178
II - Convergence: Discrete variables
23 - The elliptic functions of Weierstrass
To end this chapter with a "bonus to the brave reader", we shall return to
the general associativity formula for unconditional or absolute convergence,
and illustrate it by a example which will not profit us for the moment but
which, in contrast to the traditional gymnastic exercises, has considerable
mathematical interest.
Series like L 1/ (m 2 + n 2 ) k/2 arise in the theory of numbers, and also
appear in a closely related form in the theory of elliptic functions of a complex variable, one of the great inventions of the XIXth century, more than
ever the order of the day in "pure" mathematics after its fusion with the
theory of algebraic numbers, algebraic geometry, etc. Its "users" have fur a
long time exploited the more manipulatory parts of the theory, principally to
calculate numerically the integrals of apparently very elementary functions
(square roots of a polynomial of degree 3 or more), yet whose primitives are
not known, more precisely are not expressible in terms of elementary functions. Besides, the historical origin of the elliptic functions 61 in the XVlIIth
century was as much in mathematics as in mechanics (oscillations of a simple
pendulum), and since many interesting applications were found for them, to
mechanics, to physics, etc., they were honoured in the courses of analysis
at the Ecole polytechnique during the second half of the XIX th century, a
little beyond the level of the majority of the future artillery men 62 ... The
current developments, much closer to algebraic geometry and to number theory than to analysis, are difficult of access for the majority of professional
mathematicians who are not specialists. They are very likely not within the
reach of users and, in any case, have long since passed very far beyond the
stage of numerical calculations in the usual sense of incorporating arithmetic
calculations that machines can perform.
61 See for example C. Houzel, Analyse mathematique (Foris, Belin, 1997), pp. 290302, or, more difficult, Henry McKean and Victor Moll, Elliptic Curves (CUP,
1997), Chap. 2.
62 Jacobi also taught them at Koenigsberg about 1830, Liouville at the College de
France in 1850, Weierstrass in Berlin before and after 1870, etc. But admission
to these courses was free and there was no final examination. Adolf Kneser,
who followed Weierstrass' course in Berlin in the 18808, spoke in 1925 of "two
hundred young people" who followed Weierstrass' course on the elliptic functions
from beginning to end "in full awareness of the fact that they would not appear at
this time in any state examination, resounding testimony to the scientific mind
of this time"; Remmert, Funktionentheorie 1, p. 336. There were no internal
examinations in the German Universities at this period; one prepared for the
State examinations leading to the "professions".
II - Convergence: Discrete variables
23 - The elliptic functions of Weierstrass
To end this chapter with a "bonus to the brave reader", we shall return to
the general associativity formula for unconditional or absolute convergence,
and illustrate it by a example which will not profit us for the moment but
which, in contrast to the traditional gymnastic exercises, has considerable
mathematical interest.
Series like L 1/ (m 2 + n 2 ) k/2 arise in the theory of numbers, and also
appear in a closely related form in the theory of elliptic functions of a complex variable, one of the great inventions of the XIXth century, more than
ever the order of the day in "pure" mathematics after its fusion with the
theory of algebraic numbers, algebraic geometry, etc. Its "users" have fur a
long time exploited the more manipulatory parts of the theory, principally to
calculate numerically the integrals of apparently very elementary functions
(square roots of a polynomial of degree 3 or more), yet whose primitives are
not known, more precisely are not expressible in terms of elementary functions. Besides, the historical origin of the elliptic functions 61 in the XVlIIth
century was as much in mathematics as in mechanics (oscillations of a simple
pendulum), and since many interesting applications were found for them, to
mechanics, to physics, etc., they were honoured in the courses of analysis
at the Ecole polytechnique during the second half of the XIX th century, a
little beyond the level of the majority of the future artillery men 62 ... The
current developments, much closer to algebraic geometry and to number theory than to analysis, are difficult of access for the majority of professional
mathematicians who are not specialists. They are very likely not within the
reach of users and, in any case, have long since passed very far beyond the
stage of numerical calculations in the usual sense of incorporating arithmetic
calculations that machines can perform.
61 See for example C. Houzel, Analyse mathematique (Foris, Belin, 1997), pp. 290302, or, more difficult, Henry McKean and Victor Moll, Elliptic Curves (CUP,
1997), Chap. 2.
62 Jacobi also taught them at Koenigsberg about 1830, Liouville at the College de
France in 1850, Weierstrass in Berlin before and after 1870, etc. But admission
to these courses was free and there was no final examination. Adolf Kneser,
who followed Weierstrass' course in Berlin in the 18808, spoke in 1925 of "two
hundred young people" who followed Weierstrass' course on the elliptic functions
from beginning to end "in full awareness of the fact that they would not appear at
this time in any state examination, resounding testimony to the scientific mind
of this time"; Remmert, Funktionentheorie 1, p. 336. There were no internal
examinations in the German Universities at this period; one prepared for the
State examinations leading to the "professions".