20 Theta Functions Properties 20.7 Identities 20.9 Relations to Other Functions

§20.8 Watson’s Expansions

Keywords:: Watson’s expansions, theta functions
Permalink:: http://dlmf.nist.gov/20.8

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

Symbols:: $\mathop{\theta_{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, : base of exponential function, : integer, : complex and : nome
Permalink:: http://dlmf.nist.gov/20.8.E1
Encodings:: TeX, pMML, png

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

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