102
VIII – Cauchy Theory
on the functions f (z). The results of section (iii) apply. On the other hand,
(13) and (16) show that, for any y ∈ I and any r ∈ N,
ˆ
f (t) = O
|t|
−r exp(2πty)
as |t| −→ +∞ ;
(12.21)
this relation seemingly better than (3) suffices to ensure convergence of (18)
for y > 0. Relation (15) also shows that
ˆ
f (t)
exp(−2πty) ≤
|f (x + iy)| dx
and so
+∞
0
ˆ
f (t)
exp(−2πty)ρ(y)dy ≤
P
|f (x + iy)| ρ(y)dxdy < +∞
for all t. The function ˆ
f is, therefore, zero for the values of t for which the
integral
exp(−2πty)ρ(y)dy is divergent. For example, if
ρ(y) = y
k−2
with k real but not necessarily an integer as will now be supposed, which is
an important case in the theory of automorphic functions, then
ˆ
f (t) = 0 =⇒
+∞
0
exp(−2πty)y
k−2 dy < +∞ .
(12.22)
The convergence of the integral requires k > 1 and t > 0, which shows in
particular that, for k ≤ 1, the only holomorphic solution of (20) is f (z) = 0.
There still remains to be shown that non-zero solutions of (20) effectively
exist for k > 1. To obtain a holomorphic function on P , let us try
g(z) = (z − ¯
w)
−p
(12.23)
with Im(w) > 0. Setting w = u + iv, the change of variable x = u + ξ(y + v)
shows that
|g(x + iy)| dx =
(x − u)
2 + (y + v)
2
−p/2 dx =
=
(y + v)
1−p
ξ
2 + 1
−p/2 dξ .
This result is finite if p > 1. It remains to check that the integral
+∞
0
(y + v)
1−p y
k−2 dy , v > 0 ,
converges. This suppose that at y = 0, k > 1 and that at infinity, p > k,
which implies the condition p > 1 that has already been found. Hence (20)
has non-zero solutions for k > 1, but none for k ≤ 1. Summarizing: