DLMF:20.5.E13 (Q6780)

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DLMF:20.5.E13
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    θ 4 ( z , q ) θ 4 ( z , q ) = 4 sin ( 2 z ) n = 1 q 2 n - 1 1 - 2 q 2 n - 1 cos ( 2 z ) + q 4 n - 2 = 4 n = 1 q n 1 - q 2 n sin ( 2 n z ) . diffop Jacobi-theta 4 1 𝑧 𝑞 Jacobi-theta 4 𝑧 𝑞 4 2 𝑧 superscript subscript 𝑛 1 superscript 𝑞 2 𝑛 1 1 2 superscript 𝑞 2 𝑛 1 2 𝑧 superscript 𝑞 4 𝑛 2 4 superscript subscript 𝑛 1 superscript 𝑞 𝑛 1 superscript 𝑞 2 𝑛 2 𝑛 𝑧 {\displaystyle{\displaystyle\frac{\theta_{4}'\left(z,q\right)}{\theta_{4}\left% (z,q\right)}=4\sin\left(2z\right)\sum_{n=1}^{\infty}\frac{q^{2n-1}}{1-2q^{2n-1% }\cos\left(2z\right)+q^{4n-2}}=4\sum_{n=1}^{\infty}\frac{q^{n}}{1-q^{2n}}\sin% \left(2nz\right).}}
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    DLMF:20.5.E13
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