354
IV - Powers, Exponentials, Logarithms, Trigonometric Functions
be a polynomial in two 15 variables with complex coefficients. Suppose that
P(x, y) = 0 for all x, yEN. Then aij = 0 for all i and j, and therefore
P(x,y) = 0 for all X,y E Co
Grouping together the terms of P containing the same power of y, we have
P(x, y) = L Pj(x)yj with polynomials Pj(x) in x. Give x a value n E Nand
consider P(n, y) = L Pj(n)yj. By hypothesis, this polynomial in y vanishes
for yEN; so it has infinitely many roots. Now one of the more elementary
results in the theory of algebraic equations in one unknown is that an equation
of degree::; 152 possesses no more than 152 roots, except of course if all its
coefficients are zero. So Pj(n) = 0 for all n E N, and this, for the same reason,
proves that all the coefficients of all the Pj ' Le. those of P, are zero, qed.
The formula (7), which we now can write
(11.10)
is what will establish the theorem for z and s real. First, for z given, the
function s H Ns(z) satisfies the addition formula for the exponential functions and, for sand z real, has real values. Since N 1 (z) = 1 + z, to establish
the relation
(11.11)
Ns(z) = (1 + z)S for - 1 < z < 1, s E JR,
it is enough (nO 6, Theorem 2) to show that, for z E] - 1, 1[ given, the left
hand side is a continuous function of s on lR. We shall show that this is the
case on JR, and even for complex z, Izl < 1.
The terms of Newton's series are indeed continuous functions of s. It thus
is enough to show that it converges normally on any disc lsi::; R (Chap. III,
nO 8, Theorem 9). But on such a disc
(11.12)
Is(s - 1) ... (s - n + l)zn In!1 ::;
::; R(R + 1) ... (R + n + l)izl n In! = Vn ,
the general term of a series with positive terms independent of s; the latter
converges for I z I < 1 since
Vn+1IVn = (R + n + 2)lzI/(n + 1)
tends to Izl as n increases; whence normal convergence and (11), qed.
Third proof [general case; uses (10)]. To establish the formula
(11.13)
exp[sL(z)] = L en(s)zn
for sand z complex, Izl < 1, we shall first show that, for zED given, the two
sides are holomorphic functions of s on C, i.e. that, considered as functions
15 The case of any number of variables is treated in the same way.
IV - Powers, Exponentials, Logarithms, Trigonometric Functions
be a polynomial in two 15 variables with complex coefficients. Suppose that
P(x, y) = 0 for all x, yEN. Then aij = 0 for all i and j, and therefore
P(x,y) = 0 for all X,y E Co
Grouping together the terms of P containing the same power of y, we have
P(x, y) = L Pj(x)yj with polynomials Pj(x) in x. Give x a value n E Nand
consider P(n, y) = L Pj(n)yj. By hypothesis, this polynomial in y vanishes
for yEN; so it has infinitely many roots. Now one of the more elementary
results in the theory of algebraic equations in one unknown is that an equation
of degree::; 152 possesses no more than 152 roots, except of course if all its
coefficients are zero. So Pj(n) = 0 for all n E N, and this, for the same reason,
proves that all the coefficients of all the Pj ' Le. those of P, are zero, qed.
The formula (7), which we now can write
(11.10)
is what will establish the theorem for z and s real. First, for z given, the
function s H Ns(z) satisfies the addition formula for the exponential functions and, for sand z real, has real values. Since N 1 (z) = 1 + z, to establish
the relation
(11.11)
Ns(z) = (1 + z)S for - 1 < z < 1, s E JR,
it is enough (nO 6, Theorem 2) to show that, for z E] - 1, 1[ given, the left
hand side is a continuous function of s on lR. We shall show that this is the
case on JR, and even for complex z, Izl < 1.
The terms of Newton's series are indeed continuous functions of s. It thus
is enough to show that it converges normally on any disc lsi::; R (Chap. III,
nO 8, Theorem 9). But on such a disc
(11.12)
Is(s - 1) ... (s - n + l)zn In!1 ::;
::; R(R + 1) ... (R + n + l)izl n In! = Vn ,
the general term of a series with positive terms independent of s; the latter
converges for I z I < 1 since
Vn+1IVn = (R + n + 2)lzI/(n + 1)
tends to Izl as n increases; whence normal convergence and (11), qed.
Third proof [general case; uses (10)]. To establish the formula
(11.13)
exp[sL(z)] = L en(s)zn
for sand z complex, Izl < 1, we shall first show that, for zED given, the two
sides are holomorphic functions of s on C, i.e. that, considered as functions
15 The case of any number of variables is treated in the same way.