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}{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^[1] ：

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

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

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

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

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