82
VIII – Cauchy Theory
This is an integrable function since a > 0 and Re(s) > 0, qed.
(1) can now be justified. Since, by (3) and (4),
ˆ
ϕ(y) = Γ (s)w
−s = Γ (s) [2πi(y − a)]
−s for Re(s) > 0 ,
this function is of the order of magnitude of y
−s at infinity. It is, therefore,
integrable over R if Re(s) > 1, and as this is also the case of
ϕ(x) = e(ax)x
s−1
+
which, moreover, is then continuous everywhere, including at x = 0, Fourier’s
inversion formula shows that
Γ (s)
[2πi(y − a)]
−s e(xy)dy = x
s−1
+ e(ax)
(10.7)
for Re(s) > 1 and Im(a) > 0, provided we set | Arg(w)| < π/2. Choosing
−π < Arg(y − a) < 0 and Arg(2πi) = π/2 ,
Arg(w) = Arg [2πi(y − a)] = Arg(2πi) + Arg(y − a), and so
[2πi(y − a)]
−s = (2πi)
−s (y − a)
−s = (y − a)
−s /(2πi)
s
and (7) can be written
(y − a)
−s e(xy)dy =
(2πi)
s
Γ (s)
x
s−1
+ e(ax) for Im(a) > 0 , Re(s) > 1 ;
(10.8)
this is the formula generalizing (8.15). Similarly, we get
(y + a)
−s e(−xy)dy =
(−2πi)
s
Γ (s)
x
s−1
+ e(ax) for Im(a) > 0 ,
(10.8’)
where we have to take 0 < Arg(y + a) < π and Arg(−2πi) = −π/2.
Replacing a by z and applying Poisson’s summation formula (Chap. VII,
n
◦ 23), we get
1
(z + n) s =
(−2πi)
s
Γ (s)
n≥1
n
s−1 exp(2πinz) for Im(z) > 0
(10.9)
and Re(s) > 1, a result generalizing (8.16). The reader should not forget to
check the assumptions allowing the application of the Possion formula to a
function f : the latter is continuous and the series
f (x + n) and
ˆ
f (y + n)
converge normally on any compact set.