テータ関数
θ 1 ( v ; τ ) = − ∑ n = − ∞ ∞ e π i τ ( n + 1 / 2 ) 2 e 2 π i ( n + 1 / 2 ) ( v + 1 / 2 ) = − i e π i τ / 4 e π i v ∑ n = − ∞ ∞ e π i τ n 2 e π i τ n + 2 π i v n + π i n = − i e π i τ / 4 e π i v ∏ m = 1 ∞ ( 1 − e 2 m π i τ ) ( 1 − e 2 m π i τ e 2 π i v ) ( 1 − e ( 2 m − 2 ) π i τ e − 2 π i v ) = − i e π i τ / 4 e π i v ( 1 − e − 2 π i v ) ∏ m = 1 ∞ ( 1 − e 2 m π i τ ) ( 1 − e 2 m π i τ e 2 π i v ) ( 1 − e 2 m π i τ e − 2 π i v ) = 2 e π i τ / 4 sin  π v ∏ m = 1 ∞ ( 1 − e 2 m π i τ ) ( 1 − e 2 m π i τ e 2 π i v ) ( 1 − e 2 m π i τ e − 2 π i v ) = 2 e π i τ / 4 sin  π v ∏ m = 1 ∞ ( 1 − e 2 m π i τ ) ( 1 − 2 e 2 m π i τ cos  2 π v + e 4 m π i τ ) {\displaystyle {\begin{aligned}\vartheta _{1}(v;\tau )&=-\sum _{n=-\infty }^{\infty }{e^{{\pi }i{\tau }\left(n+1/2\right)^{2}}e^{2{\pi }i(n+1/2)(v+1/2)}}\\&=-ie^{{\pi }i{\tau }/4}e^{{\pi }iv}\sum _{n=-\infty }^{\infty }{e^{{\pi }i{\tau }n^{2}}e^{{\pi }i{\tau }n+2{\pi }ivn+{\pi }in}}\\&=-ie^{{\pi }i{\tau }/4}e^{{\pi }iv}\prod _{m=1}^{\infty }{\left(1-e^{2m{\pi }i{\tau }}\right)\left(1-e^{2m{\pi }i{\tau }}e^{2{\pi }iv}\right)\left(1-e^{(2m-2){\pi }i{\tau }}e^{-2{\pi }iv}\right)}\\&=-ie^{{\pi }i{\tau }/4}e^{{\pi }iv}(1-e^{-2{\pi }iv})\prod _{m=1}^{\infty }{\left(1-e^{2m{\pi }i{\tau }}\right)\left(1-e^{2m{\pi }i{\tau }}e^{2{\pi }iv}\right)\left(1-e^{2m{\pi }i{\tau }}e^{-2{\pi }iv}\right)}\\&=2e^{{\pi }i{\tau }/4}\sin {\pi }v\prod _{m=1}^{\infty }{\left(1-e^{2m{\pi }i{\tau }}\right)\left(1-e^{2m{\pi }i{\tau }}e^{2{\pi }iv}\right)\left(1-e^{2m{\pi }i{\tau }}e^{-2{\pi }iv}\right)}\\&=2e^{{\pi }i{\tau }/4}\sin {\pi }v\prod _{m=1}^{\infty }{\left(1-e^{2m{\pi }i{\tau }}\right)\left(1-2e^{2m{\pi }i{\tau }}\cos {2{\pi }v}+e^{4m{\pi }i{\tau }}\right)}\\\end{aligned}}}
θ 3 ( 0 , τ ) 4 − θ 4 ( 0 , τ ) 4 = ( θ 3 ( 0 , 2 τ ) 2 + θ 2 ( 0 , 2 τ ) 2 ) 2 − ( θ 3 ( 0 , 2 τ ) 2 − θ 2 ( 0 , 2 τ ) 2 ) 2 = 4 θ 3 ( 0 , 2 τ ) 2 θ 2 ( 0 , 2 τ ) 2 = θ 2 ( 0 , τ ) 4 {\displaystyle {\begin{aligned}\vartheta _{3}\left(0,\tau \right)^{4}-\vartheta _{4}\left(0,\tau \right)^{4}&=\left(\vartheta _{3}\left(0,2\tau \right)^{2}+\vartheta _{2}\left(0,2\tau \right)^{2}\right)^{2}-\left(\vartheta _{3}\left(0,2\tau \right)^{2}-\vartheta _{2}\left(0,2\tau \right)^{2}\right)^{2}\\&=4\vartheta _{3}\left(0,2\tau \right)^{2}\vartheta _{2}\left(0,2\tau \right)^{2}\\&=\vartheta _{2}\left(0,\tau \right)^{4}\end{aligned}}}