内维尔Θ函数

**內維爾 Θ 函數**（Neville Theta functions）共有四個，定義如下：

 $NevilleC(z,m)={\frac {{\sqrt {(}}2)*q(m)^{1/4}*(\sum _{k=0}^{\infty }(q(m)^{(}k*(k+1))*cos((1/2)*(2*k+1)*\pi *z/K(m))))}{{\sqrt {(}}K(m))*m^{1/4}}}$
 $NevilleThetaC(z,m)={\frac {{\sqrt {(}}2*\pi )*q(m)^{1/4}*(\sum _{k=0}^{\infty }(q(m)^{k*(k+1)}*cos((1/2)*(2*k+1)*\pi *z/K(m))))}{{\sqrt {(}}K(m))*m^{1/4}}}$
 $NevilleThetaD(z,m)={\frac {{\sqrt {(}}(1/2)*\pi )*(1+2*(\sum _{k=1}^{\infty }(q(m)^{(}k^{2})*cos(k*\pi *z/K(m)))))}{{\sqrt {(}}K(m))}}$
 $NevilleThetaN(z,m)={\frac {{\sqrt {(}}\pi )*(1+2*(\sum _{k=1}^{\infty }((-1)^{k}*q(m)^{k^{2}}*cos(k*\pi *z/K(m)))))}{{\sqrt {(}}2)*(1-m)^{(}1/4)*{\sqrt {K(m)}}}}$
 $$

其中

 * $K(m)=EllipticK({\sqrt {(}}m))$
* $K'(m)=EllipticK({\sqrt {(}}1-m))$
* $q(m)=e^{\frac {-\pi *K(m)}{K'(m)}}$

尼維爾 Θ 函數也可以通過雅可比 Θ 函數的傅立葉級數來定義，並使得尼維爾 Θ 函數可以進一步被用於定義相對應的雅可比橢圓函數。

　
: $\theta _{c}(z,m)={\frac {{\sqrt {2\pi }}\,q(m)^{1/4}}{m^{1/4}{\sqrt {K(m)}}}}\,\,\sum _{k=0}^{\infty }(q(m))^{k(k+1)}\cos \left({\frac {(2k+1)\pi z}{2K(m)}}\right)$

　
: $\theta _{d}(z,m)={\frac {\sqrt {2\pi }}{2{\sqrt {K(m)}}}}\,\,\left(1+2\,\sum _{k=1}^{\infty }(q(m))^{k^{2}}\cos \left({\frac {\pi zk}{K(m)}}\right)\right)$

　
: $\theta _{n}(z,m)={\frac {\sqrt {2\pi }}{2(1-m)^{1/4}{\sqrt {K(m)}}}}\,\,\left(1+2\sum _{k=1}^{\infty }(-1)^{k}(q(m))^{k^{2}}\cos \left({\frac {\pi zk}{K(m)}}\right)\right)$

　
: $\theta _{s}(z,m)={\frac {{\sqrt {2\pi }}\,q(m)^{1/4}}{m^{1/4}(1-m)^{1/4}{\sqrt {K(m)}}}}\,\,\sum _{k=0}^{\infty }(-1)^{k}(q(m))^{k(k+1)}\sin \left({\frac {(2k+1)\pi z}{2K(m)}}\right)$

這種定義涉及到第一類完全橢圓積分。

### 例子

利用 Maple, 將 z=2.5,m=3 代人上列公式，即得： 與 wolfram math 結果相當
：

 * $NevilleThetaC(2.5,.3)=-.65900466676738154967$
* $NevilleThetaD(2.5,.3)=0.95182196661267561994$
* $NevilleThetaN(2.5,.3)=1.0526693354651613637$
* $NevilleThetaS(2.5,.3)=0.82086879524530400536$

## 對稱關係

 * $NevilleThetaC(z,m)=NevilleThetaC(-z,m)$
* $NevilleThetaD(z,m)=NevilleThetaD(-z,m)$
* $NevilleThetaN(z,m)=NevilleThetaN(-z,m)$
* $NevilleThetaS(z,m)=-NevilleThetaS(-z,m)$

## 級數展開

 * $NevilleThetaC(z,1/2)=.9998-.3641*z^{2}+0.2466e-1*z^{4}-0.1210e-2*z^{6}+0.8707e-4*z^{8}+O(z^{1}0)$
* $NevilleThetaD(z,1/2)=.9995-.1143*z^{2}+0.2736e-1*z^{4}-0.2629e-2*z^{6}+0.1368e-3*z^{8}+O(z^{1}0)$
* $NevilleThetaN(z,1/2)=1.000+.1358*z^{2}-0.3244e-1*z^{4}+0.3093e-2*z^{6}-0.1561e-3*z^{8}+O(z^{1}0)$
* $NevilleThetaS(z,1/2)=1.000*z-.1142*z^{3}+0.2358e-2*z^{5}+0.2276e-3*z^{7}-0.2630e-4*z^{9}+O(z^{1}1)$

