Number Theory: An Introduction to Mathematics

526 XII Elliptic Functions 1 −λ(τ)=θ 014 ( 0 ;τ)/θ 004 ( 0 ;τ), (49) and from (39)–(40) that cn^2 u= 1 −sn^2 u, dn^2 u= 1 −λsn^2 ...

5 Jacobian Elliptic Functions 527 K=K(τ):=πθ 002 ( 0 ;τ)/ 2 , K′=K′(τ):=τK(τ)/i, (51) then we have Poles ofsnu,cnu,dnu: u= 2 mK+ ...

528 XII Elliptic Functions The addition formulas show that the evaluation of the Jacobian elliptic functions for arbitrary compl ...

5 Jacobian Elliptic Functions 529 Proposition 10Fo r a l l u∈Candτ∈H, sn(u;− 1 /τ)=−isn(iu;τ)/cn(iu;τ), cn(u;− 1 /τ)= 1 /cn(iu;τ ...

530 XII Elliptic Functions Hence, by (49), u′′=[1+( 1 −λ(τ))^1 /^2 ]u. By Proposition 5 also, sn(u′′; 2 τ)=−iθ 00 ( 0 ; 2 τ)θ 10 ...

6 The Modular Function 531 6 The Modular Function The function λ(τ):=θ 104 ( 0 ;τ)/θ 004 ( 0 ;τ), which was introduced in§5, is ...

532 XII Elliptic Functions where the bar denotes complex conjugation, and hence λ(− ̄τ)=λ(τ). (59) We next note that, by takingτ ...

6 The Modular Function 533 A B C A' B ' C' 0 1/2 1 0 1/2 1 2 τ-plane w-plane ' Fig. 2.w=λ(τ)mapsTontoT′. Again, sinceλ(τ− 1 ) = ...

534 XII Elliptic Functions is holomorphic at every pointzinsideT. Hence, by Cauchy’s theorem, ∫ T f(z)dz= 0. But, since logλ(z)= ...

6 The Modular Function 535 There remains the practical problem, for a givenw∈C, of determiningτ∈H such thatλ(τ)=w.If0<w<1, ...

536 XII Elliptic Functions =( 1 −e−πi/^12 )/( 1 +e−πi/^12 )=itanπ/ 24. Thus||≤tanπ/ 24 < 2 /15 and|/ 2 |^4 < 2 × 10 −^5 ...

7 Further Remarks 537 carried out this extension. (For a modern account, see Rosen [25].) However, Gauss’s claim in a letter to ...

538 XII Elliptic Functions It may be shown that a functionf, which is meromorphic in the whole complex plane, has an algebraic a ...

8 Selected References 539 Many geometrical properties of a lattice are reflected in its theta function. However, a lattice is no ...

540 XII Elliptic Functions [23] I.G. Macdonald, Affine root systems and Dedekind’sη-function,Invent. Math. 15 (1972), 91–143. [2 ...

XIII Connections with Number Theory.............................. 1 SumsofSquares In Proposition II.40 we proved Lagrange’s theo ...

542 XIII Connections with Number Theory by (36) of Chapter XII. Since the theta functions are all solutions of the partial diffe ...

1 Sums of Squares 543 Proposition 2The number of representations of a positive integer m as a sum of 2 squares of integers is eq ...

544 XIII Connections with Number Theory it follows that θ 002 ( 0 )= 1 + 4 ∑ n≥ 1 ∑ k≥ 0 {q(^4 k+^1 )n−q(^4 k+^3 )n} = 1 + 4 ∑ m ...

2 Partitions 545 For many purposes the discussion of convergence is superfluous and Proposition 3 may be regarded simply as a re ...