卡咯提的-昆达利尼函数

卡咯提的-昆达利尼函数（Carotid-Kundalili Function）定义如下

K(n, x)= cos(nxarccos(x))

与其他特殊函数的关系

• K(n,x)=\frac{ -(1/2I)(-1+exp(I(2nxarccos(x)+Pi)))}{exp((1/2I)(2nx*arccos(x)+Pi))}
• K(n,x)= \frac{(nxcos^1(x)+\pi/2)KummerM(1,2,I(2nxarccos(x)+\pi))}{exp(I(2nxarccos(x)+2\pi/2))}

*-n{x}^{2}{\it HeunB} \left( 2,0,0,0,\sqrt {2}\sqrt {1/2\,i \left( 2\,n
x \left( 1/2\,\pi -x{\it HeunC} \left( 0,1/2,0,0,1/4,{\frac {{x}^{2}}{
{x}^{2}-1}} \right) {\frac {1}{\sqrt {1-{x}^{2}}}} \right) +\pi
\right) } \right) {\it HeunC} \left( 0,1/2,0,0,1/4,{\frac {{x}^{2}}{{
x}^{2}-1}} \right) {\frac {1}{\sqrt {1-{x}^{2}}}} \left( {{\rm e}^{-1/
2\,i \left( -nx\pi \,\sqrt {1-{x}^{2}}+2\,n{x}^{2}{\it HeunC} \left( 0
,1/2,0,0,1/4,{\frac {{x}^{2}}{{x}^{2}-1}} \right) -\pi \,\sqrt {1-{x}^
{2}} \right) {\frac {1}{\sqrt {1-{x}^{2}}}}}} \right) ^{-1}+1/2\,\pi
\, \left( nx+1 \right) {\it HeunB} \left( 2,0,0,0,\sqrt {2}\sqrt {1/2
\,i \left( 2\,nx \left( 1/2\,\pi -x{\it HeunC} \left( 0,1/2,0,0,1/4,{
\frac {{x}^{2}}{{x}^{2}-1}} \right) {\frac {1}{\sqrt {1-{x}^{2}}}}
\right) +\pi \right) } \right) \left( {{\rm e}^{-1/2\,i \left( -nx
\pi \,\sqrt {1-{x}^{2}}+2\,n{x}^{2}{\it HeunC} \left( 0,1/2,0,0,1/4,{
\frac {{x}^{2}}{{x}^{2}-1}} \right) -\pi \,\sqrt {1-{x}^{2}} \right) {
\frac {1}{\sqrt {1-{x}^{2}}}}}} \right) ^{-1}

K(n,x)={\frac {-i \left( 2\,nx\arccos \left( x \right) +\pi \right)
{{\rm \it WhittakerM}\left(0,\,1/2,\,i \left( 2\,nx\arccos \left( x \right) +\pi \right) \right)}
}{4\,nx\arccos \left( x \right) +2\,\pi }}

函数展开
K(n,x) \approx {1-(1/8)n^2\pi^2x^2+(1/2)n^2\pix^3+((1/384)n^4\pi^4-(1/2)n^2)x^4+(-(1/48)n^4\pi^3+(1/12)n^2\pi)*x^5+O(x^6)}
帕德近似
帕德近似:

K(n,x) \approx \left\{ {\frac { 1800.0+ \left( - 36.4\,{n}^{4}+ 516.0 \right) x+
\left( - 46.3\,{n}^{4}- 1830.0\,{n}^{2}- 71.0 \right) {x}^{2}+
\left( 1820.0\,{n}^{2}+ 37.4\,{n}^{6}- 44.3\,{n}^{4}+ 81.9 \right) {
x}^{3}}{ 1800.0+ \left( - 36.4\,{n}^{4}+ 516.0 \right) x+ \left( -
46.3\,{n}^{4}+ 368.0\,{n}^{2}- 71.0 \right) {x}^{2}+ \left( - 7.48\,{
n}^{6}- 44.3\,{n}^{4}- 363.0\,{n}^{2}+ 81.9 \right) {x}^{3}}}
\right\}

外部链接
*[http://smallsats.org/2014/01/03/carotid-kundalini-function/ Carotid-Kundalini function]

参考文献