Elliptic Invariants

The elliptic invariants of a Weierstrass elliptic function are defined by the Eisenstein series

			(1)
			(2)

Here, the prime omits from the sums over integers, and

(3)

where and are the half-periods of the elliptic function. The Wolfram Language command WeierstrassInvariants[omega1, omega2] gives the invariants and corresponding to the half-periods and .

For the half-period ratio in the upper half-plane, use the normalization

			(4)
			(5)

and the invariants have the Fourier series

			(6)
			(7)

where is the divisor function (Apostol 1997, pp. 18-19).