内维尔Θ函数

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

 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。

:  \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)

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

## 对称关系
*NevilleThetaC(z,m)=NevilleThetaC(-z,m)
*NevilleThetaD(z,m)=NevilleThetaD(-z,m)
*NevilleThetaN(z,m)=NevilleThetaN(-z,m)
*NevilleThetaS(z,m)=-NevilleThetaS(-z,m)
## 级数展开
*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)               

## 与其他特殊函数关系
*NevilleThetaC(z,m)=\sqrt {2}\sqrt {\pi }\sqrt [4]}\sum _{k=0}^{\infty } \left(  \right) ^{k \left( k+1 \right) } \left( 1/2\,{
\frac { \left( 2\,k+1 \right) \pi \,z}+1/2\,\pi  \right) 
}

*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) }}} \right) ^
 \left( {\frac {k\pi \,z}+
1/2\,\pi  \right) 
{{\rm M}\left(1,\,2,\,2\,i \left( {\frac {k\pi \,z}+1/2\,\pi  \right) \right)}
 \left(  \right) ^{-1} \right) {
\frac {1}{\sqrt }

*NevilleThetaN(z,m)=1/2\,\sqrt {2}\sqrt {\pi } \left( 1+2\,\sum _{k=1}^{\infty } \left( -1
 \right) ^{k} \left( 
 \right) ^ \left( {\frac {k\pi \,z}+1/2\,\pi  \right) 
}{\frac {1}{\sqrt }


*NevilleThetaS(z,m)=\sqrt {2}\sqrt {\pi }\sqrt [4]}\sum _{k=0}^{\infty }1/2\, \left( -1 \right) ^{k}
 \left(  \right) ^{k
 \left( k+1 \right) } \left( 2\,k+1 \right) \pi \,z
}{\frac {1}{\sqrt }


## 平面图

## 复数3维图

## 外部链接
*Wolfram Mathworld, Neville Theta functions 

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