170
II - Convergence: Discrete variables
But there is much more. Put g(z) = liz, an analytic function on the open
subset z f:. 0 of C as we have known since nO 6. One has l/J(z) = g[J(z)].
We are thus led to ask much more generally if one again obtains an analytic
function on composing two analytic functions, for example cos(sinz) where,
on C, one defines sin z and cos z by the power series which have already
appeared several times in this chapter. Here again the response is affirmative
and the proof, once again, uses the associativity theorem, but in a case where
the set ofindices is essentially more complicated than N or NxN. This is really
the only difficulty, the calculations being, as one says in English, pedestrian.
Theorem 17. Let U and V be two open subsets of C, let J be an analytic
function defined on U with values in V, and 9 an analytic function defined
on V. Then the composed function h(z) = g[f(z)] is analytic in U, and
h'(z) = g'[J(z)].f'(z).
Consider an a E U and the point b = f(a) of V. On a neighbourhood of b
the function g(z) is a power series in z - b = Y, so that h(z) is obtained by
substituting J(z) for z in this; one thus obtains a power series in f(z) - b =
J(z) - J(a). Since J(z) is, in its turn, a power series in z - a = X whose
constant term is J(a), the difference J(z) - J(a) is a power series without
constant term in X.
The situation is thus the following: we have two convergent power series
(Le. converging at points other than the origin) in variables X and Y, the
first in X without constant term, and we wish to show that on substituting
the first for the variable Y in the second, manipulating crudely, and then
grouping together the terms containing the same powers of X, we will again
find a convergent power series. We may also assume that the second series
has zero constant term, since this poses no problem.
So - we return to the traditional notation z for the variable - let
(22.6)
be the given series, and let us first calculate formally.
We have
(22.7)
Now the formula for multiplication of series, which clearly generalises to more
than two factors, shows that
(22.8)
where one sums over all families (nb.'" np) of p integers> 0 chosen arbitrarily (for p = 2, these are ordered pairs in the sense of set theory): to
multiply p sums one by the other, one chooses arbitrarily a term in each
II - Convergence: Discrete variables
But there is much more. Put g(z) = liz, an analytic function on the open
subset z f:. 0 of C as we have known since nO 6. One has l/J(z) = g[J(z)].
We are thus led to ask much more generally if one again obtains an analytic
function on composing two analytic functions, for example cos(sinz) where,
on C, one defines sin z and cos z by the power series which have already
appeared several times in this chapter. Here again the response is affirmative
and the proof, once again, uses the associativity theorem, but in a case where
the set ofindices is essentially more complicated than N or NxN. This is really
the only difficulty, the calculations being, as one says in English, pedestrian.
Theorem 17. Let U and V be two open subsets of C, let J be an analytic
function defined on U with values in V, and 9 an analytic function defined
on V. Then the composed function h(z) = g[f(z)] is analytic in U, and
h'(z) = g'[J(z)].f'(z).
Consider an a E U and the point b = f(a) of V. On a neighbourhood of b
the function g(z) is a power series in z - b = Y, so that h(z) is obtained by
substituting J(z) for z in this; one thus obtains a power series in f(z) - b =
J(z) - J(a). Since J(z) is, in its turn, a power series in z - a = X whose
constant term is J(a), the difference J(z) - J(a) is a power series without
constant term in X.
The situation is thus the following: we have two convergent power series
(Le. converging at points other than the origin) in variables X and Y, the
first in X without constant term, and we wish to show that on substituting
the first for the variable Y in the second, manipulating crudely, and then
grouping together the terms containing the same powers of X, we will again
find a convergent power series. We may also assume that the second series
has zero constant term, since this poses no problem.
So - we return to the traditional notation z for the variable - let
(22.6)
be the given series, and let us first calculate formally.
We have
(22.7)
Now the formula for multiplication of series, which clearly generalises to more
than two factors, shows that
(22.8)
where one sums over all families (nb.'" np) of p integers> 0 chosen arbitrarily (for p = 2, these are ordered pairs in the sense of set theory): to
multiply p sums one by the other, one chooses arbitrarily a term in each