# A Few More Geometries after Ramanujan

A Few More Geometries after Ramanujan

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We do not know if Ramanujan had any plane curves in mind when working with the integral period functions , and [1], though certainly these imply many simple geometries of intrinsic relevance [2]. This Demonstration gives three additional models characterized by real period functions

K

1

K

2

K

3

T(α)=,;1;,s=3,4,or6

2

F

1

1

2

s-1

s

2

α

A new and improved integral-differential algorithm is used to prove the integral periods. Given the input of a well-chosen Hamiltonian , the certified algorithm quickly derives that the associated period function satisfies a special case of the hypergeometric differential equation

H(p,q)

T(α)

4(s-1)αT(α)-(1-3)T(α)-(1-)αT(α)=0,s=3,4,or6

2

s

2

α

∂

α

2

s

2

α

2

∂

α

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).