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IV - Powers, Exponentials, Logarithms, Trigonometric Functions
nR(R + 1) ... (R + n + 1)lzln In! = W n.
Since wn+dwn = (R + n + 2) Izl/n tends to Izl < 1, the series converges,
hence, on differentiating Newton's series 9 term-by-term with respect to 8,
we find a series normally convergent on all compact subsets of C.
It follows that 9 is a holomorphic function of 8 for all zED and that one
can calculate g'(8) by differentiating the series term-by-term.
Let us calculate this derivative. We know by (10) that g(8 + t) = g(8)g(t)
for 8, t E C; differentiating this formula with respect to t E C for 8 given, we
find g'(8 + t) = g(8)g'(t) and in particular
g'(8) = g(8)g'(0) = g(8) L c~(O)zn.
Since Cn(8) = 8(8 - 1) ... (8 - n + 1)/n! is a polynomial, c~(O) is the coefficient of 8 in its expansion, i.e. the term independent of 8 in the polynomial
en(8)/8 = (8 -1) ... (8 - n + 1)/n!, i.e. the value of this polynomial for 8 = 0,
whence
c~(O) = (_I)n-1(n -1)!/n! = (_I)n-1/n .
So we find, for Izl < 1,
an anticipated miracle from which it follows that g'(8) = L(Z)g(8). But the
function 1(8) = exp[8L(z)] also satisfies 1'(8) = L(z)/(8). Since 1 never
vanishes, we deduce that the holomorphic function g( 8) /1 (8) (Chap. III,
end of n° 20) has an identically zero derivative on C. It is therefore constant (Chap. III, nO 21, a result valid, on an open connected subset of ]R2,
for every function C 1 whose two first partial derivatives are zero). Since
1(0) = g(O) = 1 we conclude that 1 = g, which accomplishes, in every sense
of the term, a proof that might be considered too baroque, but which nevertheless yields the general result with the minimum of means: the theorems
of Chap. III, § 5 on functions of two real variables.
Let us now give some examples of applications of Newton's formula.
If S E Z the formula is also valid for any z E C if 8 > 0 (algebraic
binomial formula). For s a negative integer and z complex, Izl < 1, the
relation Ns(z)N_s(z) = 1 shows that Ns(z) = 1/(I+z)-S = (l+z)S, whence
the formula again. For z not real and s = p/q rational but not an integer, it
is more prudent to write only that Npjq(z)q = (1 + z)P, given that a complex
number has, as we shall see in nO 14, several qth roots of which, a priori,
none is more "natural" that the others, in contrast to what happens for a
real positive number.
IV - Powers, Exponentials, Logarithms, Trigonometric Functions
nR(R + 1) ... (R + n + 1)lzln In! = W n.
Since wn+dwn = (R + n + 2) Izl/n tends to Izl < 1, the series converges,
hence, on differentiating Newton's series 9 term-by-term with respect to 8,
we find a series normally convergent on all compact subsets of C.
It follows that 9 is a holomorphic function of 8 for all zED and that one
can calculate g'(8) by differentiating the series term-by-term.
Let us calculate this derivative. We know by (10) that g(8 + t) = g(8)g(t)
for 8, t E C; differentiating this formula with respect to t E C for 8 given, we
find g'(8 + t) = g(8)g'(t) and in particular
g'(8) = g(8)g'(0) = g(8) L c~(O)zn.
Since Cn(8) = 8(8 - 1) ... (8 - n + 1)/n! is a polynomial, c~(O) is the coefficient of 8 in its expansion, i.e. the term independent of 8 in the polynomial
en(8)/8 = (8 -1) ... (8 - n + 1)/n!, i.e. the value of this polynomial for 8 = 0,
whence
c~(O) = (_I)n-1(n -1)!/n! = (_I)n-1/n .
So we find, for Izl < 1,
an anticipated miracle from which it follows that g'(8) = L(Z)g(8). But the
function 1(8) = exp[8L(z)] also satisfies 1'(8) = L(z)/(8). Since 1 never
vanishes, we deduce that the holomorphic function g( 8) /1 (8) (Chap. III,
end of n° 20) has an identically zero derivative on C. It is therefore constant (Chap. III, nO 21, a result valid, on an open connected subset of ]R2,
for every function C 1 whose two first partial derivatives are zero). Since
1(0) = g(O) = 1 we conclude that 1 = g, which accomplishes, in every sense
of the term, a proof that might be considered too baroque, but which nevertheless yields the general result with the minimum of means: the theorems
of Chap. III, § 5 on functions of two real variables.
Let us now give some examples of applications of Newton's formula.
If S E Z the formula is also valid for any z E C if 8 > 0 (algebraic
binomial formula). For s a negative integer and z complex, Izl < 1, the
relation Ns(z)N_s(z) = 1 shows that Ns(z) = 1/(I+z)-S = (l+z)S, whence
the formula again. For z not real and s = p/q rational but not an integer, it
is more prudent to write only that Npjq(z)q = (1 + z)P, given that a complex
number has, as we shall see in nO 14, several qth roots of which, a priori,
none is more "natural" that the others, in contrast to what happens for a
real positive number.