§20.8 Watson’s Expansions

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Keywords:: Watson’s expansions, theta functions
Permalink:: http://dlmf.nist.gov/20.8
See also:: Annotations for Ch.20

20.8.1 $\frac{\theta_{2}\left(0,q\right)\theta_{3}\left(z,q\right)\theta_{4}\left(z,q% \right)}{\theta_{2}\left(z,q\right)}=2\sum_{n=-\infty}^{\infty}\frac{(-1)^{n}q% ^{n^{2}}e^{i2nz}}{q^{-n}e^{-iz}+q^{n}e^{iz}}.$

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, : integer, : complex and : nome
Permalink:: http://dlmf.nist.gov/20.8.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §20.8 and Ch.20

See Watson (1935a). This reference and Bellman (1961, pp. 46–47) include other expansions of this type.