94
VIII – Cauchy Theory
Differentiate (17) with respect to x. For holomorphic functions, differentiating
with respect to x is the same as differentiating with respect to z. Thus
F
1 (z) + F
1 (z) =
d
dx
u f (x, y) =
1
2πi
d
dx
1
t − z
−
1
t − ¯
z
f (t)dt
(11.18)
=
1
2πi
1
(t − z) 2 −
1
(t − z) 2
f (t)dt = G(z) + G(z)
provided – an easy application of Theorem 9 of n
◦ 7 – differentiation under
the
sign with respect to x in (7) is justified. As holomorphic functions F
1
and G have the same real parts,
G(z) = F
1 (z) + ia ,
where a is a real constant; the function
F
+ (z) = F 1 (z) + iaz
is, therefore, a primitive for G on Im(z) > 0. Since a is real,
F
+ (z) + F + (z) = u f (x, y) − 2ay .
Denoting by F (z) the function equal to F
+ (z) for Im(z) > 0 and to −F + (z)
for Im(z) < 0, the following result is finally obtained:
Theorem 11. For any continuous and bounded function f on R, there is a
function F defined and holomorphic on C − R such that
lim
y=0+
[F (x + iy) − F (x − iy)] = f (x) for all x ∈ R .
12 – The Complex Fourier Transform
(i) Generalities. Given a function on R that will be denoted ˆ
f (t) for reasons
specified later, the function
f (z) =
e(tz) ˆ
f (t)dt =
e(tx) exp(−2πty) ˆ
f (t)dt
(12.1)
is called the (inverse. . . ) complex Fourier transform of ˆ
f . This assumes that
the integral converges absolutely for a non-empty set of values of z, which for
example excludes the function exp(πt
2 ). It also excludes functions for which
(1) only converges for real z, for example rational functions, since in this
context, f (z) is expected to be defined and holomorphic on an open subset
of C.
Exercise 1. Show that for ˆ
f (t) = exp(−πt
2 ) the integral converges for all
z and is equal to exp(−πz
2 ).