74
VIII – Cauchy Theory
implying the result once again. If, however, you refuse to use Dini’s theorem,
you can check it by an explicit calculation. For this observe that
sin u ≥ sin δ = α ,
is a strictly positive constant in [δ, π − δ], so that
exp(−ρ sin u) ≤ exp(−αρ)
and uniform convergence on the interval under consideration holds again
since exp(−αρ), which does not depend on u, approaches 0 as ρ increases
indefinitely.
In fact, a somewhat more precise result can be obtained. It is sometimes
called Jordan’s lemma by the author of a famous Analysis Course and professor at the ´
Ecole Polytechnique from 1876 to 1911 ; given there were about
two hundred students per year, his students must have provided many artillerymen. It suffices to study the integral
I(R) =
π
0
exp(−R sin u)du = 2
π/2
0
and to note that between 0 and π/2,u/2 ≤ sin u ≤ u.
I(R)
π/2
0
exp(−Ru) du 1/R .
immediately follows.
The previous arguments suppose that t = 0 and fall apart if t = 0. But
in this case, we are brought back to the computations of section (i) and for
example to formula (8), which shows that, one of the following holds:
ˆ
f (0) =
2πi
Im(a)> 0
Res(f, a) + πi Res(f, ∞) ,
−2πi
Im(a)< 0
Res(f, a) − πi Res(f, ∞)
(8.18)
where the residue at infinity is given by the relation
Res(f, ∞) = − lim zf (z) = −c .
In conclusion, (10’) and (10”) continue to hold for t = 0, the case t = 0 being
a consequence of (18). For example,
R
e(−tx)
x − a
dx =
0
i f t > 0 ,
πi
if t = 0 ,
2πie(−ta) if t < 0.
(Im(a) > 0)
(8.19)