查看“︁内维尔Θ函数”︁的源代码

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

<math> 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}}                           </math>
<math>  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}   }     </math>
<math> 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))  }                           </math>
<math>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)}   } </math>
<math>                            </math>

其中 
*<math>K(m)=EllipticK(\sqrt(m))</math>
*<math>K'(m)=EllipticK(\sqrt(1-m))</math>
*<math>q(m)=e^\frac{-\pi*K(m)}{K'(m)}</math>

尼维尔Θ函数也可以通过[[Θ函数|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]]。

: <math> \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) 
</math>
: <math> \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)
</math>
: <math> \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)
</math>
: <math> \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)
</math>
这种定义涉及到[[椭圆积分|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<ref>{{Cite web |url=http://www.wolframalpha.com/input/?i=NevilleThetaC%282.5%2C0.3%29 |title=wolfram math 计算结果 |accessdate=2015-03-09 |archive-date=2020-06-14 |archive-url=https://web.archive.org/web/20200614013658/https://www.wolframalpha.com/input/?i=NevilleThetaC(2.5,0.3) |dead-url=no }}</ref>
：
*<math>NevilleThetaC(2.5, .3)=-.65900466676738154967        </math>
*<math> NevilleThetaD(2.5, .3)=0.95182196661267561994       </math>
*<math>NevilleThetaN(2.5, .3)=1.0526693354651613637        </math>
*<math>NevilleThetaS(2.5, .3)=0.82086879524530400536        </math>

==对称关系==
*<math>NevilleThetaC(z,m)=NevilleThetaC(-z,m)</math>
*<math>NevilleThetaD(z,m)=NevilleThetaD(-z,m)</math>
*<math>NevilleThetaN(z,m)=NevilleThetaN(-z,m)</math>
*<math>NevilleThetaS(z,m)=-NevilleThetaS(-z,m)</math>
==级数展开==
*<math>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)</math>
*<math>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)                </math>
*<math>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)               </math>
*<math>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)               </math>

==与其他特殊函数关系==
*<math>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}}}
</math>
*<math>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) }}}</math>

*<math>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) }}}
</math>

*<math>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) }}}
</math>

==平面图==
{|
|[[File:NevilleThetaC.png|thumb|Neville ThetaC  function  Maple plot]]
|[[File:NevilleThetaD.png|thumb|Neville ThetaD  function Maple plot]]
|[[File:NevilleThetaN.png|thumb|Neville ThetaD function Maple plot]]
|[[File:NevilleThetaS.png|thumb|Neville ThetaS  function Maple plot]]
|}

==复数3维图==
{|
|[[File:NevilleThetaC Maple complex plot 01.png|200px]]
|[[File:NevilleThetaD Maple complex plot.png|200px]]
|[[File:NevilleThetaN Maple complex plot.png|200px]]
|[[File:NevilleThetaS Maple complex plot.png|200px]]
|}


==外部链接==
*[http://mathworld.wolfram.com/NevilleThetaFunctions.html Wolfram Mathworld, Neville Theta functions] {{Wayback|url=http://mathworld.wolfram.com/NevilleThetaFunctions.html |date=20200319041732 }}

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

[[Category:特殊函数]]