336
IV - Powers, Exponentials, Logarithms, Trigonometric Functions
These formulae allow us to understand why Cauchy's function 5
(5.19)
f(x) = exp(-1/x 2 ), x:l 0, f(O) = 0
is indefinitely differentiable on JR, including at x = 0, that all its derivatives
vanish at the origin, and therefore that it cannot be represented by Maclaurin's formula. The Chain Rule and the relation exp' = exp, obvious from the
power series as we have already said several times, show first, that for x :I 0,
the function f has successive derivatives given by
J'(x) = 2f(x)/x 3 , !"(x) = 4f(x)/x 6 - 6f(x)/x 4 , •..
so of the form f(k)(x) = Pk(l/x) exp( -1/x 2 ) where Pk is a polynomial.
Putting 1/ x = y we get
when x tends to 0, y2 tends to +00 and since every power, integer or not,
of y is negligible with respect to exp(y2) by (16), we see that each derivative
f(k)(x) tends to 0 with x. To show that f is Coo even on a neighbourhood
of 0, it thus is enough to show directly that the successive derivatives of f all
vanish at O.
First, f'(O) = limf(x)/x = limexp( -1/x 2 )/x = 0 as we have just seen.
Thus the first derivative f' exists for all x and is continuous, including at o.
The second derivative, if it exists, is the limit of the ratio f'(x)/x = 2f(x)/x 4 ,
so we find a zero second derivative, and so on indefinitely.
The situation will be the same for the function
(5.20)
g(x) = exp( -1/x 2 ) for x > 0, = 0 for x :::; o.
This time the derivatives are all zero for x :::; o. If all COO functions were
analytic, aeroplanes could never take off; they would roll eternally on their
take-off strips, assumed rigorously flat, and, in flight, would be incapable
of changing direction or altitude, since if a function is real-analytic on an
interval I and constant on a neighbourhood of a point t of I, it is constant
on I. It is the existence of functions such as (20) which allows one to join Coo
functions (for example constants) defined on closed disjoint intervals perfectly
smoothly. This remark will play an important role in Chap. V, nO 29, when
we shall prove the existence of Coo functions having arbitrarily prescribed
successive derivatives at a given point.
6 - Characterisations of the exponential, power and logarithmic
functions
For any a > 0, the exponential function f(x) = aX satisfies the identity
5 Here we use the fact, proved later, that the function exp(x) = 1 + x + ... is a
particular function eXPa(x), with a = e = exp(l) > 1. The following argument
can be applied equally to every other exponential function of base a > 1.
IV - Powers, Exponentials, Logarithms, Trigonometric Functions
These formulae allow us to understand why Cauchy's function 5
(5.19)
f(x) = exp(-1/x 2 ), x:l 0, f(O) = 0
is indefinitely differentiable on JR, including at x = 0, that all its derivatives
vanish at the origin, and therefore that it cannot be represented by Maclaurin's formula. The Chain Rule and the relation exp' = exp, obvious from the
power series as we have already said several times, show first, that for x :I 0,
the function f has successive derivatives given by
J'(x) = 2f(x)/x 3 , !"(x) = 4f(x)/x 6 - 6f(x)/x 4 , •..
so of the form f(k)(x) = Pk(l/x) exp( -1/x 2 ) where Pk is a polynomial.
Putting 1/ x = y we get
when x tends to 0, y2 tends to +00 and since every power, integer or not,
of y is negligible with respect to exp(y2) by (16), we see that each derivative
f(k)(x) tends to 0 with x. To show that f is Coo even on a neighbourhood
of 0, it thus is enough to show directly that the successive derivatives of f all
vanish at O.
First, f'(O) = limf(x)/x = limexp( -1/x 2 )/x = 0 as we have just seen.
Thus the first derivative f' exists for all x and is continuous, including at o.
The second derivative, if it exists, is the limit of the ratio f'(x)/x = 2f(x)/x 4 ,
so we find a zero second derivative, and so on indefinitely.
The situation will be the same for the function
(5.20)
g(x) = exp( -1/x 2 ) for x > 0, = 0 for x :::; o.
This time the derivatives are all zero for x :::; o. If all COO functions were
analytic, aeroplanes could never take off; they would roll eternally on their
take-off strips, assumed rigorously flat, and, in flight, would be incapable
of changing direction or altitude, since if a function is real-analytic on an
interval I and constant on a neighbourhood of a point t of I, it is constant
on I. It is the existence of functions such as (20) which allows one to join Coo
functions (for example constants) defined on closed disjoint intervals perfectly
smoothly. This remark will play an important role in Chap. V, nO 29, when
we shall prove the existence of Coo functions having arbitrarily prescribed
successive derivatives at a given point.
6 - Characterisations of the exponential, power and logarithmic
functions
For any a > 0, the exponential function f(x) = aX satisfies the identity
5 Here we use the fact, proved later, that the function exp(x) = 1 + x + ... is a
particular function eXPa(x), with a = e = exp(l) > 1. The following argument
can be applied equally to every other exponential function of base a > 1.