§3. First concepts of analytic functions
173
Newton would have explained to you that, to do this without wasting
energy, you write
fez) = alZ + ... ,
g(Z) = bIZ + ...
where the unwritten terms are of degree> 1 in z. Then
h(z) = al(blz + ... ) + ...
with the same commentary. Thus h(z) = aIblz + ... , and since h'(O) must
be the coefficient of z, one obtains
h'(O) = /,(O)g'(O) = /,(O)g'[/(O)].
The general case where one works at a point a of U and the point b = f(a)
of V reduces immediately to this: on a neighbourhood of a, fez) - f(a) is a
series without constant term in X = z - a, the coefficient of X being /' (a)
by Taylor's formula; on a neighbourhood of b, g(z) is a series in Y = z - b,
the coefficient of Y being g'(b) = g'[f(a)]. In consequence, in the series in X
obtained in substituting the series in X which represents fez) - f(a) for Y,
the coefficient of X = z - a is the number f'(a)g'[f(a)], qed.
The inventor of power series, Newton, clearly knew all these results, more
precisely he used them constantly in his calculations as if they were self evident, and as if it was not necessary to worry about convergence and even
less so to formulate general theorems. He is, for example, capable of calculating the quotient of two power series by the method of division applicable to
decimal fractions 57 , x playing the role of 1/10 as we mentioned above in the
quotient 1/(1 + q). He is capable of inverting a power series 58 , i.e. deducing
from a relation
a relation
x = y + b2 y2 + b3y3 + ...
and, for example, of working out the exponential series by inverting the logarithmic series, or the series for the sine by inverting that for arcsin.
57 I am amazed that it has occurred to no one (if you except N. Mercator with his
quadmture of the hyperbola) to fit the doctrine recently established for decimal
numbers in similar fashion to variables, especially since the way is then open to
more striking consequences. First page of the English translation of the Newton's
manuscript mentioned below. For division, one has to widen the definition of
power series a little and allow a finite number of negative powers of z:
1/(z2 - z3) = z-2 + Z-l + 1 + z + ...
After all, conventional arithmetic does not confine itself to considering numbers
with zero integer part.
58 There is of course a general theorem, but it appeared two centuries later.
173
Newton would have explained to you that, to do this without wasting
energy, you write
fez) = alZ + ... ,
g(Z) = bIZ + ...
where the unwritten terms are of degree> 1 in z. Then
h(z) = al(blz + ... ) + ...
with the same commentary. Thus h(z) = aIblz + ... , and since h'(O) must
be the coefficient of z, one obtains
h'(O) = /,(O)g'(O) = /,(O)g'[/(O)].
The general case where one works at a point a of U and the point b = f(a)
of V reduces immediately to this: on a neighbourhood of a, fez) - f(a) is a
series without constant term in X = z - a, the coefficient of X being /' (a)
by Taylor's formula; on a neighbourhood of b, g(z) is a series in Y = z - b,
the coefficient of Y being g'(b) = g'[f(a)]. In consequence, in the series in X
obtained in substituting the series in X which represents fez) - f(a) for Y,
the coefficient of X = z - a is the number f'(a)g'[f(a)], qed.
The inventor of power series, Newton, clearly knew all these results, more
precisely he used them constantly in his calculations as if they were self evident, and as if it was not necessary to worry about convergence and even
less so to formulate general theorems. He is, for example, capable of calculating the quotient of two power series by the method of division applicable to
decimal fractions 57 , x playing the role of 1/10 as we mentioned above in the
quotient 1/(1 + q). He is capable of inverting a power series 58 , i.e. deducing
from a relation
a relation
x = y + b2 y2 + b3y3 + ...
and, for example, of working out the exponential series by inverting the logarithmic series, or the series for the sine by inverting that for arcsin.
57 I am amazed that it has occurred to no one (if you except N. Mercator with his
quadmture of the hyperbola) to fit the doctrine recently established for decimal
numbers in similar fashion to variables, especially since the way is then open to
more striking consequences. First page of the English translation of the Newton's
manuscript mentioned below. For division, one has to widen the definition of
power series a little and allow a finite number of negative powers of z:
1/(z2 - z3) = z-2 + Z-l + 1 + z + ...
After all, conventional arithmetic does not confine itself to considering numbers
with zero integer part.
58 There is of course a general theorem, but it appeared two centuries later.