62
VIII – Cauchy Theory
Third proof. Let us directly show that g(z) has a power series expansion
on the interior of any compact disc D ⊂ U . This reduces to the case where D
is the disc |z| ≤ R. As z → f (t, z) is holomorphic and hence analytic, there
is a Taylor series expansion on D
f (t, z) =
f
(n) (t, 0)z
[n]
(7.6)
with z
[n] = z
n /n! and
f
(n) (t, 0)R
[n] =
f (t, Re(u)) e(−nu)du
(7.7)
for t ∈ I (Fourier series. . . ). Since, according to assumption (c), |f (t, z)| ≤
p D (t) for t ∈ I and z ∈ D, it follows that
f
(n) (t, 0)
≤ p D (t)/R
[n]
(7.8)
for all n and all t ∈ I. For |z| = qR with q < 1 and all r ∈ N, the Taylor
series
f
(n+r) (t, 0)z
[n] = f
(r) (t, z)
(7.9)
is, therefore, dominated for all t ∈ I, z ∈ D and p ∈ N by the series
p D (t)q
[n] R
n /R
[n+r] = p D (t)R
−r
q
n (n + r)!/n! ,
(7.10)
which converges since (n + r)!/n! n
r for large n. If K ⊂ I is compact, the
continuous function f (t, z) is bounded on K ×D and p D (t) can be replaced in
(10) by a constant independent of (t, z) ∈ K ×D, so that the Taylor series (9)
converges normally on K ×D. As a result, the function f
(r) (t, z) is continuous
on K × D for any K, hence on I × D, and so on I × U since the argument
can be applied to any compact disc D ⊂ U .
As p D (t) is also a factor in (10),
n≥0
f
(n+r) (t, 0)z
[n]
≤ M r p D (t) ,
(7.12)
where M r is a constant. Series (9) can, therefore (Chap. V, § 7, Theorem 20
generalized to an arbitrary measure), be integrated term by term on I, and
so
f
(r) (t, z)dμ(t) =
a n+r z
[n] o` u a n =
f
(n) (t, 0)dμ(t) .
For r = 0, this shows that g(z) =
a n z
[n] , and, therefore, that g has a power
series expansion and that, moreover,
f
(r) (t, z)dμ(t) =
a n+r z
[n] = g
(r) (z) .
This is relation (2). That f
(r) (t, z) satisfies condition (c), a fact of local
nature, follows immediately from (12).
VIII – Cauchy Theory
Third proof. Let us directly show that g(z) has a power series expansion
on the interior of any compact disc D ⊂ U . This reduces to the case where D
is the disc |z| ≤ R. As z → f (t, z) is holomorphic and hence analytic, there
is a Taylor series expansion on D
f (t, z) =
f
(n) (t, 0)z
[n]
(7.6)
with z
[n] = z
n /n! and
f
(n) (t, 0)R
[n] =
f (t, Re(u)) e(−nu)du
(7.7)
for t ∈ I (Fourier series. . . ). Since, according to assumption (c), |f (t, z)| ≤
p D (t) for t ∈ I and z ∈ D, it follows that
f
(n) (t, 0)
≤ p D (t)/R
[n]
(7.8)
for all n and all t ∈ I. For |z| = qR with q < 1 and all r ∈ N, the Taylor
series
f
(n+r) (t, 0)z
[n] = f
(r) (t, z)
(7.9)
is, therefore, dominated for all t ∈ I, z ∈ D and p ∈ N by the series
p D (t)q
[n] R
n /R
[n+r] = p D (t)R
−r
q
n (n + r)!/n! ,
(7.10)
which converges since (n + r)!/n! n
r for large n. If K ⊂ I is compact, the
continuous function f (t, z) is bounded on K ×D and p D (t) can be replaced in
(10) by a constant independent of (t, z) ∈ K ×D, so that the Taylor series (9)
converges normally on K ×D. As a result, the function f
(r) (t, z) is continuous
on K × D for any K, hence on I × D, and so on I × U since the argument
can be applied to any compact disc D ⊂ U .
As p D (t) is also a factor in (10),
n≥0
f
(n+r) (t, 0)z
[n]
≤ M r p D (t) ,
(7.12)
where M r is a constant. Series (9) can, therefore (Chap. V, § 7, Theorem 20
generalized to an arbitrary measure), be integrated term by term on I, and
so
f
(r) (t, z)dμ(t) =
a n+r z
[n] o` u a n =
f
(n) (t, 0)dμ(t) .
For r = 0, this shows that g(z) =
a n z
[n] , and, therefore, that g has a power
series expansion and that, moreover,
f
(r) (t, z)dμ(t) =
a n+r z
[n] = g
(r) (z) .
This is relation (2). That f
(r) (t, z) satisfies condition (c), a fact of local
nature, follows immediately from (12).