## 與其他特殊函數關係

 * $NevilleThetaC(z,m)={\sqrt {2}}{\sqrt {\pi }}{\sqrt[{4}]{{\rm {e}}^{-{\frac {\pi \,{\it {EllipticK}}\left({\sqrt {1-m}}\right)}{{\it {EllipticK}}\left({\sqrt {m}}\right)}}}}}\sum _{k=0}^{\infty }\left({{\rm {e}}^{-{\frac {\pi \,{\it {EllipticK}}\left({\sqrt {1-m}}\right)}{{\it {EllipticK}}\left({\sqrt {m}}\right)}}}}\right)^{k\left(k+1\right)}\left(1/2\,{\frac {\left(2\,k+1\right)\pi \,z}{{\it {EllipticK}}\left({\sqrt {m}}\right)}}+1/2\,\pi \right){{\rm {M}}\left(1,\,2,\,2\,i\left(1/2\,{\frac {\left(2\,k+1\right)\pi \,z}{{\it {EllipticK}}\left({\sqrt {m}}\right)}}+1/2\,\pi \right)\right)}\left({{\rm {e}}^{i\left(1/2\,{\frac {\left(2\,k+1\right)\pi \,z}{{\it {EllipticK}}\left({\sqrt {m}}\right)}}+1/2\,\pi \right)}}\right)^{-1}{\frac {1}{\sqrt {{\it {EllipticK}}\left({\sqrt {m}}\right)}}}{\frac {1}{\sqrt[{4}]{m}}}$
* $NevilleThetaD(z,n)=1/2\,{\sqrt {2}}{\sqrt {\pi }}\left(1+2\,\sum _{k=1}^{\infty }\left({{\rm {e}}^{-{\frac {\pi \,{\it {EllipticK}}\left({\sqrt {1-m}}\right)}{{\it {EllipticK}}\left({\sqrt {m}}\right)}}}}\right)^{{k}^{2}}\left({\frac {k\pi \,z}{{\it {EllipticK}}\left({\sqrt {m}}\right)}}+1/2\,\pi \right){{\rm {M}}\left(1,\,2,\,2\,i\left({\frac {k\pi \,z}{{\it {EllipticK}}\left({\sqrt {m}}\right)}}+1/2\,\pi \right)\right)}\left({{\rm {e}}^{i\left({\frac {k\pi \,z}{{\it {EllipticK}}\left({\sqrt {m}}\right)}}+1/2\,\pi \right)}}\right)^{-1}\right){\frac {1}{\sqrt {{\it {EllipticK}}\left({\sqrt {m}}\right)}}}$
 * $NevilleThetaN(z,m)=1/2\,{\sqrt {2}}{\sqrt {\pi }}\left(1+2\,\sum _{k=1}^{\infty }\left(-1\right)^{k}\left({{\rm {e}}^{-{\frac {\pi \,{\it {EllipticK}}\left({\sqrt {1-m}}\right)}{{\it {EllipticK}}\left({\sqrt {m}}\right)}}}}\right)^{{k}^{2}}\left({\frac {k\pi \,z}{{\it {EllipticK}}\left({\sqrt {m}}\right)}}+1/2\,\pi \right){{\rm {M}}\left(1,\,2,\,2\,i\left({\frac {k\pi \,z}{{\it {EllipticK}}\left({\sqrt {m}}\right)}}+1/2\,\pi \right)\right)}\left({{\rm {e}}^{i\left({\frac {k\pi \,z}{{\it {EllipticK}}\left({\sqrt {m}}\right)}}+1/2\,\pi \right)}}\right)^{-1}\right){\frac {1}{\sqrt[{4}]{1-m}}}{\frac {1}{\sqrt {{\it {EllipticK}}\left({\sqrt {m}}\right)}}}$
 * $NevilleThetaS(z,m)={\sqrt {2}}{\sqrt {\pi }}{\sqrt[{4}]{{\rm {e}}^{-{\frac {\pi \,{\it {EllipticK}}\left({\sqrt {1-m}}\right)}{{\it {EllipticK}}\left({\sqrt {m}}\right)}}}}}\sum _{k=0}^{\infty }1/2\,\left(-1\right)^{k}\left({{\rm {e}}^{-{\frac {\pi \,{\it {EllipticK}}\left({\sqrt {1-m}}\right)}{{\it {EllipticK}}\left({\sqrt {m}}\right)}}}}\right)^{k\left(k+1\right)}\left(2\,k+1\right)\pi \,z{{\rm {M}}\left(1,\,2,\,{\frac {i\pi \,z\left(2\,k+1\right)}{{\it {EllipticK}}\left({\sqrt {m}}\right)}}\right)}\left({\it {EllipticK}}\left({\sqrt {m}}\right)\right)^{-1}\left({{\rm {e}}^{\frac {1/2\,i\pi \,z\left(2\,k+1\right)}{{\it {EllipticK}}\left({\sqrt {m}}\right)}}}\right)^{-1}{\frac {1}{\sqrt[{4}]{1-m}}}{\frac {1}{\sqrt[{4}]{m}}}{\frac {1}{\sqrt {{\it {EllipticK}}\left({\sqrt {m}}\right)}}}$

## 平面圖

## 複數 3 維圖

 

## 外部連結

* Wolfram Mathworld, Neville Theta functions （頁面存檔備份，存於網際網路檔案館）

## 參考文獻

* Milton Abramowitz and Irene Stegun,Handbook of Mathematical Functions, p578, National Bureau of Standards, 1972.