ヤコビの二平方定理
F ( 1 4 ) = θ 1 ′ ( 1 4 ) θ 2 ( 1 4 ) − θ 1 ( 1 4 ) θ 2 ′ ( 1 4 ) θ 2 ( 1 4 ) 2 = 2 θ 1 ′ ( 1 4 ) θ 1 ( 1 4 ) {\displaystyle {\begin{aligned}F\left({\tfrac {1}{4}}\right)&={\frac {\vartheta _{1}'\left({\tfrac {1}{4}}\right)\vartheta _{2}\left({\tfrac {1}{4}}\right)-\vartheta _{1}\left({\tfrac {1}{4}}\right)\vartheta _{2}'\left({\tfrac {1}{4}}\right)}{\vartheta _{2}\left({\tfrac {1}{4}}\right)^{2}}}={\frac {2\vartheta _{1}'\left({\tfrac {1}{4}}\right)}{\vartheta _{1}\left({\tfrac {1}{4}}\right)}}\\\end{aligned}}}
であり、テータ関数の対数微分の公式により
F ( 1 4 ) = 2 π cot  π 4 + 8 π ∑ n = 1 ∞ q 2 n 1 − q 2 n sin  π n 2 = 2 π + 8 π ∑ k = 0 ∞ q 2 ( 2 k + 1 ) 1 − q 2 ( 2 k + 1 ) ( − 1 ) k ( n → 2 k + 1 ) = 2 π + 8 π ∑ k = 0 ∞ ( − 1 ) k ∑ j = 1 ∞ q 2 j ( 2 k + 1 ) {\displaystyle {\begin{aligned}F\left({\tfrac {1}{4}}\right)&=2\pi \cot {\frac {\pi }{4}}+8\pi \sum _{n=1}^{\infty }{\frac {q^{2n}}{1-q^{2n}}}\sin {\frac {\pi {n}}{2}}\\&=2\pi +8\pi \sum _{k=0}^{\infty }{\frac {q^{2(2k+1)}}{1-q^{2(2k+1)}}}(-1)^{k}\qquad (n\to 2k+1)\\&=2\pi +8\pi \sum _{k=0}^{\infty }(-1)^{k}\sum _{j=1}^{\infty }q^{2j(2k+1)}\\\end{aligned}}}