368
IV - Powers, Exponentials, Logarithms, Trigonometric Functions
He then explains, by the example y3 - 2y - 5 = 0, his method (inspired
by Viete and Oughtred) for finding the roots of an equation. Here there is
a visible change of sign, and so a root of the function, between 2 and 2, l.
Newton put y = 2 + p, whence p3 + 6p2 + lOp - 1 = 0; since the square and
the cube are small relative to p, we have, approximately, p = 1/10. Now one
puts p = 1/10 + q, whence a new equation q3 + 6.3q2 + 1l.23q + 0.061 = 0,
or almost 11.23q + 0.061 = 0, whence q = -0.0054 + r, with a new equation
for r, etc. In the case of the relation y3 + a 2 y - 2a 3 + axy - x 3 = 0 he
explains the method which we mentioned in Chap. II, nO 22, and, from the
expansion found, deduces a formula for calculating the area bounded by the
curve and the verticals 0 and x, namely ax - x 2 /8 + x 3 /192a + ... , "an
expansion which approximates the truth more rapidly as x becomes smaller' .
He shows how by inverting the series for 10g(1 + x) one can calculate the
abscissa corresponding to a given value y of the area, and so finds in passing
the first terms of the exponential series, though without realising, and for a
good reason, that he had just discovered what would be the most important
function in analysis up to the present day. He obtained the length of the arc
of the semi-circle x 2 + y2 = x, Y > 0, contained between the point (1,0) and
the point of ordinate y (it is clearly ! arcsin y since the radius of the circle
is !) by first calculating the derivative of the arc with respect to y (by the
geometric infinitesimal argument that had become standard) and expanding
that by his binomial formula; by "integrating" he obtained the power series
for arcsin y which, later, he inverted to find that of sin x and, by the same
type of argument, of cos x.
To follow the history would take us too far; everything that he expounded
in the manuscript of 1669 is taken up again and developed in the De Methodis
Serierum et Fluxionum of 1670-1671 where, this time, he explains in detail
the role of "fluents" and "fluxions" in his calculus, as we have indicated in
Chap. III, nO 14. And he had other occasions to explain himself during his
correspondence with Leibniz in 1676-1677.
13 - The exponential function as a limit
We saw in n° 10 that
expx = lim(l + x/n)n = lim(l + hX)l/h
for x real. The dominated convergence theorem enables one to go further by
imitating the calculation which led us to the series for 10g(1 + x):
Theorem 11. Let (zp) be a sequence of complex numbers tending to a
limit z. Then
00
(13.1)
lim (1 + zp/p)P = L zn In! = exp(z),
n=O
and in particular
IV - Powers, Exponentials, Logarithms, Trigonometric Functions
He then explains, by the example y3 - 2y - 5 = 0, his method (inspired
by Viete and Oughtred) for finding the roots of an equation. Here there is
a visible change of sign, and so a root of the function, between 2 and 2, l.
Newton put y = 2 + p, whence p3 + 6p2 + lOp - 1 = 0; since the square and
the cube are small relative to p, we have, approximately, p = 1/10. Now one
puts p = 1/10 + q, whence a new equation q3 + 6.3q2 + 1l.23q + 0.061 = 0,
or almost 11.23q + 0.061 = 0, whence q = -0.0054 + r, with a new equation
for r, etc. In the case of the relation y3 + a 2 y - 2a 3 + axy - x 3 = 0 he
explains the method which we mentioned in Chap. II, nO 22, and, from the
expansion found, deduces a formula for calculating the area bounded by the
curve and the verticals 0 and x, namely ax - x 2 /8 + x 3 /192a + ... , "an
expansion which approximates the truth more rapidly as x becomes smaller' .
He shows how by inverting the series for 10g(1 + x) one can calculate the
abscissa corresponding to a given value y of the area, and so finds in passing
the first terms of the exponential series, though without realising, and for a
good reason, that he had just discovered what would be the most important
function in analysis up to the present day. He obtained the length of the arc
of the semi-circle x 2 + y2 = x, Y > 0, contained between the point (1,0) and
the point of ordinate y (it is clearly ! arcsin y since the radius of the circle
is !) by first calculating the derivative of the arc with respect to y (by the
geometric infinitesimal argument that had become standard) and expanding
that by his binomial formula; by "integrating" he obtained the power series
for arcsin y which, later, he inverted to find that of sin x and, by the same
type of argument, of cos x.
To follow the history would take us too far; everything that he expounded
in the manuscript of 1669 is taken up again and developed in the De Methodis
Serierum et Fluxionum of 1670-1671 where, this time, he explains in detail
the role of "fluents" and "fluxions" in his calculus, as we have indicated in
Chap. III, nO 14. And he had other occasions to explain himself during his
correspondence with Leibniz in 1676-1677.
13 - The exponential function as a limit
We saw in n° 10 that
expx = lim(l + x/n)n = lim(l + hX)l/h
for x real. The dominated convergence theorem enables one to go further by
imitating the calculation which led us to the series for 10g(1 + x):
Theorem 11. Let (zp) be a sequence of complex numbers tending to a
limit z. Then
00
(13.1)
lim (1 + zp/p)P = L zn In! = exp(z),
n=O
and in particular