Elliptic Rational Functions - Particular Values

Particular Values

We may write the first few elliptic rational functions as:

where
where
R_4(\xi,x)=R_2(R_2(\xi,\xi),R_2(\xi,x))=\frac {(1+t)(1+\sqrt{t})^2x^4-2(1+t)(1+\sqrt{t})x^2+1} {(1+t)(1-\sqrt{t})^2x^4-2(1+t)(1-\sqrt{t})x^2+1}
etc.

See Lutovac (2001, ยง 13) for further explicit expressions of order n=5 and .

The corresponding discrimination factors are:

etc.

The corresponding zeroes are where n is the order and j is the number of the zero. There will be a total of n zeroes for each order.

From the inversion relationship, the corresponding poles may be found by

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