Number Theory: An Introduction to Mathematics

3 Elliptic Functions 509

By symmetry,y 2 (λ)=K(λ′)is also a solution. It follows that the ‘Wronskian’

W=λλ′(y 2 dy 1 /dλ−y 1 dy 2 /dλ)

has derivative zero and so is constant. But, writing

K′(λ)=K( 1 −λ), E′(λ)=E( 1 −λ),

we have

2 W=K′(E−λ′K)+K(E′−λK′)=KE′+K′(E−K).

To evaluate this constant we letλ→0. Puttingx=sin^2 θ, we obtain

K(λ)=

∫π/ 2

0

( 1 −λsin^2 θ)−^1 /^2 dθ, E(λ)=

∫π/ 2

0

( 1 −λsin^2 θ)^1 /^2 dθ

and hence, asλ→0,

K(λ)→π/ 2 ,E(λ)→π/ 2 ,E(λ′)→ 1.

Moreover

K(λ′)[E(λ)−K(λ)]→ 0 ,

since

K(λ)−E(λ)=λ

∫ 1

0

x[4x( 1 −x)( 1 −λx)]−^1 /^2 dx=O(λ)

and

0 ≤K(λ′)≤

∫π/ 2

0

[1−( 1 −λ)]−^1 /^2 dθ=O(λ−^1 /^2 ).

It follows that 2W=π/2.

Ifλ= 1 /2, thenλ′=λand (5) takes the simple form

K( 1 / 2 )[2E( 1 / 2 )−K( 1 / 2 )]=π/ 2.

By the remarks preceding the statement of Proposition 1, the left side can be
evaluated by theAGMalgorithm. In this wayπhas recently been calculated to
millions of decimal places. (It will be recalled that the valueλ= 1 /2 occurred in
the rectification of the lemniscate.)

3 Elliptic Functions

According to Jacobi, the theory of elliptic functions was conceived on 23 December
1751, the day on which the Berlin Academy asked Euler to report on theProduzioni