デデキントのイータ関数
η ( τ ) = e π i τ / 12 ∏ m = 1 ∞ ( 1 − e 2 π i τ m ) = e π i τ / 12 ∏ m = 1 ∞ ( 1 − e 2 π i τ m / 3 ) ( 1 + e 2 π i τ m / 3 + e 4 π i τ m / 3 ) = 2 3 e π i τ / 12 cos  π 6 ∏ m = 1 ∞ ( 1 − e 2 π i τ m / 3 ) ( 1 + 2 cos  π 3 e 2 π i τ m / 3 + e 4 π i τ m / 3 ) = 1 3 θ 2 ( 1 6 , τ 3 ) {\displaystyle {\begin{aligned}\eta (\tau )&=e^{\pi {i}\tau /12}\prod _{m=1}^{\infty }(1-e^{2\pi {i}\tau {m}})\\&=e^{\pi {i}\tau /12}\prod _{m=1}^{\infty }(1-e^{2\pi {i}\tau {m}/3})(1+e^{2\pi {i}\tau {m}/3}+e^{4\pi {i}\tau {m}/3})\\&={\frac {2}{\sqrt {3}}}e^{\pi {i\tau }/12}\cos {\frac {\pi }{6}}\prod _{m=1}^{\infty }(1-e^{2\pi {i}\tau {m}/3})\left(1+2\cos {\frac {\pi }{3}}e^{2\pi {i}\tau {m}/3}+e^{4\pi {}i\tau {m}/3}\right)\\&={\frac {1}{\sqrt {3}}}\vartheta _{2}\left({\frac {1}{6}},{\frac {\tau }{3}}\right)\\\end{aligned}}}
である。