内维尔Θ函数

内维尔Θ函數（Neville Theta functions）共有四个，定义如下：

$N e v i l l e C (z, m) = \frac{\sqrt{(} 2) * q (m)^{1 / 4} * (\sum_{k = 0}^{\infty} (q (m)^{(} k * (k + 1)) * c o s ((1 / 2) * (2 * k + 1) * π * z / K (m))))}{\sqrt{(} K (m)) * m^{1 / 4}}$ $N e v i l l e T h e t a C (z, m) = \frac{\sqrt{(} 2 * π) * q (m)^{1 / 4} * (\sum_{k = 0}^{\infty} (q (m)^{k * (k + 1)} * c o s ((1 / 2) * (2 * k + 1) * π * z / K (m))))}{\sqrt{(} K (m)) * m^{1 / 4}}$ $N e v i l l e T h e t a D (z, m) = \frac{\sqrt{(} (1 / 2) * π) * (1 + 2 * (\sum_{k = 1}^{\infty} (q (m)^{(} k^{2}) * c o s (k * π * z / K (m)))))}{\sqrt{(} K (m))}$ $N e v i l l e T h e t a N (z, m) = \frac{\sqrt{(} π) * (1 + 2 * (\sum_{k = 1}^{\infty} ((- 1)^{k} * q (m)^{k^{2}} * c o s (k * π * z / K (m)))))}{\sqrt{(} 2) * (1 - m)^{(} 1 / 4) * \sqrt{K (m)}}$

其中

$K (m) = E l l i p t i c K (\sqrt{(} m))$
$K^{'} (m) = E l l i p t i c K (\sqrt{(} 1 - m))$
$q (m) = e^{\frac{- π * K (m)}{K^{'} (m)}}$

尼维尔Θ函数也可以通过雅可比Θ函数的傅里叶级数来定义，并使得尼维尔Θ函数可以进一步被用于定义相对应的雅可比椭圆函数。

θ_{c} (z, m) = \frac{\sqrt{2 π} q (m)^{1 / 4}}{m^{1 / 4} \sqrt{K (m)}} \sum_{k = 0}^{\infty} (q (m))^{k (k + 1)} \cos (\frac{(2 k + 1) π z}{2 K (m)})

θ_{d} (z, m) = \frac{\sqrt{2 π}}{2 \sqrt{K (m)}} (1 + 2 \sum_{k = 1}^{\infty} (q (m))^{k^{2}} \cos (\frac{π z k}{K (m)}))

θ_{n} (z, m) = \frac{\sqrt{2 π}}{2 (1 - m)^{1 / 4} \sqrt{K (m)}} (1 + 2 \sum_{k = 1}^{\infty} (- 1)^{k} (q (m))^{k^{2}} \cos (\frac{π z k}{K (m)}))

θ_{s} (z, m) = \frac{\sqrt{2 π} q (m)^{1 / 4}}{m^{1 / 4} (1 - m)^{1 / 4} \sqrt{K (m)}} \sum_{k = 0}^{\infty} (- 1)^{k} (q (m))^{k (k + 1)} \sin (\frac{(2 k + 1) π z}{2 K (m)})

这种定义涉及到第一类完全椭圆积分。

例子

利用Maple,将z=2.5,m=3 代人上列公式，即得：与wolfram math结果相当^[1] ：

$N e v i l l e T h e t a C (2.5, .3) = - .65900466676738154967$
$N e v i l l e T h e t a D (2.5, .3) = 0.95182196661267561994$
$N e v i l l e T h e t a N (2.5, .3) = 1.0526693354651613637$
$N e v i l l e T h e t a S (2.5, .3) = 0.82086879524530400536$

对称关系

$N e v i l l e T h e t a C (z, m) = N e v i l l e T h e t a C (- z, m)$
$N e v i l l e T h e t a D (z, m) = N e v i l l e T h e t a D (- z, m)$
$N e v i l l e T h e t a N (z, m) = N e v i l l e T h e t a N (- z, m)$
$N e v i l l e T h e t a S (z, m) = - N e v i l l e T h e t a S (- z, m)$

级数展开

$N e v i l l e T h e t a C (z, 1 / 2) = .9998 - .3641 * z^{2} + 0.2466 e - 1 * z^{4} - 0.1210 e - 2 * z^{6} + 0.8707 e - 4 * z^{8} + O (z^{1} 0)$
$N e v i l l e T h e t a D (z, 1 / 2) = .9995 - .1143 * z^{2} + 0.2736 e - 1 * z^{4} - 0.2629 e - 2 * z^{6} + 0.1368 e - 3 * z^{8} + O (z^{1} 0)$
$N e v i l l e T h e t a N (z, 1 / 2) = 1.000 + .1358 * z^{2} - 0.3244 e - 1 * z^{4} + 0.3093 e - 2 * z^{6} - 0.1561 e - 3 * z^{8} + O (z^{1} 0)$
$N e v i l l e T h e t a S (z, 1 / 2) = 1.000 * z - .1142 * z^{3} + 0.2358 e - 2 * z^{5} + 0.2276 e - 3 * z^{7} - 0.2630 e - 4 * z^{9} + O (z^{1} 1)$

