§2. Series expansions
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of the real and imaginary parts of s, they are C 1 and satisfy the Cauchy
condition Dd = -iD2/ (Chap. II, n° 19; see also nOs 19 to 22 of Chap. III).
This is obvious for the left hand side
f(s) = exp[sL(z)] = L L(z)ns[nJ
since it is analytic on C. The case of the right hand side is a little less easy16.
We showed in Chap. III, nO 22, as a consequence of Theorem 23 on termby-term differentiation of a series of C 1 functions (in the real sense) on an
open subset U of C, that if these functions are holomorphic and if the series of
derivatives in the complex sense converges uniformly (for example, normally)
on any compact K C U, then the sum of the series is again holomorphic, and
its derivative is obtained by differentiating the series term-by-term. Although
we then declared this result to be of "no interest", it will serve us here. The
reader is asked to consider the following arguments as a simple exercise since
we have already obtained the result above in a much easier way.
It is clear that the complex derivative of the function Cn (s) =
s(s - 1) ... (s - n + l)/n! is
c~(s) = LS(s -1) ... (s - p)' ... (s - n + l)/n!.
For lsi ~ R, one thus has
ic~(s)1 ~ L R(R + 1) ... 1 ... (R + n + l)/n!;
since 1 ~ R + P for all p 2:: 0 so long as R 2:: 1, and since the sum considered
has n terms, we have
Ic~(s)1 ~ nR(R + 1) ... (R + n + l)/n!.
When we differentiate the series
term-by-term with respect to s we find a series whose general term is majorised in modulus on the disc lsi ~ R by
16 We have seen above that, for z given, Newton's series converges normally on
every disc 181 ~ R. Since its terms are analytic functions of 8 - polynomials -,
the general theorems of Chap. VII show, if we deploy them, that the sum of the
series is analytic on C. As for proving ''without knowing anything" that Newton's
series is an everywhere convergent power series in 8, this would demand intricate
explicit calculations. Note also that, for the rest of the proof, one need only know
that the power series of /(8) is, like every power series, holomorphic on its disc
of convergence; Theorem 23 of Chap. III, nO 22 on sequences of C 1 functions
would allow one instead to give a direct proof, since on differentiating a power
series in 8 with respect to the real or imaginary coordinates of 8 one finds the
derived power series, up to a factor equal to 1 or to i; it is then enough to know
that the latter has the same radius of convergence as the given series.
355
of the real and imaginary parts of s, they are C 1 and satisfy the Cauchy
condition Dd = -iD2/ (Chap. II, n° 19; see also nOs 19 to 22 of Chap. III).
This is obvious for the left hand side
f(s) = exp[sL(z)] = L L(z)ns[nJ
since it is analytic on C. The case of the right hand side is a little less easy16.
We showed in Chap. III, nO 22, as a consequence of Theorem 23 on termby-term differentiation of a series of C 1 functions (in the real sense) on an
open subset U of C, that if these functions are holomorphic and if the series of
derivatives in the complex sense converges uniformly (for example, normally)
on any compact K C U, then the sum of the series is again holomorphic, and
its derivative is obtained by differentiating the series term-by-term. Although
we then declared this result to be of "no interest", it will serve us here. The
reader is asked to consider the following arguments as a simple exercise since
we have already obtained the result above in a much easier way.
It is clear that the complex derivative of the function Cn (s) =
s(s - 1) ... (s - n + l)/n! is
c~(s) = LS(s -1) ... (s - p)' ... (s - n + l)/n!.
For lsi ~ R, one thus has
ic~(s)1 ~ L R(R + 1) ... 1 ... (R + n + l)/n!;
since 1 ~ R + P for all p 2:: 0 so long as R 2:: 1, and since the sum considered
has n terms, we have
Ic~(s)1 ~ nR(R + 1) ... (R + n + l)/n!.
When we differentiate the series
term-by-term with respect to s we find a series whose general term is majorised in modulus on the disc lsi ~ R by
16 We have seen above that, for z given, Newton's series converges normally on
every disc 181 ~ R. Since its terms are analytic functions of 8 - polynomials -,
the general theorems of Chap. VII show, if we deploy them, that the sum of the
series is analytic on C. As for proving ''without knowing anything" that Newton's
series is an everywhere convergent power series in 8, this would demand intricate
explicit calculations. Note also that, for the rest of the proof, one need only know
that the power series of /(8) is, like every power series, holomorphic on its disc
of convergence; Theorem 23 of Chap. III, nO 22 on sequences of C 1 functions
would allow one instead to give a direct proof, since on differentiating a power
series in 8 with respect to the real or imaginary coordinates of 8 one finds the
derived power series, up to a factor equal to 1 or to i; it is then enough to know
that the latter has the same radius of convergence as the given series.