ヤコビの虚数変換式
ヤコビの楕円関数はテータ関数の比により表される。楕円関数の周期を K , i K ′ {\displaystyle K,iK'} とすると
τ = i K ′ K {\displaystyle \tau ={\frac {iK'}{K}}}
k = ( θ 2 ( 0 , τ ) θ 3 ( 0 , τ ) ) 2 {\displaystyle k=\left({\frac {\vartheta _{2}(0,\tau )}{\vartheta _{3}(0,\tau )}}\right)^{2}}
sn  ( u , k ) = θ 3 ( 0 , τ ) θ 1 ( u / 2 K , τ ) θ 2 ( 0 , τ ) θ 4 ( u / 2 K , τ ) {\displaystyle \operatorname {sn} (u,k)={\frac {\vartheta _{3}(0,\tau )\vartheta _{1}(u/2K,\tau )}{\vartheta _{2}(0,\tau )\vartheta _{4}(u/2K,\tau )}}}
cn  ( u , k ) = θ 4 ( 0 , τ ) θ 2 ( u / 2 K , τ ) θ 2 ( 0 , τ ) θ 4 ( u / 2 K , τ ) {\displaystyle \operatorname {cn} (u,k)={\frac {\vartheta _{4}(0,\tau )\vartheta _{2}(u/2K,\tau )}{\vartheta _{2}(0,\tau )\vartheta _{4}(u/2K,\tau )}}}