352
IV - Powers, Exponentials, Logarithms, Trigonometric Functions
First varation. We look for the Maclaurin series of f at the origin. We have
(same reference)
J'(z) = exp'[sL(z)JsL'(z) = sf(z)L'(z).
Since L'(z) = 1 - z + z2 - ... = (1 + Z)-l on D it follows that
(11.4)
(1 + z)f'(z) = sf(z)
on D. Differentiating this relation n -1 times, using Leibniz' formula, we find
(1 + z)f(n)(z) + (n - l)f(n-l)(z) = sf(n-l)(z),
from which the numbers an = f(n)(o) satisfy an = (s - n + l)an-l. Since
ao = 1, one finds an = s(s-l) ... (s-n+1); the Maclaurin series L f(n) (O)z[n]
of f at the origin is thus Newton's series. Since f is analytic, it is represented
by its Maclaurin series on a neighbourhood of 0 (Chap. II, nO 19); the two
sides of (2) coinciding on a neighbourhood of 0, they are equal on all the disc
D where they are analytic (Chap. II, n° 20: analytic continuation), qed.
Second variation. Consider Newton's power series g(z). This also is an
analytic function on D. Its derivative is obtained by the usual procedure
[Chap. II, equ. (19.4)], whence
g'(z) = s + s(s - l)z + s(s - l)(s - 2)z[2] + ...
On multiplying by 1 + z, and taking account of the relation
between the binomial coefficients, we again obtain the differential equation
(11.4')
(1 + z)g'(z) = sg(z).
Since fez) never vanishes, being an exponential, the function
g(z)/ fez) = h(z) is analytic on D (Chap. II, nO 22) and its derivative can
be calculated by the standard formula. The relations (4) and (4') now show
that h'(z) = 0 and thus that all the successive derivatives of h are zero. Now
we showed in Chap. II, nO 20 (principle of analytic continuation) that, on an
open connected set, for example D, such a function is constant 14 - one even
needs much less (vanishing of successive derivatives at a single point). Since
h(O) = 1, we have fez) = g(z) on D, qed.
14 Instead of invoking the principle of analytic continuation, one might observe that
h, being analytic, is holomorphic and that the relation h' = 0 means that its first
partial derivatives with respect to the real coordinates x and y of z are zero; it
then remains to cite Chap. III, nO 21, equ. (21.11), which generalises the mean
value theorem.
IV - Powers, Exponentials, Logarithms, Trigonometric Functions
First varation. We look for the Maclaurin series of f at the origin. We have
(same reference)
J'(z) = exp'[sL(z)JsL'(z) = sf(z)L'(z).
Since L'(z) = 1 - z + z2 - ... = (1 + Z)-l on D it follows that
(11.4)
(1 + z)f'(z) = sf(z)
on D. Differentiating this relation n -1 times, using Leibniz' formula, we find
(1 + z)f(n)(z) + (n - l)f(n-l)(z) = sf(n-l)(z),
from which the numbers an = f(n)(o) satisfy an = (s - n + l)an-l. Since
ao = 1, one finds an = s(s-l) ... (s-n+1); the Maclaurin series L f(n) (O)z[n]
of f at the origin is thus Newton's series. Since f is analytic, it is represented
by its Maclaurin series on a neighbourhood of 0 (Chap. II, nO 19); the two
sides of (2) coinciding on a neighbourhood of 0, they are equal on all the disc
D where they are analytic (Chap. II, n° 20: analytic continuation), qed.
Second variation. Consider Newton's power series g(z). This also is an
analytic function on D. Its derivative is obtained by the usual procedure
[Chap. II, equ. (19.4)], whence
g'(z) = s + s(s - l)z + s(s - l)(s - 2)z[2] + ...
On multiplying by 1 + z, and taking account of the relation
between the binomial coefficients, we again obtain the differential equation
(11.4')
(1 + z)g'(z) = sg(z).
Since fez) never vanishes, being an exponential, the function
g(z)/ fez) = h(z) is analytic on D (Chap. II, nO 22) and its derivative can
be calculated by the standard formula. The relations (4) and (4') now show
that h'(z) = 0 and thus that all the successive derivatives of h are zero. Now
we showed in Chap. II, nO 20 (principle of analytic continuation) that, on an
open connected set, for example D, such a function is constant 14 - one even
needs much less (vanishing of successive derivatives at a single point). Since
h(O) = 1, we have fez) = g(z) on D, qed.
14 Instead of invoking the principle of analytic continuation, one might observe that
h, being analytic, is holomorphic and that the relation h' = 0 means that its first
partial derivatives with respect to the real coordinates x and y of z are zero; it
then remains to cite Chap. III, nO 21, equ. (21.11), which generalises the mean
value theorem.