A Few More Geometries after Ramanujan

Out[]=

display

toric sections of Hamiltonian, H(p,q)

derivation of the Picard–Fuchs equation

period T(α)

2

F

1

1/3, 2/3; 1;

2

α



2

F

1

1/4, 3/4; 1;

2

α



2

F

1

1/6, 5/6; 1;

2

α



energy α

0.5

α = -

4

27

3



2

p

+

2

q



+

4

27

2

p

2

3

2

q

-

2

p



+

2

p

+

2

q

We do not know if Ramanujan had any plane curves in mind when working with the integral period functions

K

1

,

K

2

and

K

3

[1], though certainly these imply many simple geometries of intrinsic relevance [2]. This Demonstration gives three additional models characterized by real period functions

T(α)=

2

F

1

2

,

s-1

s

;1;

2

α

,s=3,4,or6

.

A new and improved integral-differential algorithm is used to prove the integral periods. Given the input of a well-chosen Hamiltonian

H(p,q)

, the certified algorithm quickly derives that the associated period function

T(α)

satisfies a special case of the hypergeometric differential equation

4(s-1)αT(α)-

2

s

(1-3

2

α

)

∂

α

T(α)-

2

s

(1-

2

α

)α

2

∂

α

T(α)=0,s=3,4,or6

.

In prospectus, the functions HEllipticPFDE[Hqp] and DihedralPFDE[Hλϕ] defined in this Demonstration return valid results on infinitely many more calls. These algorithms help to extend the regime of easy integral-differential calculations far beyond the three simple examples depicted here (see Details).