Number Theory: An Introduction to Mathematics

486 XI Uniform Distribution and Ergodic Theory (ii) For any givenz∈ X, the closure of the set(Tnz)n≥ 1 is a nonempty closed in- ...

5 Recurrence 487 Suppose we have defined pointsx 1 ,...,xk, positive integersn 1 ,...,nk,and ε 1 ,...,εk∈( 0 ,ε 0 )such that, fo ...

488 XI Uniform Distribution and Ergodic Theory i.e.z 1 =zn+ 1 =z 2 n+ 1 =···=zpn+ 1 .Sincez∈X, there is a positive integermsuch ...

6 Further Remarks 489 The converse of Proposition 10 is proved by Kemperman [27]. For the history of the problem of mean motion, ...

490 XI Uniform Distribution and Ergodic Theory Applications of ergodic theory to classical mechanics are discussed in the books ...

7 Selected References 491 [14] I. Dupain and V.T. S ́os, On the discrepancy of (nα) sequences,Topics in classical number theory( ...

492 XI Uniform Distribution and Ergodic Theory [41] H. Poincar ́e, Sur la th ́eorie cin ́etique des gaz,Oeuvres, t. X, pp. 246–2 ...

XII Elliptic Functions............................................ Our discussion of elliptic functions may be regarded as an es ...

494 XII Elliptic Functions If we putb^2 =a^2 ( 1 −k^2 ),wherek( 0 <k< 1 )is theeccentricityof the ellipse, this takes the ...

1 Elliptic Integrals 495 An example is the determination of the arc length of alemniscate. This curve, which was studied by Jaco ...

496 XII Elliptic Functions where σ=α^2 cos^2 θ+β^2 sin^2 θ. If we now put u^2 =( 1 −r^2 )/( 1 −σr^2 ), thenr^2 =( 1 −u^2 )/( 1 − ...

1 Elliptic Integrals 497 In general, supposeg(x)=(x−α)h(x),wherehis a cubic. If h(x)=h 0 (x−α)^3 +h 1 (x−α)^2 +h 2 (x−α)+h 3 and ...

498 XII Elliptic Functions Consider now the integralJn(γ). In the same way as before, for any integerm≥1, d{(x−γ)−mw}/dx=−m(x−γ) ...

1 Elliptic Integrals 499 The range ofλcan be further restricted by linear changes of variables. The trans- formationy=( 1 −λx)/( ...

500 XII Elliptic Functions Ifgis a cubic or quartic with only real roots, this can be achieved by a linear frac- tional transfor ...

1 Elliptic Integrals 501 Suppose now thatgis a real cubic or quartic with a pair of conjugate complex roots. Then we can write g ...

502 XII Elliptic Functions If we puty={(x+d 1 )/(x+d 2 )}^2 ,then R(x)=R 1 (y)+R 2 (y)y^1 /^2 , where again the rational functio ...

2 The Arithmetic-Geometric Mean 503 Thusa 1 ,b 1 satisfy the same hypotheses asa,band the procedure can be repeated. If we defin ...

504 XII Elliptic Functions then tdt/dθ=(a^2 −b^2 )sinθcosθ=[(a^2 −t^2 )(t^2 −b^2 )]^1 /^2 and J= ∫π/ 2 0 φ((a^2 sin^2 θ+b^2 cos^ ...

2 The Arithmetic-Geometric Mean 505 and bn≤(an^2 sin^2 θ+b^2 ncos^2 θ)^1 /^2 ≤an. Consequently, by lettingn→∞we obtain K(a,b)=π/ ...