内维尔Θ函数

内维尔Θ函數（Neville Theta functions）共有四个，定义如下： N e v i l l e C ( z , m ) = ( 2 ) ∗ q ( m ) 1 / 4 ∗ ( ∑ k = 0 ∞ ( q ( m ) ( k ∗ ( k + 1 ) ) ∗ c o s ( ( 1 / 2 ) ∗ ( 2 ∗ k + 1 ) ∗ π ∗ z / K ( m ) ) ) ) ( K ( m ) ) ∗ m 1 / 4 {\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}}}} N e v i l l e T h e t a C ( z , m ) = ( 2 ∗ π ) ∗ q ( m ) 1 / 4 ∗ ( ∑ k = 0 ∞ ( q ( m ) k ∗ ( k + 1 ) ∗ c o s ( ( 1 / 2 ) ∗ ( 2 ∗ k + 1 ) ∗ π ∗ z / K ( m ) ) ) ) ( 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}}}} N e v i l l e T h e t a D ( z , m ) = ( ( 1 / 2 ) ∗ π ) ∗ ( 1 + 2 ∗ ( ∑ k = 1 ∞ ( q ( m ) ( k 2 ) ∗ c o s ( k ∗ π ∗ z / K ( m ) ) ) ) ) ( K ( m ) ) {\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))}}} N e v i l l e T h e t a N ( z , m ) = ( π ) ∗ ( 1 + 2 ∗ ( ∑ k = 1 ∞ ( ( − 1 ) k ∗ q ( m ) k 2 ∗ c o s ( k ∗ π ∗ z / K ( m ) ) ) ) ) ( 2 ) ∗ ( 1 − m ) ( 1 / 4 ) ∗ 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 } 其中 K ( m ) = E l l i p t i c K ( ( m ) ) {\displaystyle K(m)=EllipticK({\sqrt {(}}m))} K ′ ( m ) = E l l i p t i c K ( ( 1 − m ) ) {\displaystyle K'(m)=EllipticK({\sqrt {(}}1-m))} q ( m ) = e − π ∗ K ( m ) K ′ ( m ) {\displaystyle 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。 θ c ( z , m ) = 2 π q ( m ) 1 / 4 m 1 / 4 K ( m ) ∑ k = 0 ∞ ( q ( m ) ) k ( k + 1 ) cos ⁡ ( ( 2 k + 1 ) π z 2 K ( m ) ) {\displaystyle \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)} θ d ( z , m ) = 2 π 2 K ( m ) ( 1 + 2 ∑ k = 1 ∞ ( q ( m ) ) k 2 cos ⁡ ( π z k K ( m ) ) ) {\displaystyle \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)} θ n ( z , m ) = 2 π 2 ( 1 − m ) 1 / 4 K ( m ) ( 1 + 2 ∑ k = 1 ∞ ( − 1 ) k ( q ( m ) ) k 2 cos ⁡ ( π z k K ( m ) ) ) {\displaystyle \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)} θ s ( z , m ) = 2 π q ( m ) 1 / 4 m 1 / 4 ( 1 − m ) 1 / 4 K ( m ) ∑ k = 0 ∞ ( − 1 ) k ( q ( m ) ) k ( k + 1 ) sin ⁡ ( ( 2 k + 1 ) π z 2 K ( m ) ) {\displaystyle \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。 利用Maple,将z=2.5,m=3 代人上列公式，即得： 与wolfram math结果相当 ： N e v i l l e T h e t a C ( 2.5 , .3 ) = − .65900466676738154967 {\displaystyle NevilleThetaC(2.5,.3)=-.65900466676738154967} N e v i l l e T h e t a D ( 2.5 , .3 ) = 0.95182196661267561994 {\displaystyle NevilleThetaD(2.5,.3)=0.95182196661267561994} N e v i l l e T h e t a N ( 2.5 , .3 ) = 1.0526693354651613637 {\displaystyle NevilleThetaN(2.5,.3)=1.0526693354651613637} N e v i l l e T h e t a S ( 2.5 , .3 ) = 0.82086879524530400536 {\displaystyle NevilleThetaS(2.5,.3)=0.82086879524530400536} N e v i l l e T h e t a C ( z , m ) = N e v i l l e T h e t a C ( − z , m ) {\displaystyle NevilleThetaC(z,m)=NevilleThetaC(-z,m)} N e v i l l e T h e t a D ( z , m ) = N e v i l l e T h e t a D ( − z , m ) {\displaystyle NevilleThetaD(z,m)=NevilleThetaD(-z,m)} N e v i l l e T h e t a N ( z , m ) = N e v i l l e T h e t a N ( − z , m ) {\displaystyle