# §23.7 Quarter Periods

 23.7.1 $\displaystyle\wp\left(\tfrac{1}{2}\omega_{1}\right)$ $\displaystyle=e_{1}+\sqrt{(e_{1}-e_{3})(e_{1}-e_{2})}=e_{1}+\omega_{1}^{-2}(K% \left(k\right))^{2}k^{\prime},$ 23.7.2 $\displaystyle\wp\left(\tfrac{1}{2}\omega_{2}\right)$ $\displaystyle=e_{2}-i\sqrt{(e_{1}-e_{2})(e_{2}-e_{3})}=e_{2}-i\omega_{1}^{-2}(% K\left(k\right))^{2}kk^{\prime},$ 23.7.3 $\displaystyle\wp\left(\tfrac{1}{2}\omega_{3}\right)$ $\displaystyle=e_{3}-\sqrt{(e_{1}-e_{3})(e_{2}-e_{3})}=e_{3}-\omega_{1}^{-2}(K% \left(k\right))^{2}k,$

where $k,k^{\prime}$ and the square roots are real and positive when the lattice is rectangular; otherwise they are determined by continuity from the rectangular case.