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)(2k+1)\piz/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)(2k+1)\piz/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\piz/K(m))))) }{\sqrt(K(m)) }
NevilleThetaN(z, m)=\frac{\sqrt(\pi)(1+2(\sum_{k=1}^{\infty}((-1)^kq(m)^{k^2}cos(k\piz/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{-\piK(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-.3641z^2+0.2466e-1z^4-0.1210e-2z^6+0.8707e-4*z^8+O(z^10)
NevilleThetaD(z,1/2)= .9995-.1143z^2+0.2736e-1z^4-0.2629e-2z^6+0.1368e-3*z^8+O(z^10)
NevilleThetaN(z,1/2)= 1.000+.1358z^2-0.3244e-1z^4+0.3093e-2z^6-0.1561e-3*z^8+O(z^10)
NevilleThetaS(z,1/2)= 1.000z-.1142z^3+0.2358e-2z^5+0.2276e-3z^7-0.2630e-4z^9+O(z^11)

与其他特殊函数关系
*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) }}}

平面图
复数3维图
外部链接
*[http://mathworld.wolfram.com/NevilleThetaFunctions.html Wolfram Mathworld, Neville Theta functions]

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