§3. First concepts of analytic functions
181
contains a disc with centre 0 and radius r > 0 and that Pi is contained in
a disc with centre 0 and radius R > O. The distance to the origin of any
point u of Pi is thus between r and R. By homothety, the distance to the
origin of any point of Pn lies between nr and nR, whence nr < Iwl < nR
for every period w E Ln. In (2), the partial sum S(Ln) of the moduli thus
lies between 8n/(nR)k and 8n/(nr)k, i.e. is of the same order of magnitude
(::=::) as l/n k - i . The condition of convergence that we seek is thus k > 2, i.e.
k ? 3 since k is an integer.
It happens that the elliptic function which "controls" all the others corresponds to k = 2: the fk are, up to numerical factors, its derivatives in the
sense of nO 19, as is "obvious" from the formulae, though one does not differentiate an infinite sum without taking precautions (but see Chap. VII for
series of analytic functions). One proves this by modifying (1) so as to make
the series converge, as was done in nO 21 to make the series with general term
l/(x - n) converge.
For this, one remarks that, for Iwl > lui,
(23.3)
1
(u - w)2
1
1
w2 . (1 - u/w)2
1
2' (1 + 2u/w + 3u 2 /w 2 + 4u 3 /w 3 + ... )
w
by (19.12). The quasi-geometric series (3) shows that, for Iwllarge, the general term of (1) is approximately equal to l/w k , the reason why (1) does not
converge for k = 2. The solution now lies in diminishing the order of magnitude of the general term by subtracting l/w 2 from it, for w -I- 0, in other
words we introduce the famous function
(23.4)
p(u) = l/u 2 + L [1/(u - w)2 - 1/w2]
w#O
of Weierstrass (it already appeared in Eisenstein), with a p which smacks
of the gothic, of the italic and of the cursive, chosen by the inventor65 and
retained by posterity. Formula (3) above shows that, for lu/wl < 1/2 for
example, i.e. for "almost all" the w E L, the set of periods, the general term
of (4) is of the same order of magnitude as l/w 3 , whence convergence ..
More precisely, let us work as above in a disc lui < R and, to decide on the
convergence of the series, eliminate from the series the finite number of terms
for which Iwl ~ R. For all u such that lui < R, there then exists a number
q < 1 such that one has lu/wl < q in the retained terms; the difference
(23.5)
w- 2 [2u/w + 3(U/w)2 + ... ] =
Lw- 2 (n + l)(u/w)n
n
65 His biography in the DSB tells us that in the course of his fourteen years of
high-school teaching he had to teach mathematics, physics, German, botany,
geography, history, gymnastics "and even calligraphy" .
181
contains a disc with centre 0 and radius r > 0 and that Pi is contained in
a disc with centre 0 and radius R > O. The distance to the origin of any
point u of Pi is thus between r and R. By homothety, the distance to the
origin of any point of Pn lies between nr and nR, whence nr < Iwl < nR
for every period w E Ln. In (2), the partial sum S(Ln) of the moduli thus
lies between 8n/(nR)k and 8n/(nr)k, i.e. is of the same order of magnitude
(::=::) as l/n k - i . The condition of convergence that we seek is thus k > 2, i.e.
k ? 3 since k is an integer.
It happens that the elliptic function which "controls" all the others corresponds to k = 2: the fk are, up to numerical factors, its derivatives in the
sense of nO 19, as is "obvious" from the formulae, though one does not differentiate an infinite sum without taking precautions (but see Chap. VII for
series of analytic functions). One proves this by modifying (1) so as to make
the series converge, as was done in nO 21 to make the series with general term
l/(x - n) converge.
For this, one remarks that, for Iwl > lui,
(23.3)
1
(u - w)2
1
1
w2 . (1 - u/w)2
1
2' (1 + 2u/w + 3u 2 /w 2 + 4u 3 /w 3 + ... )
w
by (19.12). The quasi-geometric series (3) shows that, for Iwllarge, the general term of (1) is approximately equal to l/w k , the reason why (1) does not
converge for k = 2. The solution now lies in diminishing the order of magnitude of the general term by subtracting l/w 2 from it, for w -I- 0, in other
words we introduce the famous function
(23.4)
p(u) = l/u 2 + L [1/(u - w)2 - 1/w2]
w#O
of Weierstrass (it already appeared in Eisenstein), with a p which smacks
of the gothic, of the italic and of the cursive, chosen by the inventor65 and
retained by posterity. Formula (3) above shows that, for lu/wl < 1/2 for
example, i.e. for "almost all" the w E L, the set of periods, the general term
of (4) is of the same order of magnitude as l/w 3 , whence convergence ..
More precisely, let us work as above in a disc lui < R and, to decide on the
convergence of the series, eliminate from the series the finite number of terms
for which Iwl ~ R. For all u such that lui < R, there then exists a number
q < 1 such that one has lu/wl < q in the retained terms; the difference
(23.5)
w- 2 [2u/w + 3(U/w)2 + ... ] =
Lw- 2 (n + l)(u/w)n
n
65 His biography in the DSB tells us that in the course of his fourteen years of
high-school teaching he had to teach mathematics, physics, German, botany,
geography, history, gymnastics "and even calligraphy" .