\[E\left(k\right)=K\left(k\right)+\frac{2\pi^{2}}{K\left(k\right)}\,\frac{\sum_{%
n=1}^{\infty}(-1)^{n}n^{2}q^{n^{2}}}{1+2\sum_{n=1}^{\infty}(-1)^{n}q^{n^{2}}},\]