§ 3. Some Applications of Cauchy’s Method
95
Formula
+∞
0
t
s−1 e(zt)dt = Γ (s)(−2πiz)
−s , Re(s) > 0 , Im(z) > 0 ,
which amounts to re-writing relations (10.2) to (10.4) differently, is another
example; here the function ˆ
f is t
s−1
+ .
As |e(tz)| = exp(−2πty) and
exp(y) ≤ exp(a
) + exp(b
) if a
≤ y ≤ b
.
If (1) converges absolutely for y = a
and y = b
> a
, it converges for
a
≤ y ≤ b
. The set of values of y such that
ˆ
f (t)
exp(−2πty)dt < +∞
(12.2)
is, therefore, an interval I = (a, b) of a priori arbitrary nature; f (z) is defined
on the horizontal strip B : y ∈ I. For any compact interval I
= [a
, b
] ⊂
I, the function being integrated is dominated by the integrable function
56
|f (t)|(e
−2πa
t + e
−2πb
t ) on the closed strip Im(z) ∈ I
since a
, b
∈ I. In
other words, integral (1) converges normally
57 on the closed strip B
: y ∈ I
.
As a result (theorem 9 of n
◦ 7),
(1) f is continuous and bounded on B
, and so is continuous but not necessarily bounded on B since I is the union of the intervals I
,
(2) f is holomorphic on the open strip y ∈]a
, b
[, hence on the interior
a < y < b of the strip B which is the union of these open strips,
(3) the derivatives of f can be computed by differentiating under the
sign.
To make sure that condition (2) is satisfied in a given open interval I =
]a, b[, it suffices to suppose that
a < y < b =⇒ sup
t∈ R
ˆ
f (t)
exp(−2πty) < +∞ .
(12.3)
Indeed, if this condition holds, for given y, a
and b
can be chosen so that
a < a
< y < b
< b. As (3) is satisfied for y = a
and y = b
, for large |t| ,
ˆ
f (t) = O [exp(2πa
t)] , ˆ
f (t) = O [exp(2πb
t)] ,
and so
56 Recall that“ integrable ” means “ absolutely integrable ”, except in very rare cases
when it is specified to be otherwise.
57 An integral
f (x, y)dμ(x), defined for y ∈ E, converges normally on A ⊂ E if
there is a μ-integrable positive function pA(x) such that |f (x, y)| ≤ pA(x) for
all y ∈ A and all x. This is clearly analogous to the normal convergence of a
sequence of functions.
95
Formula
+∞
0
t
s−1 e(zt)dt = Γ (s)(−2πiz)
−s , Re(s) > 0 , Im(z) > 0 ,
which amounts to re-writing relations (10.2) to (10.4) differently, is another
example; here the function ˆ
f is t
s−1
+ .
As |e(tz)| = exp(−2πty) and
exp(y) ≤ exp(a
) + exp(b
) if a
≤ y ≤ b
.
If (1) converges absolutely for y = a
and y = b
> a
, it converges for
a
≤ y ≤ b
. The set of values of y such that
ˆ
f (t)
exp(−2πty)dt < +∞
(12.2)
is, therefore, an interval I = (a, b) of a priori arbitrary nature; f (z) is defined
on the horizontal strip B : y ∈ I. For any compact interval I
= [a
, b
] ⊂
I, the function being integrated is dominated by the integrable function
56
|f (t)|(e
−2πa
t + e
−2πb
t ) on the closed strip Im(z) ∈ I
since a
, b
∈ I. In
other words, integral (1) converges normally
57 on the closed strip B
: y ∈ I
.
As a result (theorem 9 of n
◦ 7),
(1) f is continuous and bounded on B
, and so is continuous but not necessarily bounded on B since I is the union of the intervals I
,
(2) f is holomorphic on the open strip y ∈]a
, b
[, hence on the interior
a < y < b of the strip B which is the union of these open strips,
(3) the derivatives of f can be computed by differentiating under the
sign.
To make sure that condition (2) is satisfied in a given open interval I =
]a, b[, it suffices to suppose that
a < y < b =⇒ sup
t∈ R
ˆ
f (t)
exp(−2πty) < +∞ .
(12.3)
Indeed, if this condition holds, for given y, a
and b
can be chosen so that
a < a
< y < b
< b. As (3) is satisfied for y = a
and y = b
, for large |t| ,
ˆ
f (t) = O [exp(2πa
t)] , ˆ
f (t) = O [exp(2πb
t)] ,
and so
56 Recall that“ integrable ” means “ absolutely integrable ”, except in very rare cases
when it is specified to be otherwise.
57 An integral
f (x, y)dμ(x), defined for y ∈ E, converges normally on A ⊂ E if
there is a μ-integrable positive function pA(x) such that |f (x, y)| ≤ pA(x) for
all y ∈ A and all x. This is clearly analogous to the normal convergence of a
sequence of functions.