內維爾Θ函數

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。

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

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

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

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

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

- $$NevilleThetaC(z,1/2)=.9998-.3641*z^2+0.2466e-1*z^4-0.1210e-2*z^6+0.8707e-4*z^8+O(z^10)$$
- $$NevilleThetaD(z,1/2)= .9995-.1143*z^2+0.2736e-1*z^4-0.2629e-2*z^6+0.1368e-3*z^8+O(z^10)$$
- $$NevilleThetaN(z,1/2)= 1.000+.1358*z^2-0.3244e-1*z^4+0.3093e-2*z^6-0.1561e-3*z^8+O(z^10)$$
- $$NevilleThetaS(z,1/2)= 1.000*z-.1142*z^3+0.2358e-2*z^5+0.2276e-3*z^7-0.2630e-4*z^9+O(z^11)$$

## 與其他特殊函數關係

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

## 外部連結

- Wolfram Mathworld, Neville Theta functions