卡咯提的-昆達利尼函數

卡咯提的-昆達利尼函數（Carotid-Kundalili Function）定義如下^[1]

Carotid-Kundalini function

${\displaystyle K(n,x)=cos(n*x*arccos(x))}$

與其他特殊函數的關係

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

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

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

函數展開

${\displaystyle K(n,x)\approx {1-(1/8)*n^{2}*\pi ^{2}*x^{2}+(1/2)*n^{2}*\pi *x^{3}+((1/384)*n^{4}*\pi ^{4}-(1/2)*n^{2})*x^{4}+(-(1/48)*n^{4}*\pi ^{3}+(1/12)*n^{2}*\pi )*x^{5}+O(x^{6})}}$

帕德近似

帕德近似:

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

外部連結

• Carotid-Kundalini function （頁面存檔備份，存於網際網路檔案館）

參考文獻

1. Weisstein, Eric W. "Carotid-Kundalini Function." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Carotid-KundaliniFunction.html （頁面存檔備份，存於網際網路檔案館）