内维尔Θ函數(Neville Theta functions)共有四个,定义如下:
其中
![{\displaystyle K(m)=EllipticK({\sqrt {(}}m))}](//wikimedia.org/api/rest_v1/media/math/render/svg/0ac87d0c1ba79c2ce46deabc07aacd79e88e6e97)
![{\displaystyle K'(m)=EllipticK({\sqrt {(}}1-m))}](//wikimedia.org/api/rest_v1/media/math/render/svg/dcd4a325c520d74c5b21e49d7defaebdd7c425bb)
![{\displaystyle q(m)=e^{\frac {-\pi *K(m)}{K'(m)}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/8918826055198e2845c73835be5df29864c82497)
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。
![{\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)}](//wikimedia.org/api/rest_v1/media/math/render/svg/23c24bd9586d43e40359544742532cde9e54542f)
![{\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)}](//wikimedia.org/api/rest_v1/media/math/render/svg/d839d12fde64c9c4137ad8884f6823b3517f5003)
![{\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)}](//wikimedia.org/api/rest_v1/media/math/render/svg/9b54ebbb8587e676aac5348e5d2879dddfd0dcf9)
![{\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)}](//wikimedia.org/api/rest_v1/media/math/render/svg/886ed86bc171fc62c4d5e250ff58b4afeef6bebb)
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[1]
:
![{\displaystyle NevilleThetaC(2.5,.3)=-.65900466676738154967}](//wikimedia.org/api/rest_v1/media/math/render/svg/bc316d0955ef744fdac43dc56e7dbe4a30204d1a)
![{\displaystyle NevilleThetaD(2.5,.3)=0.95182196661267561994}](//wikimedia.org/api/rest_v1/media/math/render/svg/3e06097b4de2b9299913881fc051a20d62978b3f)
![{\displaystyle NevilleThetaN(2.5,.3)=1.0526693354651613637}](//wikimedia.org/api/rest_v1/media/math/render/svg/fe1438e87396f8da53160c50e6f5d498d227d482)
![{\displaystyle NevilleThetaS(2.5,.3)=0.82086879524530400536}](//wikimedia.org/api/rest_v1/media/math/render/svg/d4225d1fe8124877f3e299365b52bfc7c5942552)
![{\displaystyle NevilleThetaC(z,m)=NevilleThetaC(-z,m)}](//wikimedia.org/api/rest_v1/media/math/render/svg/4e16917f9266d44eb2038d379914647954b0fb21)
![{\displaystyle NevilleThetaD(z,m)=NevilleThetaD(-z,m)}](//wikimedia.org/api/rest_v1/media/math/render/svg/282ce0380ae0ebc06c78a926e67892839f427aab)
![{\displaystyle NevilleThetaN(z,m)=NevilleThetaN(-z,m)}](//wikimedia.org/api/rest_v1/media/math/render/svg/49fd87e2301c7e7d28f1d786a8f19add9794e68e)
![{\displaystyle NevilleThetaS(z,m)=-NevilleThetaS(-z,m)}](//wikimedia.org/api/rest_v1/media/math/render/svg/f6ab8bcf0264fb62447e2f53415a8884112857f6)
![{\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)}](//wikimedia.org/api/rest_v1/media/math/render/svg/b4c26472b1b44b9fb3ba14792fc1c00be1100feb)
![{\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)}](//wikimedia.org/api/rest_v1/media/math/render/svg/9f978de6076ba239eddfc87bc2453c5f7297e320)
![{\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)}](//wikimedia.org/api/rest_v1/media/math/render/svg/2fedb252b39c8778c6ab5d942e818505a4efa1a2)
![{\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)}](//wikimedia.org/api/rest_v1/media/math/render/svg/19c2db314db2ffe319047feb15b31b2913ffc436)
![{\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}}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/a41fca93eb297b5ce98334c3e1a295fed4cc3d62)
![{\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)}}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/88317564f63cf29b2d19475362cfd99dc66f5fc3)
![{\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)}}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/d0c3d45ccf4eed42fcac01b5a00ba0cbe98c4baa)
![{\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)}}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/034646b9b7bf8483712fc3347b82228097fe004f)
Neville ThetaC function Maple plot
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Neville ThetaD function Maple plot
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Neville ThetaD function Maple plot
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Neville ThetaS function Maple plot
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- Milton Abramowitz and Irene Stegun,Handbook of Mathematical Functions, p578, National Bureau of Standards, 1972.