§ 3. Some Applications of Cauchy’s Method
91
all neighbourhoods of 1, so that it does not contradict Dirichlet’s principle
for the unit disc. . .
For f integrable, apart from the function,
u f (x, y) = F
+ (z) − F
− (¯ z) ,
the function
v f (x, y) = −i
F
+ (z) + F
− (¯ z)
(11.11)
may be introduced; if f is real, clearly F
− (¯ z) = −F + (z), so that u f and v f
are, up to a factor of 2, the real and imaginary parts of F
+ (z) on the upper
half-plane. Changing the computations leading to (4) and (5),
v f (x, y) =
1
π
t − x
(t − x) 2 + y 2 f (t)dt =
1
π
t
t 2 + y 2 f (x + t)dt
(11.12)
= −i
e(xu)sgn(u) exp(−2πy|u|) ˆ
f (u)du
follows. Let us investigate the limit behaviour of this function when y > 0
approaches 0, but by supposing now that f ∈ F
1 (R). As done above for u f ,
it is possible to pass to the limit under the
sign in the Fourier integral, and
so
lim
y=0+
v f (x, y) = −i
e(xu)sgn(u) ˆ
f (u)du .
(11.13)
Let us now consider the first integral (12), which can also be written
πv f (0, y) =
t
t 2 + y 2 f (t)dt =
+∞
0
t
2
t 2 + y 2 g(t)dt/t
for x = 0, where g(t) = f (t) − f (−t). If f is differentiable at the origin, then
f (t) = f (0) + f
(0)t + o(t) , f(−t) = f (0) − f
(0)t + o(t)
and, therefore, g(t) = 2f
(0)t + o(t); hence the function g(t)/t is integrable
in the neighbourhood of 0, as well as at infinity like f . As y approaches
0, the function t
2 /(t
2 + y
2 ) approaches 1 while always remaining ≤ 1. The
elementary version of the dominated convergence theorem can, therefore, be
applied again, which shows that
lim πv f (0, y) =
t>0
g(t)dt/t = lim
r=0
|t|>r
f (t)dt/t .
Following traditions, set
p.v
+∞
−∞
f (t)dt/t = lim
r=0
|t|>r
f (t)dt/t .
(11.14)