# 内维尔Θ函数

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

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

## 目录

### 例子

• ${\displaystyle NevilleThetaC(2.5,.3)=-.65900466676738154967}$
• ${\displaystyle NevilleThetaD(2.5,.3)=0.95182196661267561994}$
• ${\displaystyle NevilleThetaN(2.5,.3)=1.0526693354651613637}$
• ${\displaystyle NevilleThetaS(2.5,.3)=0.82086879524530400536}$

## 对称关系

• ${\displaystyle NevilleThetaC(z,m)=NevilleThetaC(-z,m)}$
• ${\displaystyle NevilleThetaD(z,m)=NevilleThetaD(-z,m)}$
• ${\displaystyle NevilleThetaN(z,m)=NevilleThetaN(-z,m)}$
• ${\displaystyle NevilleThetaS(z,m)=-NevilleThetaS(-z,m)}$

## 级数展开

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

## 与其他特殊函数关系

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

## 平面图

 Neville ThetaC function Maple plot Neville ThetaD function Maple plot Neville ThetaD function Maple plot Neville ThetaS function Maple plot

## 参考文献

• Milton Abramowitz and Irene Stegun,Handbook of Mathematical Functions, p578, National Bureau of Standards, 1972.
• ^ wolfram math 计算结果