テータ関数
テータ関数は次のように定義される関数のことを指す。
θ ( z , τ ) := ∑ n = − ∞ ∞ e π i n 2 τ + 2 π i n z . {\displaystyle \vartheta (z,\tau ):=\sum _{n=-\infty }^{\infty }e^{\pi in^{2}\tau +2\pi inz}.}
テータ関数を z の関数と見た場合、周期 1 の周期関数である。
θ ( z + 1 , τ ) = θ ( z , τ ) . {\displaystyle \vartheta (z+1,\tau )=\vartheta (z,\tau ).}
一般には以下の等式を満たす。
θ ( z + m τ + n , τ ) = e − π i m 2 τ − 2 π i m z θ ( z , τ ) . {\displaystyle \vartheta (z+m\tau +n,\tau )=e^{-\pi im^{2}\tau -2\pi imz}\vartheta (z,\tau ).}
θ 1 ( v + τ ; τ ) = − ∑ n = − ∞ ∞ e π i τ ( n + 1 2 ) 2 + 2 π i ( n + 1 2 ) ( v + 1 2 ) + 2 π i ( n + 1 2 ) τ = − ∑ n = − ∞ ∞ e π i τ ( n + 1 + 1 2 ) 2 + 2 π i ( n + 1 + 1 2 ) ( v + 1 2 ) − π i τ − 2 π i ( v + 1 2 ) = − ∑ n = − ∞ ∞ e π i τ ( n + 1 2 ) 2 + 2 π i ( n + 1 2 ) ( v + 1 2 ) − π i τ − 2 π i ( v + 1 2 ) = ∑ n = − ∞ ∞ e π i τ ( n + 1 2 ) 2 + 2 π i ( n + 1 2 ) ( v + 1 2 ) − π i τ − 2 π i v = − e − π i τ e − 2 π i v θ 1 ( v ; τ ) {\displaystyle {\begin{aligned}\vartheta _{1}(v+\tau ;\tau )&=-\sum _{n=-\infty }^{\infty }{e^{{\pi }i{\tau }\left(n+{\frac {1}{2}}\right)^{2}+2{\pi }i(n+{\frac {1}{2}})(v+{\frac {1}{2}})+2{\pi }i(n+{\frac {1}{2}})\tau }}\\&=-\sum _{n=-\infty }^{\infty }{e^{{\pi }i{\tau }\left(n+1+{\frac {1}{2}}\right)^{2}+2{\pi }i(n+1+{\frac {1}{2}})(v+{\frac {1}{2}})-{\pi }i{\tau }-2{\pi }i(v+{\frac {1}{2}})}}\\&=-\sum _{n=-\infty }^{\infty }{e^{{\pi }i{\tau }\left(n+{\frac {1}{2}}\right)^{2}+2{\pi }i(n+{\frac {1}{2}})(v+{\frac {1}{2}})-{\pi }i{\tau }-2{\pi }i(v+{\frac {1}{2}})}}\\&=\sum _{n=-\infty }^{\infty }{e^{{\pi }i{\tau }\left(n+{\frac {1}{2}}\right)^{2}+2{\pi }i(n+{\frac {1}{2}})(v+{\frac {1}{2}})-{\pi }i{\tau }-2{\pi }iv}}\\&=-e^{-{\pi }i\tau }e^{-2{\pi }i{v}}\vartheta _{1}(v;\tau )\end{aligned}}}
θ 4 ( v ± 1 2 , τ ) = θ 3 ( v ; τ ) {\displaystyle \vartheta _{4}\left(v\pm {\frac {1}{2}},\tau \right)=\vartheta _{3}(v;\tau )}
θ 1 ( v ± τ 2 , τ ) = ± i e − π i τ / 4 e ∓ π i v θ 4 ( v ; τ ) {\displaystyle \vartheta _{1}\left(v\pm {\frac {\tau }{2}},\tau \right)=\pm {i}e^{-\pi {i\tau }/4}e^{\mp \pi {i}v}\vartheta _{4}(v;\tau )}
θ 2 ( v ± τ 2 , τ ) = e − π i τ / 4 e ∓ π i v θ 3 ( v ; τ ) {\displaystyle \vartheta _{2}\left(v\pm {\frac {\tau }{2}},\tau \right)=e^{-\pi {i\tau }/4}e^{\mp \pi {i}v}\vartheta _{3}(v;\tau )}
θ 3 ( v ± τ 2 , τ ) = e − π i τ / 4 e ∓ π i v θ 2 ( v ; τ ) {\displaystyle \vartheta _{3}\left(v\pm {\frac {\tau }{2}},\tau \right)=e^{-\pi {i\tau }/4}e^{\mp \pi {i}v}\vartheta _{2}(v;\tau )}