NevilleThetaN(z,m)=NevilleThetaN(-z,m)} N e v i l l e T h e t a S ( z , m ) = − N e v i l l e T h e t a S ( − z , m ) {\displaystyle NevilleThetaS(z,m)=-NevilleThetaS(-z,m)} N e v i l l e T h e t a C ( z , 1 / 2 ) = .9998 − .3641 ∗ z 2 + 0.2466 e − 1 ∗ z 4 − 0.1210 e − 2 ∗ z 6 + 0.8707 e − 4 ∗ z 8 + O ( z 1 0 ) {\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)} N e v i l l e T h e t a D ( z , 1 / 2 ) = .9995 − .1143 ∗ z 2 + 0.2736 e − 1 ∗ z 4 − 0.2629 e − 2 ∗ z 6 + 0.1368 e − 3 ∗ 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)} N e v i l l e T h e t a N ( z , 1 / 2 ) = 1.000 + .1358 ∗ z 2 − 0.3244 e − 1 ∗ z 4 + 0.3093 e − 2 ∗ z 6 − 0.1561 e − 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)} N e v i l l e T h e t a S ( z , 1 / 2 ) = 1.000 ∗ z − .1142 ∗ z 3 + 0.2358 e − 2 ∗ z 5 + 0.2276 e − 3 ∗ z 7 − 0.2630 e − 4 ∗ z 9 + O ( z 1 1 ) {\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)} N e v i l l e T h e t a C ( z , m ) = 2 π e − π E l l i p t i c K ( 1 − m ) E l l i p t i c K ( m ) 4 ∑ k = 0 ∞ ( e − π E l l i p t i c K ( 1 − m ) E l l i p t i c K ( m ) ) k ( k + 1 ) ( 1 / 2 ( 2 k + 1 ) π z E l l i p t i c K ( m ) + 1 / 2 π ) M ( 1 , 2 , 2 i ( 1 / 2 ( 2 k + 1 ) π z E l l i p t i c K ( m ) + 1 / 2 π ) ) ( e i ( 1 / 2 ( 2 k + 1 ) π z E l l i p t i c K ( m ) + 1 / 2 π ) ) − 1 1 E l l i p t i c K ( m ) 1 m 4 {\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}}}} N e v i l l e T h e t a D ( z , n ) = 1 / 2 2 π ( 1 + 2 ∑ k = 1 ∞ ( e − π E l l i p t i c K ( 1 − m ) E l l i p t i c K ( m ) ) k 2 ( k π z E l l i p t i c K ( m ) + 1 / 2 π ) M ( 1 , 2 , 2 i ( k π z E l l i p t i c K ( m ) + 1 / 2 π ) ) ( e i ( k π z E l l i p t i c K ( m ) + 1 / 2 π ) ) − 1 ) 1 E l l i p t i c K ( 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)}}}} N e v i l l e T h e t a N ( z , m ) = 1 / 2 2 π ( 1 + 2 ∑ k = 1 ∞ ( − 1 ) k ( e − π E l l i p t i c K ( 1 − m ) E l l i p t i c K ( m ) ) k 2 ( k π z E l l i p t i c K ( m ) + 1 / 2 π ) M ( 1 , 2 , 2 i ( k π z E l l i p t i c K ( m ) + 1 / 2 π ) ) ( e i ( k π z E l l i p t i c K ( m ) + 1 / 2 π ) ) − 1 ) 1 1 − m 4 1 E l l i p t i c K ( m ) {\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)}}}} N e v i l l e T h e t a S ( z , m ) = 2 π e − π E l l i p t i c K ( 1 − m ) E l l i p t i c K ( m ) 4 ∑ k = 0 ∞ 1 / 2 ( − 1 ) k ( e − π E l l i p t i c K ( 1 − m ) E l l i p t i c K ( m ) ) k ( k + 1 ) ( 2 k + 1 ) π z M ( 1 , 2 , i π z ( 2 k + 1 ) E l l i p t i c K ( m ) ) ( E l l i p t i c K ( m ) ) − 1 ( e 1 / 2 i π z ( 2 k + 1 ) E l l i p t i c K ( m ) ) − 1 1 1 − m 4 1 m 4 1 E l l i p t i c K ( m ) {\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)}}}} Wolfram Mathworld, Neville Theta functions （页面存档备份，存于互联网档案馆） Milton Abramowitz and Irene Stegun,Handbook of Mathematical Functions, p578, National Bureau of Standards, 1972. wolfram math 计算结果. [2015-03-09]. （原始内容存档于2020-06-14）.