184
II - Convergence: Discrete variables
with constants c( w). If one takes for w one of the periods of base WI, W2, it is
clear that W /2 is not a period, which legitimates the formula
p(w/2) = p( -w/2) + c(w).
Now the function p is clearly even. So c(w) = 0 in this case, qed.
Although we cannot justify it entirely here, we would not hide from the
reader the fundamental property of the function p: it satisfies the differential
equation
(23.11)
where a2 and a4 are the coefficients (8). The proof is very simple, up to a
"detail". By using the power series found above for p and p' = -213, one
finds, by a small calculation a la Newton that, in the difference between the
two sides of (11), the negative powers of u as well as the constant terms cancel. This means that this difference, which a priori is analytic in
But this difference is clearly a function that is also doubly periodic like p
and p' and it can have no more poles than those of p and p', Le. the periods;
if then 0 is not a pole, the other periods will not be either. In other words,
the difference is an elliptic function which is holomorphic or analytic in all
the "detail" - a general theorem of Joseph Liouville (Chap. VII) which says
that such a function is necessarily constant and so zero if its power series
around u = 0 has zero constant term.
Exercise. Apart from several negative powers of u, each of the two sides
of (11) is a power series in u 2 whose coefficients can in principle be calculated
from the coefficients an of p( u). On writing that these two power series are
identical, one obtains strange algebraic relations between the an, Le. between
the Eisenstein series G 2k (L). Perform, a la Newton, this calculation for the
coefficients of u 2 and of u 4 .
The equation (11) allows one to integrate the square roots of polynomials
of third degree if one knows that one can always choose the lattice of periods
so that the coefficients a2 and a4 take values given in advance 67 . Correct, but
very far from obvious (Chap. XII, nO 18).
Traditionally one writes (11) in the form
67 One has to exclude the case, which can be treated elementarily, where the polynomial 4X 3 - 20a2 X - 28a4 has a double root. For a polynomial of the form
X 3 + pX + q, this means 4p3 + 27q2 = 0 (write that it and its derivative have a
common root).
II - Convergence: Discrete variables
with constants c( w). If one takes for w one of the periods of base WI, W2, it is
clear that W /2 is not a period, which legitimates the formula
p(w/2) = p( -w/2) + c(w).
Now the function p is clearly even. So c(w) = 0 in this case, qed.
Although we cannot justify it entirely here, we would not hide from the
reader the fundamental property of the function p: it satisfies the differential
equation
(23.11)
where a2 and a4 are the coefficients (8). The proof is very simple, up to a
"detail". By using the power series found above for p and p' = -213, one
finds, by a small calculation a la Newton that, in the difference between the
two sides of (11), the negative powers of u as well as the constant terms cancel. This means that this difference, which a priori is analytic in
But this difference is clearly a function that is also doubly periodic like p
and p' and it can have no more poles than those of p and p', Le. the periods;
if then 0 is not a pole, the other periods will not be either. In other words,
the difference is an elliptic function which is holomorphic or analytic in all
the "detail" - a general theorem of Joseph Liouville (Chap. VII) which says
that such a function is necessarily constant and so zero if its power series
around u = 0 has zero constant term.
Exercise. Apart from several negative powers of u, each of the two sides
of (11) is a power series in u 2 whose coefficients can in principle be calculated
from the coefficients an of p( u). On writing that these two power series are
identical, one obtains strange algebraic relations between the an, Le. between
the Eisenstein series G 2k (L). Perform, a la Newton, this calculation for the
coefficients of u 2 and of u 4 .
The equation (11) allows one to integrate the square roots of polynomials
of third degree if one knows that one can always choose the lattice of periods
so that the coefficients a2 and a4 take values given in advance 67 . Correct, but
very far from obvious (Chap. XII, nO 18).
Traditionally one writes (11) in the form
67 One has to exclude the case, which can be treated elementarily, where the polynomial 4X 3 - 20a2 X - 28a4 has a double root. For a polynomial of the form
X 3 + pX + q, this means 4p3 + 27q2 = 0 (write that it and its derivative have a
common root).