§ 3. Some Applications of Cauchy’s Method
63
§ 3. Some Applications of Cauchy’s Method
In this § aimed at showing that even a limited knowledge of Cauchy theory allows us to do mathematics that does not merely amount to irrelevant
exercises, the following notation will be systematically used:
L
1 (R) will denote the set of functions defined and absolutely integrable
over R with respect to the usual measure dx; the reader is free to interpret
this notation in the sense of Lebesgue theory. In fact, as is done in Lebesgue
theory, we will often say “ integrable ” instead of “ absolutely integrable ”,
even if this means warning the reader when we will encounter semi-convergent
integrals (Chapter V, § 7);
F
1 (R) will denote the set of continuous functions f
41 on R such that both
f ∈ L
1 (R) and ˆ
f ∈ L
1 (R) hold; Fourier’s inversion formula applies to these
functions (Chapter VII, § 6, n
◦ 30, Theorem 26).
Recall that, for us, the Fourier transform is defined by the formula
ˆ
f (y) =
f (x)e(−xy)dx ,
where integration is over R and where, for every z ∈ C, as in Chapter VII,
e(z) = exp(2πiz) .
So e(−x) = e(x) for x ∈ R and
|e(z)| = exp(−2πy) , y = Im(z)
for all z ∈ C.
Expressions such as the following will be frequently used:
f (z) = O (g(z)) at infinity on U ,
where U is a subset of C; this means (Chapter II, n
◦ 3) that there exists
M > 0 and R > 0 such that
z ∈ U & |z| ≥ R =⇒ |f (z)| ≤ M |g(z)| .
Similar conventions apply to relations o, and ∼. We will also write
f (z) g(z) on U
if there are constants m, M ≥ 0 such that
m |g(z)| ≤ |f (z)| ≤ M |g(z)| for all z ∈ U
41 A somewhat superfluous condition: in Lebesgue theory, any f ∈ L
1 (R) whose
Fourier transform is integrable is shown to be equal “ almost everywhere ” to a
continuous function given by the inversion formula.