (+,- - e' zt(t2- l)v- dt
and this is Hankel's generalisation of Poisson's integral.
Next let us consider the second type of contour. Take the contour to lie
wholly outside the circle I t= 1, and then (t2- 1)v- is expansible in a series
of descending powers of t, uniformly convergent on the contour; thus we have
(t2 _)"- = _ ( I2 + )t2v-1-2M
n=o m r (g -. )
and in the series the phase of t lies between - 3v1 and + vr.
Assumingt the permissibility of integrating term-by-term, we have
r(-l+, i+) o zV r Q -V + m)f(-i+ i+)
z eizt (t2 - 1)"- dt = t21-2-m eiZt dt.
J co m==0 Wlir(i-) J= 2
But
j~ ~(-~1~~, 1~~+~) r(0+)
t2v-1-2m eizt dt -(_)+I e-^vri Z2m-2vj (_ )2v-1-2m e-U d,
ci GJt oo exp ic
where a is the phase of z (between + -7r); and, by a well-known formula+, the
last integral equals - 27i/F (2m - 2v + 1).
Hence
z eizt ('2 )v di2
(-1+,+ e (t 1)-+ dt = o!r ( - v) r (2m- 2 + )
2v + 7ri e- vri (Q-)
r(- ( )
when we use the duplication formula~ to express F (2nm - 2v + 1) in terms of
'( - v + m) and r(- +r+ 1).
I If v - 1 is a negative integer, the simplest way of evaluating the integral is to calculate the
residue of the integrand at u= 1.
t To justify the term-by-term integration, observe that ( ' ) eZt dt I is convergent; let
its value be K. Since the expansion of (2 - 1)V- converges uniformly, it follows that, when we
are given a positive number e, we can find an integer Mo independent of t, such that the remainder
after M1 terms of the expansion does not exceed E/K in absolute value when 1M l o. We then
have at once
((-1+,, ^ _ i)'- _ r(i- +rm) [(-1+' +)e t2-l-2"~ d
I ____________ iI C et 2izt 2v-1-22
|(- + izt~t2 )-et- 1) -t t-, -nd~
D i 'M=o mi c-v) i
<e l f(;l+,l+) l eizt d =e,
and the required result follows from the definition of the sum of an infinite series.
+ Cf. Modern Analysis, ~ 12-22. ~ Cf. Modern Analysis, ~ 12-15.
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