XII Elliptic Functions............................................
Our discussion of elliptic functions may be regarded as an essay in revisionism, since
we do not use Liouville’s theorem, Riemann surfaces or the Weierstrassian functions.
We wish to show that the methods used by the founding fathers of the subject provide
a natural and rigorous approach, which is very well suited for applications.
The work is arranged so that the initial sections are mutually independent, although
motivation for each section is provided by those which precede it. To some extent we
have also separated the discussion for realand for complex parameters, so that those
interested only in the real case may skip the complex one.
1 Elliptic Integrals............................................
After the development of the integral calculus in the second half of the 17th century,
it was natural to apply it to the determination of the arc length of an ellipse since, by
Kepler’s first law, the planets move in elliptical orbits with the sun at one focus.
An ellipse is described in rectangular coordinates by an equation
x^2 /a^2 +y^2 /b^2 = 1 ,whereaandbare thesemi-axesof the ellipse(a>b> 0 ). It is also given parametri-
cally by
x=asinθ,y=bcosθ( 0 ≤θ≤ 2 π).The arc lengths(Θ)fromθ=0toθ=Θis given by
s(Θ)=∫Θ
0[(dx/dθ)^2 +(dy/dθ)^2 ]^1 /^2 dθ=
∫Θ
0(a^2 cos^2 θ+b^2 sin^2 θ)^1 /^2 dθ=
∫Θ
0[a^2 −(a^2 −b^2 )sin^2 θ]^1 /^2 dθ.W.A. Coppel, Number Theory: An Introduction to Mathematics, Universitext,
DOI: 10.1007/978-0-387-89486-7_12, © Springer Science + Business Media, LLC 2009
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