# §20.6 Power Series

Assume

 20.6.1 $\left|\pi z\right|<\min\left|z_{m,n}\right|,$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $m$: integer, $n$: integer and $z$: complex Permalink: http://dlmf.nist.gov/20.6.E1 Encodings: TeX, pMML, png See also: Annotations for §20.6 and Ch.20

where $z_{m,n}$ is given by (20.2.5) and the minimum is for $m,n\in\mathbb{Z}$, except $m=n=0$. Then

 20.6.2 $\displaystyle\theta_{1}\left(\pi z\middle|\tau\right)$ $\displaystyle=\pi z\theta_{1}'\left(0\middle|\tau\right)\exp\left(-\sum_{j=1}^% {\infty}\frac{1}{2j}\delta_{2j}(\tau)z^{2j}\right),$ 20.6.3 $\displaystyle\theta_{2}\left(\pi z\middle|\tau\right)$ $\displaystyle=\theta_{2}\left(0\middle|\tau\right)\exp\left(-\sum_{j=1}^{% \infty}\frac{1}{2j}\alpha_{2j}(\tau)z^{2j}\right),$ 20.6.4 $\displaystyle\theta_{3}\left(\pi z\middle|\tau\right)$ $\displaystyle=\theta_{3}\left(0\middle|\tau\right)\exp\left(-\sum_{j=1}^{% \infty}\frac{1}{2j}\beta_{2j}(\tau)z^{2j}\right),$ 20.6.5 $\displaystyle\theta_{4}\left(\pi z\middle|\tau\right)$ $\displaystyle=\theta_{4}\left(0\middle|\tau\right)\exp\left(-\sum_{j=1}^{% \infty}\frac{1}{2j}\gamma_{2j}(\tau)z^{2j}\right).$

Here the coefficients are given by

 20.6.6 $\displaystyle\delta_{2j}(\tau)$ $\displaystyle=\left.\sum_{n=-\infty}^{\infty}\sum_{\begin{subarray}{c}m=-% \infty\\ \left|m\right|+\left|n\right|\neq 0\end{subarray}}^{\infty}\right.(m+n\tau)^{-% 2j},$ ⓘ Defines: $\delta_{2j}(\tau)$: coefficient (locally) Symbols: $m$: integer, $n$: integer and $\tau$: lattice parameter Referenced by: §20.6, §20.6 Permalink: http://dlmf.nist.gov/20.6.E6 Encodings: TeX, pMML, png See also: Annotations for §20.6 and Ch.20 20.6.7 $\displaystyle\alpha_{2j}(\tau)$ $\displaystyle=\sum_{n=-\infty}^{\infty}\sum_{m=-\infty}^{\infty}(m-\tfrac{1}{2% }+n\tau)^{-2j},$ ⓘ Defines: $\alpha_{2j}(\tau)$: coefficient (locally) Symbols: $m$: integer, $n$: integer and $\tau$: lattice parameter Permalink: http://dlmf.nist.gov/20.6.E7 Encodings: TeX, pMML, png See also: Annotations for §20.6 and Ch.20 20.6.8 $\displaystyle\beta_{2j}(\tau)$ $\displaystyle=\sum_{n=-\infty}^{\infty}\sum_{m=-\infty}^{\infty}(m-\tfrac{1}{2% }+(n-\tfrac{1}{2})\tau)^{-2j},$ ⓘ Defines: $\beta_{2j}(\tau)$: coefficient (locally) Symbols: $m$: integer, $n$: integer and $\tau$: lattice parameter Referenced by: §20.6 Permalink: http://dlmf.nist.gov/20.6.E8 Encodings: TeX, pMML, png See also: Annotations for §20.6 and Ch.20 20.6.9 $\displaystyle\gamma_{2j}(\tau)$ $\displaystyle=\sum_{n=-\infty}^{\infty}\sum_{m=-\infty}^{\infty}(m+(n-\tfrac{1% }{2})\tau)^{-2j},$ ⓘ Defines: $\gamma_{2j}(\tau)$: coefficient (locally) Symbols: $m$: integer, $n$: integer and $\tau$: lattice parameter Permalink: http://dlmf.nist.gov/20.6.E9 Encodings: TeX, pMML, png See also: Annotations for §20.6 and Ch.20

and satisfy

 20.6.10 $\displaystyle\alpha_{2j}(\tau)$ $\displaystyle=2^{2j}\delta_{2j}(2\tau)-\delta_{2j}(\tau),$ $\displaystyle\beta_{2j}(\tau)$ $\displaystyle=2^{2j}\gamma_{2j}(2\tau)-\gamma_{2j}(\tau).$ ⓘ Symbols: $\tau$: lattice parameter, $\delta_{2j}(\tau)$: coefficient, $\alpha_{2j}(\tau)$: coefficient, $\beta_{2j}(\tau)$: coefficient and $\gamma_{2j}(\tau)$: coefficient Referenced by: §20.6 Permalink: http://dlmf.nist.gov/20.6.E10 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §20.6 and Ch.20

In the double series the order of summation is important only when $j=1$. For further information on $\delta_{2j}$ see §23.9: since the double sums in (20.6.6) and (23.9.1) are the same, we have $\delta_{2n}=c_{n}/(2n-1)$ when $n\geq 2$.