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))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac87d0c1ba79c2ce46deabc07aacd79e88e6e97)
![{\displaystyle K'(m)=EllipticK({\sqrt {(}}1-m))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dcd4a325c520d74c5b21e49d7defaebdd7c425bb)
![{\displaystyle q(m)=e^{\frac {-\pi *K(m)}{K'(m)}}}](https://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)}](https://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)}](https://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)}](https://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)}](https://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。
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[1]
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![{\displaystyle NevilleThetaC(2.5,.3)=-.65900466676738154967}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc316d0955ef744fdac43dc56e7dbe4a30204d1a)
![{\displaystyle NevilleThetaD(2.5,.3)=0.95182196661267561994}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e06097b4de2b9299913881fc051a20d62978b3f)
![{\displaystyle NevilleThetaN(2.5,.3)=1.0526693354651613637}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe1438e87396f8da53160c50e6f5d498d227d482)
![{\displaystyle NevilleThetaS(2.5,.3)=0.82086879524530400536}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4225d1fe8124877f3e299365b52bfc7c5942552)
对称关系
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级数展开
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与其他特殊函数关系
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平面图
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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复数3维图
外部链接
参考文献
- 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).