492 17. REAL AND COMPLEX ANALYSISLegendre, Jacobi, and Abel. The foundation for further work in integration was laid
by Legendre, who invented the term elliptic integral. Off and on for some 40 years
between 1788 and 1828, he thought about integrals like those of Fagnano and Euler,
classified them, computed their values, and studied their properties. He found their
algebraic addition formulas and thereby reduced the division problem for these
integrals to the solution of algebraic equations. Interestingly, he found that whereas
the division problem requires solving an equation of degree ç for the circle, it
requires solving an equation of degree n^2 for the ellipse. After publishing his results
as exercises in integral calculus in 1811, he wrote a comprehensive treatise in the
1820s. As he was finishing the third volume of this treatise he discovered a new set of
transformations of elliptic integrals that made their computation easier. (He already
knew one set of such transformations.) Just after the treatise appeared in 1827,
he found to his astonishment that Jacobi had discovered the same transformations,
along with others, and had connected them with the division problem. Jacobi's
results in turn were partially duplicated by those of Abel.
Abel, who admired Gauss, was proud of having achieved the division of the
lemniscate into 17 equal parts,^2 just as Gauss had done for the circle. The secret
for the circle was to use the algebraic addition formula for trigonometric functions.
For the lemniscate, as Legendre had shown, the equation was of higher degree. Abel
was able to solve it by using complex variables, and in the process, he discovered
that the inverse functions of the elliptic integrals, when regarded as functions of
a complex variable, were doubly periodic. The double period accounted for the
fact that the division equation was of degree n^2 rather than n. Without complex
variables, the theory of elliptic integrals would have been a disconnected collection
of particular results. With them, a great simplicity and unity was achieved. Abel
went on to study algebraic addition formulas for very general integrals of the type
J R(x,y{x))dx,where R(x, y) is a rational function of ÷ and y and y(x) satisfies a polynomial
equation P(x,y(x)) = 0. Such integrals are now called Abelian integrals in his
honor. In particular, he established that for each polynomial P(x,y) there was
a number p, now called the genus of P(x, y), such that a sum of any number of
integrals R(x, y) with different limits could be expressed in terms of just ñ integrals,
whose limits of integration were rational functions of those in the given sum. For
elliptic integrals, ñ = 1, and that is the content of the algebraic addition formulas
discovered by Legendre. For a more complicated integral, say
/
.^ dx,
V<z(z)
where q(x) is a polynomial of degree 5 or higher, the genus may be higher. If
P(x, y) = y^2 — q(x), where the polynomial q is of degree 2p + 1 or 2p + 2, the genus
is p.
After Abel's premature death, Jacobi continued to develop algebraic function
theory. In 1832, he realized that for algebraic integrals of higher genus, the inverse
functions could not be well defined, since there were ñ integrals and only one equa-
tion connecting them to the variable in terms of which they were to be expressed.
He therefore had the idea of adjoining extra equations in order to get well-behaved(^2) Or, more generally, a Fermat prime number of parts.