MATH 68: Lecture Notes 8 June 4 Part I Elliptic functions Evaluating cos s Consider cos s First we can substitute u cos s and arrive at u u p3 u, where p 3 is a third-degree polynomial We can also write cos s cos s, then substitute u cos s to get u u p4 u, where p 4 is a degree-4 polynomial Without any proofs, this shows that p 3 / d and p 4 / d are interchangable after some manipulation For the definite integral u u u u u u, substitute v u to get v 3/4 v / dv β 4, where the last step follows from the reflection formula 4 3 4 [ ] 4 π 3 4 4 [ 4] π,
Building blocks for elliptic functions Recall that an elliptic function f is one that is meromorphic and doubly periodic with perio ω and ω such that ω, ω C and ω ω / R; that is, fz fz + ω fz + ω Denote by P the parallelogram with sides ω and ω that tiles the comple plane By periodicity, we conclude f dz, P and that f has at least poles, or is constant an entire doubly periodic function is constant Furthermore, f has every value the same number of times, and that if f has k zeros, a,, a k, then it has k poles, b,, b k, counting multiplicity Now we turn to finding the elementary building blocks for f Jacobi Θ function Assume that ω, ω τ, Iτ > Denote p e iπz, and the nome q e iπτ Then Θz n n p n q nn Furthermore, suppose Θ, Θ is entire, Θz + Θz, and Θz + τ e πiz Θz Then Θ is unique up to a constant We can argue from zf z πi fz dz a j b j, a j and b j the zeros and poles as before, and from periodicity that P aj b j m + nτ, m, n Z Finally, f can then be written fz e πinz Θz aj Θz bj Weierstrass function Suppose is an elliptic function with perio ω and ω, and suppose that is even, with a double pole at can be written z z + O z, z is normalized so that the coefficient of z is is unique, because the difference of two functions with all of the above properties is bounded, entire, and at z, and so it must be everywhere since an entire doubly periodic function is constant Then we have z 3 + Oz, 4 z 6 4 z + which give the equation 4 3 a + b
elliptic integral 3 + a + bd, After some reasoning, it can be shown that f is a rational function of and, and z z + z + w ω, ω where the sum is taken over ω mω + nω, ω To reinforce the connection to ellipses, we have: elliptic curve y 3 + a + b connected to the arc length of an ellipse 3 elliptic function 4 3 + a + a Part II Hypergeometric functions 3 Preliminary considerations Before discussing the hypergeometric function it is important to consider the difference between a series cn and a function c n z n In particular, the thing to keep in mind is that the function may not converge everywhere and may require analytic continuation 4 Hypergeometric series 4 Definition Recall that a geometric series c n is one such that The generalization to hypergeometric series is c n+ c n constant R a rational function This gives c n+ c n Rn, c n z n a n a n a p n z n b n b n b q n n! This gives the hypergeometric function: a, a pf q a, a,, a p ; b, b,, b q ; z p F,, a p q ; z b, b,, b q n a n a n a p n z n b n b n b q n n! Proposition p F q converges absolutely for all z if p q, for z < if p q +, and for only z if p > q + Proof Use the ratio test 3
4 Eamples 3 4 5 e z p F p a,, a p ; a,, a p ; z F ; ; z n z a F a; ; z z n n z 3F,, ;, ; z Φ e t dt F π π ; 3 ; z ν J ν z ν + F ; ν + ; z 4 6 43 Particular cases e t t dt e n n n! n F a; c; z Ma, c; z e F, ; ; is called Kummer s confluent hypergeometric function of the first kind F a, b; c; z is Gauss hypergeometric function, or just the hypergeometric function Fact p F q solves [zdzd + b zd + b q zzd + a zd + a p ]wz, D d dz, in that the equation is an identity, but the function may not converge In particular: F zw + c zw aw This has a regular singular point at, and a non-regular singular point at F z zw + [c a + b + z]w abw This has regular singular points at,, The confluent hypergeometric function can be obtained from the hypergeometric function In the hypergeometric equation, change to w z w z b, bz z Taking the limit as b, the singular points,, and 4
5 Further generalizations 5 q-analogue A q-analogue is obtained when we replace a parameter a qa q The original a is recovered by taking the limit q The only limitation on the q-analogue is that the substitution be natural For eample, Askey polynomials are a q-analogue of Wilson polynomials 5 p-adic version of calculus Usually, calculus is done by completing the rationals: Q R We can also complete the rationals with the p-adic norm: Q p Q p When p an identity is + + + For a N, p prime, define order p a to be the highest power of p that divides a Then a p n b p p orderpa orderpb n 5