388
IV - Powers, Exponentials, Logarithms, Trigonometric Functions
(15.7)
x = y + b2y2 + ... = g(y),
Iyl < R,
(a case to which one reduces immediately if b1 -I 0), there exists one and
only one convergent power series
(15.8)
y = x + a2x 2 + ... = f(x),
Ixl < R',
such that g[f(x)] = x, J[g(y)] = y. In terms of analytic functions: (7) represents an analytic function in a disc Iyl < R, with g(O) = 0 and g'(O) -I 0,
and we have to prove the existence of one and only one analytic function f
on a neighbourhood of 0, such that f(O) = 0 and which, for Ixl sufficiently
small, is the inverse map of g. There are analogous theorems in the differential calculus of several real variables as we have seen in Chap. III, nO 24, and
they immediately set the problem into the framework of the theory of holomorphic functions, so also analytic if we know that the words "holomorphic"
and "analytic" are synonymous.
But if one looks for a direct proof, the situation becomes complicated.
One first adopts the point of view of formal series of Chap. II, nO 22: on
substituting (8) into (7), and expanding the powers of y and identifying the
result with x, one obtains algebraic relations between the an which enable
one, theoretically, to calculate them one-by-one; this is clearly what Newton
did for the first terms. This done, which is quite easy, the crucial problem
remains: to show that if the coefficients of (7) are O(qn) for a number q> 0
[a necessary and sufficient condition for the radius of convergence of (7) to
be > 0], one similarly has bn = O(qln) for some q' > O. This demands
non-obvious estimates of bn as a function of ap , estimates analogous to, and
more difficult than, those we used in Chap. II, nO 22, Theorem 17 Ii propos
compositions of analytic functions. Neither Dieudonne, nor Remmert, serious
people, have dared to expound the subject in their books; but see Serge Lang,
Complex Analysis (Springer, 1999), Chap. II.
16 - Hyperbolic functions
·When we have a reasonably civilised curve in the plane ]R2 it is generally
possible to find an interval I c ]R and two continuous real functions x(t) and
yet) on I so that, as t traverses I, the point (x(t), yet)) traverses all the curve,
maybe several times (or even infinitely many times in the case of Peano); in
other words: the function t ~ (x(t), yet)) maps I onto the set of points of
the curve. This is called a pammetric representation of the given curve. The
level of civilisation of the curve will be higher as one can further require the
functions x and y to have continuous derivatives. Peano's curve which passes
through all the points of a square is at level zero.
The trigonometric or circular functions allow one to parametrise a circumference: as t E ]R varies, the point (cos t, sin t) describes the circle x 2 + y2 = 1.
If a and b are real nonzero constants, the point (acost,bsint) describes the
IV - Powers, Exponentials, Logarithms, Trigonometric Functions
(15.7)
x = y + b2y2 + ... = g(y),
Iyl < R,
(a case to which one reduces immediately if b1 -I 0), there exists one and
only one convergent power series
(15.8)
y = x + a2x 2 + ... = f(x),
Ixl < R',
such that g[f(x)] = x, J[g(y)] = y. In terms of analytic functions: (7) represents an analytic function in a disc Iyl < R, with g(O) = 0 and g'(O) -I 0,
and we have to prove the existence of one and only one analytic function f
on a neighbourhood of 0, such that f(O) = 0 and which, for Ixl sufficiently
small, is the inverse map of g. There are analogous theorems in the differential calculus of several real variables as we have seen in Chap. III, nO 24, and
they immediately set the problem into the framework of the theory of holomorphic functions, so also analytic if we know that the words "holomorphic"
and "analytic" are synonymous.
But if one looks for a direct proof, the situation becomes complicated.
One first adopts the point of view of formal series of Chap. II, nO 22: on
substituting (8) into (7), and expanding the powers of y and identifying the
result with x, one obtains algebraic relations between the an which enable
one, theoretically, to calculate them one-by-one; this is clearly what Newton
did for the first terms. This done, which is quite easy, the crucial problem
remains: to show that if the coefficients of (7) are O(qn) for a number q> 0
[a necessary and sufficient condition for the radius of convergence of (7) to
be > 0], one similarly has bn = O(qln) for some q' > O. This demands
non-obvious estimates of bn as a function of ap , estimates analogous to, and
more difficult than, those we used in Chap. II, nO 22, Theorem 17 Ii propos
compositions of analytic functions. Neither Dieudonne, nor Remmert, serious
people, have dared to expound the subject in their books; but see Serge Lang,
Complex Analysis (Springer, 1999), Chap. II.
16 - Hyperbolic functions
·When we have a reasonably civilised curve in the plane ]R2 it is generally
possible to find an interval I c ]R and two continuous real functions x(t) and
yet) on I so that, as t traverses I, the point (x(t), yet)) traverses all the curve,
maybe several times (or even infinitely many times in the case of Peano); in
other words: the function t ~ (x(t), yet)) maps I onto the set of points of
the curve. This is called a pammetric representation of the given curve. The
level of civilisation of the curve will be higher as one can further require the
functions x and y to have continuous derivatives. Peano's curve which passes
through all the points of a square is at level zero.
The trigonometric or circular functions allow one to parametrise a circumference: as t E ]R varies, the point (cos t, sin t) describes the circle x 2 + y2 = 1.
If a and b are real nonzero constants, the point (acost,bsint) describes the