# 拉梅函数

Lame function Maple animation plot

${\displaystyle {\frac {d^{2}w}{dz^{2}}}+(A+v(v+1)k^{2}sn^{2}(z,k))w=0}$+ 此拉梅方程的正则奇点在复数平面的${\displaystyle 2pK+(2q+1)*iK'}$ 其中 p,q ∈Z,K代表模数为k的完全椭圆积分，K'代表模数为${\displaystyle k'={\sqrt {1-k^{2}}}}$的完全椭圆积分。

${\displaystyle {\frac {d^{2}\Lambda }{ds^{2}}}+{\frac {1}{2}}*({\frac {1}{2}}+{\frac {1}{s-1}}+{\frac {1}{s-h}})*{\frac {d\Lambda }{ds}}-{\frac {n(n+1)s+H}{4s(s-1)(s-h)}}*\Lambda =0}$

${\displaystyle h=k^{-2}={\frac {a^{2}-c^{2}}{a^{2}-b^{2}}}}$,

${\displaystyle H=hA}$

${\displaystyle h>1}$ 此傅克型方程有四个正则奇点${\displaystyle 0,1,h,\infty }$

${\displaystyle {\frac {d^{2}\Lambda }{dz^{2}}}+[H-n(n+1)\wp (z)]\Lambda =0}$

${\displaystyle \zeta ={\frac {1}{2}}\pi -amz}$

${\displaystyle [1-(kcos\zeta )^{2}]{\frac {d^{2}\Lambda }{d\Lambda ^{2}}}}$${\displaystyle +k^{2}cos\zeta \sin \zeta {\frac {d\Lambda }{d\zeta }}+[h-n(n+1)(kcos\zeta )^{2}]\Lambda =0}$

## 拉梅方程的本征值

${\displaystyle a_{v}^{2m}(k^{2})}$  2K
${\displaystyle a_{v}^{2m+1}(k^{2})}$  4K
${\displaystyle b_{v}^{2m}(k^{2})}$  4K
${\displaystyle b_{v}^{2m+1}(k^{2})}$  2K

## 拉梅函数

${\displaystyle a_{v}^{2m}(k^{2})}$  2K ${\displaystyle Ec_{v}^{2m}(z,k^{2})}$
${\displaystyle a_{v}^{2m+1}(k^{2})}$  4K ${\displaystyle Ec_{v}^{2m+1}(z,k^{2})}$
${\displaystyle b_{v}^{2m}(k^{2})}$  4K ${\displaystyle Es_{v}^{2m+1}(z,k^{2})}$
${\displaystyle b_{v}^{2m+1}(k^{2})}$  2K ${\displaystyle Es_{v}^{2m+2}(z,k^{2})}$

## 拉梅函数是Heun函数的特例

Heun方程 ${\displaystyle gh:={\frac {d^{2}(y(z)}{dz^{2}}}+({\frac {\gamma }{z}}+{\frac {\delta }{z-1}}+{\frac {\epsilon }{z-a}})*{\frac {d(y(z)}{dz}}+(\alpha *\beta *z-q)*y(z)/(z*(z-1)*(z-a))=0}$

${\displaystyle {\frac {d^{2}(y(z)}{dz^{2}}}+(1/2*(1/z+1/(z-1)+1/(z-a)))*{\frac {d(y(z)}{dz}}+(1/4)*(a*h-\nu *(\nu +1)*z)*y(z)/(z*(z-1)*(z-a))=0}$

## 拉梅方程的Heun函数解

${\displaystyle +_{C}2*{\sqrt {(}}z)*HeunG(a,1/4+(1/4*(-h+1))*a,1+(1/2)*\nu ,1/2-(1/2)*\nu ,3/2,1/2,z)}$  其中二个HeunG函数是线性无关的。

## 拉梅函数的幂级数展开

${\displaystyle y(z)=\sum _{v=0}^{\infty }a_{v}*z^{\rho +v}}$

${\displaystyle y(z)={1.2+2.3*{\sqrt {(}}z)-.600*h*z-(.383*(a*h-1.*a-1.))*z^{(}3/2)/a+(0.500e-1*(-4.*a*h-4.*h+a*h^{2}+2.*\nu ^{2}+2.*\nu ))*z^{2}/a+(0.192e-1*(-10.*a^{2}*h+9.*a^{2}+6.*a-10.*a*h+9.+a^{2}*h^{2}+6.*\nu *a+6.*\nu ^{2}*a))*z^{(}5/2)/a^{2}+O(z^{3})}}$

## 参考文献

1. ^ 王竹溪 第572页
2. ^ Whittaker p554
3. Erdelyi p55
4. ^ Erdelyi p 56
5. ^ Frank Oliver p685
6. ^ Frank, p684
7. ^ Frank Oliver,p713
8. ^ 王竹溪 第573页
• 王竹溪 郭敦仁 《特殊函数概论》 北京大学出版 2000
• Whittaker and Watson, A Course of Modern Analysis 1920， Cambridge University Press
• Erdelyi, Higher Transcendental Functions Vol III