与其他特殊函数关系

$N e v i l l e T h e t a C (z, m) = \sqrt{2} \sqrt{π} \sqrt[4]{e^{- \frac{π 𝐸 𝑙 𝑙 𝑖 𝑝 𝑡 𝑖 𝑐 𝐾 (\sqrt{1 - m})}{𝐸 𝑙 𝑙 𝑖 𝑝 𝑡 𝑖 𝑐 𝐾 (\sqrt{m})}}} \sum_{k = 0}^{\infty} {(e^{- \frac{π 𝐸 𝑙 𝑙 𝑖 𝑝 𝑡 𝑖 𝑐 𝐾 (\sqrt{1 - m})}{𝐸 𝑙 𝑙 𝑖 𝑝 𝑡 𝑖 𝑐 𝐾 (\sqrt{m})}})}^{k (k + 1)} (1 / 2 \frac{(2 k + 1) π z}{𝐸 𝑙 𝑙 𝑖 𝑝 𝑡 𝑖 𝑐 𝐾 (\sqrt{m})} + 1 / 2 π) M (1, 2, 2 i (1 / 2 \frac{(2 k + 1) π z}{𝐸 𝑙 𝑙 𝑖 𝑝 𝑡 𝑖 𝑐 𝐾 (\sqrt{m})} + 1 / 2 π)) {(e^{i (1 / 2 \frac{(2 k + 1) π z}{𝐸 𝑙 𝑙 𝑖 𝑝 𝑡 𝑖 𝑐 𝐾 (\sqrt{m})} + 1 / 2 π)})}^{- 1} \frac{1}{\sqrt{𝐸 𝑙 𝑙 𝑖 𝑝 𝑡 𝑖 𝑐 𝐾 (\sqrt{m})}} \frac{1}{\sqrt[4]{m}}$
$N e v i l l e T h e t a D (z, n) = 1 / 2 \sqrt{2} \sqrt{π} (1 + 2 \sum_{k = 1}^{\infty} {(e^{- \frac{π 𝐸 𝑙 𝑙 𝑖 𝑝 𝑡 𝑖 𝑐 𝐾 (\sqrt{1 - m})}{𝐸 𝑙 𝑙 𝑖 𝑝 𝑡 𝑖 𝑐 𝐾 (\sqrt{m})}})}^{k^{2}} (\frac{k π z}{𝐸 𝑙 𝑙 𝑖 𝑝 𝑡 𝑖 𝑐 𝐾 (\sqrt{m})} + 1 / 2 π) M (1, 2, 2 i (\frac{k π z}{𝐸 𝑙 𝑙 𝑖 𝑝 𝑡 𝑖 𝑐 𝐾 (\sqrt{m})} + 1 / 2 π)) {(e^{i (\frac{k π z}{𝐸 𝑙 𝑙 𝑖 𝑝 𝑡 𝑖 𝑐 𝐾 (\sqrt{m})} + 1 / 2 π)})}^{- 1}) \frac{1}{\sqrt{𝐸 𝑙 𝑙 𝑖 𝑝 𝑡 𝑖 𝑐 𝐾 (\sqrt{m})}}$

$N e v i l l e T h e t a N (z, m) = 1 / 2 \sqrt{2} \sqrt{π} (1 + 2 \sum_{k = 1}^{\infty} {(- 1)}^{k} {(e^{- \frac{π 𝐸 𝑙 𝑙 𝑖 𝑝 𝑡 𝑖 𝑐 𝐾 (\sqrt{1 - m})}{𝐸 𝑙 𝑙 𝑖 𝑝 𝑡 𝑖 𝑐 𝐾 (\sqrt{m})}})}^{k^{2}} (\frac{k π z}{𝐸 𝑙 𝑙 𝑖 𝑝 𝑡 𝑖 𝑐 𝐾 (\sqrt{m})} + 1 / 2 π) M (1, 2, 2 i (\frac{k π z}{𝐸 𝑙 𝑙 𝑖 𝑝 𝑡 𝑖 𝑐 𝐾 (\sqrt{m})} + 1 / 2 π)) {(e^{i (\frac{k π z}{𝐸 𝑙 𝑙 𝑖 𝑝 𝑡 𝑖 𝑐 𝐾 (\sqrt{m})} + 1 / 2 π)})}^{- 1}) \frac{1}{\sqrt[4]{1 - m}} \frac{1}{\sqrt{𝐸 𝑙 𝑙 𝑖 𝑝 𝑡 𝑖 𝑐 𝐾 (\sqrt{m})}}$

$N e v i l l e T h e t a S (z, m) = \sqrt{2} \sqrt{π} \sqrt[4]{e^{- \frac{π 𝐸 𝑙 𝑙 𝑖 𝑝 𝑡 𝑖 𝑐 𝐾 (\sqrt{1 - m})}{𝐸 𝑙 𝑙 𝑖 𝑝 𝑡 𝑖 𝑐 𝐾 (\sqrt{m})}}} \sum_{k = 0}^{\infty} 1 / 2 {(- 1)}^{k} {(e^{- \frac{π 𝐸 𝑙 𝑙 𝑖 𝑝 𝑡 𝑖 𝑐 𝐾 (\sqrt{1 - m})}{𝐸 𝑙 𝑙 𝑖 𝑝 𝑡 𝑖 𝑐 𝐾 (\sqrt{m})}})}^{k (k + 1)} (2 k + 1) π z M (1, 2, \frac{i π z (2 k + 1)}{𝐸 𝑙 𝑙 𝑖 𝑝 𝑡 𝑖 𝑐 𝐾 (\sqrt{m})}) {(𝐸 𝑙 𝑙 𝑖 𝑝 𝑡 𝑖 𝑐 𝐾 (\sqrt{m}))}^{- 1} {(e^{\frac{1 / 2 i π z (2 k + 1)}{𝐸 𝑙 𝑙 𝑖 𝑝 𝑡 𝑖 𝑐 𝐾 (\sqrt{m})}})}^{- 1} \frac{1}{\sqrt[4]{1 - m}} \frac{1}{\sqrt[4]{m}} \frac{1}{\sqrt{𝐸 𝑙 𝑙 𝑖 𝑝 𝑡 𝑖 𝑐 𝐾 (\sqrt{m})}}$