テータ関数
θ 4 ( v , τ ) = ∑ n = − ∞ ∞ ( − 1 ) n e π i τ n 2 + 2 π i n v = ∑ n = − ∞ ∞ e π i τ ( 2 n ) 2 + 2 π i ( 2 n ) v − ∑ n = − ∞ ∞ e π i τ ( 2 n + 1 ) 2 + 2 π i ( 2 n + 1 ) v ( n ↦ 2 n , 2 n + 1 ) = θ 3 ( 2 v , 4 τ ) − θ 2 ( 2 v , 4 τ ) {\displaystyle {\begin{aligned}\vartheta _{4}\left(v,\tau \right)&=\sum _{n=-\infty }^{\infty }(-1)^{n}e^{\pi {i}\tau {n^{2}}+2\pi {i}nv}\\&=\sum _{n=-\infty }^{\infty }e^{\pi {i}\tau {(2n)^{2}}+2\pi {i}(2n)v}-\sum _{n=-\infty }^{\infty }e^{\pi {i}\tau {(2n+1)^{2}}+2\pi {i}(2n+1)v}\qquad ({n}\mapsto {2n,2n+1})\\&=\vartheta _{3}\left(2v,4\tau \right)-\vartheta _{2}\left(2v,4\tau \right)\end{aligned}}}
θ 3 ( v , τ ) θ 4 ( v , τ ) = ∑ n ′ = − ∞ ∞ ∑ m ′ = − ∞ ∞ ( − 1 ) n ′ − m ′ e 2 π i τ ( n ′ 2 + m ′ 2 ) + 4 π i n ′ v ( n ↦ n ′ + m ′ , m ↦ n ′ − m ′ ) = ( ∑ n ′ = − ∞ ∞ ( − 1 ) n ′ e 2 π i τ n ′ 2 + 4 π i n ′ v ) ( ∑ m ′ = − ∞ ∞ ( − 1 ) m ′ e 2 π i τ m ′ 2 ) = θ 4 ( 2 v , 2 τ ) θ 4 ( 0 , 2 τ ) {\displaystyle {\begin{aligned}\vartheta _{3}(v,\tau )\vartheta _{4}(v,\tau )&=\sum _{n'=-\infty }^{\infty }\sum _{m'=-\infty }^{\infty }(-1)^{n'-m'}e^{2\pi {i\tau }(n'^{2}+m'^{2})+4\pi {in'v}}\qquad ({n}\mapsto {n'+m'},{m}\mapsto {n'-m'})\\&=\left(\sum _{n'=-\infty }^{\infty }(-1)^{n'}e^{2\pi {i\tau }n'^{2}+4\pi {in'v}}\right)\left(\sum _{m'=-\infty }^{\infty }(-1)^{m'}e^{2\pi {i\tau }m'^{2}}\right)\\&=\vartheta _{4}(2v,2\tau )\vartheta _{4}(0,2\tau )\end{aligned}}}
となり、左辺を得る。第二式は第一式に v ′ = v + τ 2 {\displaystyle v'=v+\textstyle {\frac {\tau }{2}}} を代入して得られる。
θ 4 ( v ± τ 2 , τ ) = ± i e − π i τ / 4 e ∓ π i v θ 1 ( v ; τ ) {\displaystyle \vartheta _{4}\left(v\pm {\frac {\tau }{2}},\tau \right)=\pm {i}e^{-\pi {i\tau }/4}e^{\mp \pi {i}v}\vartheta _{1}(v;\tau )}
次の恒等式はヤコビの虚数変換式という。
θ 1 ( v τ , − 1 τ ) = i e − i π / 4 τ 1 / 2 e π i v 2 / τ θ 1 ( v , τ ) , θ 2 ( v τ , − 1 τ ) = e − i π / 4 τ 1 / 2 e π i v 2 / τ θ 4 ( v , τ ) , θ 3 ( v τ , − 1 τ ) = e − i π / 4 τ 1 / 2 e π i v 2 / τ θ 3 ( v , τ ) , θ 4 ( v τ , − 1 τ ) = e − i π / 4 τ 1 / 2 e π i v 2 / τ θ 2 ( v , τ ) .
θ 3 ( x + y , τ ) θ 3 ( x − y , τ ) θ 3 2 ( 0 , τ ) = ( θ 3 ( 2 x , 2 τ ) θ 3 ( 2 y , 2 τ ) + θ 2 ( 2 x , 2 τ ) θ 2 ( 2 y , 2 τ ) ) ( θ 3 2 ( 0 , 2 τ ) + θ 2 2 ( 0 , 2 τ ) ) = ( θ 3 ( 2 x , 2 τ ) θ 2 ( 0 , 2 τ ) + θ 2 ( 2 x , 2 τ ) θ 3 ( 0 , 2 τ ) ) ( θ 2 ( 2 y , 2 τ ) θ 3 ( 0 , 2 τ ) + θ 3 ( 2 y , 2 τ ) θ 2 ( 0 , 2 τ ) ) + ( θ 3 ( 2 x , 2 τ ) θ 3 ( 0 , 2 τ ) − θ 2 ( 2 x , 2 τ ) θ 2 ( 0 , 2 τ ) ) ( θ 3 ( 2 y , 2 τ ) θ 3 ( 0 , 2 τ ) − θ 2 ( 2 y , 2 τ ) θ 2 ( 0 , 2 τ ) ) = θ 2 2 ( x , τ ) θ 2 2 ( y , τ ) + θ 4 2 ( x , τ ) θ 4 2 ( y , τ ) {\displaystyle {\begin{aligned}\vartheta _{3}\left(x+y,\tau \right)\vartheta _{3}\left(x-y,\tau \right)\vartheta _{3}^{\;2}(0,\tau )&=\left(\vartheta _{3}\left(2x,2\tau \right)\vartheta _{3}\left(2y,2\tau \right)+\vartheta _{2}\left(2x,2\tau \right)\vartheta _{2}\left(2y,2\tau \right)\right)\left(\vartheta _{3}^{\;2}\left(0,2\tau \right)+\vartheta _{2}^{\;2}\left(0,2\tau \right)\right)\\&=\left(\vartheta _{3}\left(2x,2\tau \right)\vartheta _{2}\left(0,2\tau \right)+\vartheta _{2}\left(2x,2\tau \right)\vartheta _{3}\left(0,2\tau \right)\right)\left(\vartheta _{2}\left(2y,2\tau \right)\vartheta _{3}\left(0,2\tau \right)+\vartheta _{3}\left(2y,2\tau \right)\vartheta _{2}\left(0,2\tau \right)\right)\\&\quad +\left(\vartheta _{3}\left(2x,2\tau \right)\vartheta _{3}\left(0,2\tau \right)-\vartheta _{2}\left(2x,2\tau \right)\vartheta _{2}\left(0,2\tau \right)\right)\left(\vartheta _{3}\left(2y,2\tau \right)\vartheta _{3}\left(0,2\tau \right)-\vartheta _{2}\left(2y,2\tau \right)\vartheta _{2}\left(0,2\tau \right)\right)\\&=\vartheta _{2}^{\;2}\left(x,\tau \right)\vartheta _{2}^{\;2}\left(y,\tau \right)+\vartheta _{4}^{\;2}\left(x,\tau \right)\vartheta _{4}^{\;2}\left(y,\tau \right)\end{aligned}}}