§23.7 Quarter Periods

Notes:: See Lawden (1989, p. 182).
Keywords:: Weierstrass elliptic functions
Permalink:: http://dlmf.nist.gov/23.7

23.7.1 $\mathop{\wp\/}\nolimits\!\left(\tfrac{1}{2}\omega _{1}\right)=e_{1}+\sqrt{(e_{1}-e_{3})(e_{1}-e_{2})}=e_{1}+\omega _{1}^{{-2}}(\mathop{K\/}\nolimits\!\left(k\right))^{2}k^{{\prime}},$

Symbols:: $\mathop{\wp\/}\nolimits\!\left(z\right)$ (= $\mathop{\wp\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\mathop{\wp\/}\nolimits$ -function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathbb{L}$ : lattice, $e_{j}$ : zeros, $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators, $k$ : modulus and $k^{{\prime}}$ : complementary modulus
A&S Ref:: 18.3.18
Permalink:: http://dlmf.nist.gov/23.7.E1
Encodings:: TeX, pMML, png

23.7.2 $\mathop{\wp\/}\nolimits\!\left(\tfrac{1}{2}\omega _{2}\right)=e_{2}-i\sqrt{(e_{1}-e_{2})(e_{2}-e_{3})}=e_{2}-i\omega _{1}^{{-2}}(\mathop{K\/}\nolimits\!\left(k\right))^{2}kk^{{\prime}},$

Symbols:: $\mathop{\wp\/}\nolimits\!\left(z\right)$ (= $\mathop{\wp\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\mathop{\wp\/}\nolimits$ -function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathbb{L}$ : lattice, $e_{j}$ : zeros, $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators, $k$ : modulus and $k^{{\prime}}$ : complementary modulus
A&S Ref:: 18.3.18
Permalink:: http://dlmf.nist.gov/23.7.E2
Encodings:: TeX, pMML, png

23.7.3 $\mathop{\wp\/}\nolimits\!\left(\tfrac{1}{2}\omega _{3}\right)=e_{3}-\sqrt{(e_{1}-e_{3})(e_{2}-e_{3})}=e_{3}-\omega _{1}^{{-2}}(\mathop{K\/}\nolimits\!\left(k\right))^{2}k,$

Symbols:: $\mathop{\wp\/}\nolimits\!\left(z\right)$ (= $\mathop{\wp\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\mathop{\wp\/}\nolimits$ -function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathbb{L}$ : lattice, $e_{j}$ : zeros, $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators and $k$ : modulus
A&S Ref:: 18.3.18
Permalink:: http://dlmf.nist.gov/23.7.E3
Encodings:: TeX, pMML, png

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.