ヤコビの虚数変換式
θ 3 ( v , τ ) {\displaystyle \vartheta _{3}(v,\tau )} の虚数変換式の両辺の比を f ( v , τ ) {\displaystyle f(v,\tau )} して恒等的に f ( v , τ ) = 1 {\displaystyle f(v,\tau )=1} であることを証明する。テータ関数の二重周期性により
f ( v , τ ) = − i τ e π i v 2 / τ θ 3 ( v , τ ) θ 3 ( v τ , − 1 τ ) {\displaystyle f(v,\tau )={\frac {{\sqrt {-i\tau }}e^{{\pi }iv^{2}/\tau }\vartheta _{3}\left(v,\tau \right)}{\vartheta _{3}\left({\frac {v}{\tau }},-{\frac {1}{\tau }}\right)}}}
f ( v + τ , τ ) = − i τ e π i ( v + τ ) 2 / τ θ 3 ( v + τ , τ ) θ 3 ( v τ + 1 , − 1 τ ) = − i τ e π i v 2 / τ + 2 π i v + π i τ e − π i τ − 2 π i v θ 3 ( v , τ ) θ 3 ( v τ , − 1 τ ) = f ( v , τ ) {\displaystyle f(v+\tau ,\tau )={\frac {{\sqrt {-i\tau }}e^{{\pi }i(v+\tau )^{2}/\tau }\vartheta _{3}\left(v+\tau ,\tau \right)}{\vartheta _{3}\left({\frac {v}{\tau }}+1,-{\frac {1}{\tau }}\right)}}={\frac {{\sqrt {-i\tau }}e^{{\pi }iv^{2}/\tau +2{\pi }iv+{\pi }i\tau }e^{-{\pi }i\tau -2{\pi }iv}\vartheta _{3}\left(v,\tau \right)}{\vartheta _{3}\left({\frac {v}{\tau }},-{\frac {1}{\tau }}\right)}}=f(v,\tau )}
であるから、 f ( v , τ ) {\displaystyle f(v,\tau )} は v {\displaystyle v} の関数として二重周期を持つ。また、テータ関数は極を持たず、零点は
θ 3 ( 1 ± 2 m 2 + ( 1 ± 2 n ) τ 2 , τ ) = 0 {\displaystyle \vartheta _{3}\left({\frac {1\pm {2m}}{2}}+{\frac {(1\pm {2n})\tau }{2}},\tau \right)=0}
テータ関数の虚数変換式により
τ ′ = − 1 τ = i K K ′ {\displaystyle \tau '=-{\frac {1}{\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  ( i u , k ) = θ 3 ( 0 , τ ) θ 1 ( i u / 2 K , τ ) θ 2 ( 0 , τ ) θ 4 ( i u / 2 K , τ ) = i θ 3 ( 0 , τ ′ ) θ 1 ( u / 2 K ′ , τ ′ ) θ 4 ( 0 , τ ′ ) θ 2 ( u / 2 K ′ , τ ′ ) = i sn  ( u , k ′ ) cn  ( u , k ′ ) {\displaystyle \operatorname {sn} (iu,k)={\frac {\vartheta _{3}(0,\tau )\vartheta _{1}(iu/2K,\tau )}{\vartheta _{2}(0,\tau )\vartheta _{4}(iu/2K,\tau )}}={\frac {i\vartheta _{3}(0,\tau ')\vartheta _{1}(u/2K',\tau ')}{\vartheta _{4}(0,\tau ')\vartheta _{2}(u/2K',\tau ')}}=i{\frac {\operatorname {sn} (u,k')}{\operatorname {cn} (u,k')}}}