Number Theory: An Introduction to Mathematics

506 XII Elliptic Functions In this case ψ(t 1 )=q 1 ±p 1 [(p 12 −a 12 )(p 12 −b^21 )]^1 /^2 /(p^21 −t 12 ), where p 1 =( 1 / 2 ) ...

2 The Arithmetic-Geometric Mean 507 Using (1)–(3), complete elliptic integrals of all three kinds can be calculated by theAGMalg ...

508 XII Elliptic Functions It follows that πK(a,c)/ 2 K(a,b)=log( 4 a 1 /c 0 )− ∑∞ n= 1 2 −nlog(an/an+ 1 ). (4) Finally, to dete ...

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 ...

510 XII Elliptic Functions Matematicheof Count Fagnano, a copy of which had been sent them by the author. The papers which arous ...

3 Elliptic Functions 511 we obtain 2 x 1 x 2 (x 1 x 2 ′+x′ 1 x 2 )′−(x 1 x′ 2 +x 1 ′x 2 )^2 − 2 x 1 x 2 x 1 ′x 2 ′ =x^21 { 2 x 2 ...

512 XII Elliptic Functions since S(u)^2 S′(v)^2 −S′(u)^2 S(v)^2 =S(u)^2 gλ[S(v)]−S(v)^2 gλ[S(u)] = 4 S(u)S(v)[S(u)−S(v)][1−λS(u) ...

3 Elliptic Functions 513 then (dz/dt)^2 =( 1 + √ λ)^2 { 4 λ 0 z^3 − 4 ( 1 +λ 0 )z^2 + 4 z}, where λ 0 = 4 √ λ/( 1 + √ λ)^2. (19) ...

514 XII Elliptic Functions it follows thatT≤K(λ),where K(λ):= ∫ 1 0 dx/gλ(x)^1 /^2. HenceS′(t)vanishes for sometsuch that 0<t ...

3 Elliptic Functions 515 u(z)=φ(z)^1 /^2 , where the square root is chosen so thatRu(z)>0. ThenS[u(z)]=z. Differentiating and ...

516 XII Elliptic Functions where α′={γ′+(γ′^2 +δ′^2 )^1 /^2 }^1 /^2 / √ 2 , 2 α′β′=δ′. Thusα′andβ′are positive for 0<y<1, ...

4 Theta Functions 517 as the standard elliptic integral of the second kind, and Π(u,a):=(λ/ 2 ) ∫u 0 S′(a)S(v)dv/[1−λS(a)S(v)] ( ...

518 XII Elliptic Functions whereq,z∈Candz=0. Both series on the right converge if|q|<1, both diverge if|q|>1, and at most ...

4 Theta Functions 519 Moreover, there exists a constantA>0, depending onqbut not onnorN, such that |cnN|≤A|q|n 2 . For we can ...

520 XII Elliptic Functions One important property of the theta function is almost already known to us: Proposition 3Fo r a l lv∈ ...

4 Theta Functions 521 Since the zeros ofθ(v;τ)are the pointsv= 1 / 2 +τ/ 2 +mτ+n, the zeros ofθα,β(v) are the points v=(β+ 1 )/ ...

522 XII Elliptic Functions By differentiating with respect tovand then puttingv =0, we obtain in addition θ 11 ′( 0 )= 2 πiq^1 / ...

4 Theta Functions 523 Proof From the definition ofθ 00 , θ 00 (v;τ)θ 00 (w;τ)= ∑ j,k eπiτ(j (^2) +k (^2) ) e^2 πivje^2 πiwk= ∑ j ...

524 XII Elliptic Functions Proof Consider the second relation. If we use the first and fourth relations of Propo- sition 4 to ev ...

5 Jacobian Elliptic Functions 525 Proposition 8Fo r a l lv∈Candτ∈H, {θ 00 (v)/θ 01 (v)}′=πiθ 102 ( 0 )θ 10 (v)θ 11 (v)/θ 012 (v) ...