Aquileo | DLMF: Figure 23.4.8 ‣ §23.4 Graphics ‣ Weierstrass Elliptic Functions ‣ Chapter 23 Weierstrass Elliptic and Modular Functions

Figure 23.4.8: $\wp\left(x+iy\right)$ with $\omega_{1}=K\left(k\right)$ , $\omega_{3}=i{K^{\prime}}\left(k\right)$ for $-2K\left(k\right)\leq x\leq 2K\left(k\right)$ , $0\leq y\leq 6{K^{\prime}}\left(k\right)$ , $k^{2}=0.9$ . (The scaling makes the lattice appear to be square.)

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Annotations:

Symbols:: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, ${K^{\prime}}\left(\NVar{k}\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$ : imaginary unit, $\mathbb{L}$ : lattice, : real part of , : imaginary part of , $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators and : modulus
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