卡咯提的-昆达利尼函数（Carotid-Kundalili Function）定义如下[1]
K ( n , x ) = c o s ( n ∗ x ∗ a r c c o s ( x ) ) {\displaystyle K(n,x)=cos(n*x*arccos(x))}
K ( n , x ) = − i ( 2 n x arccos ( x ) + π ) W h i t t a k e r M ( 0 , 1 / 2 , i ( 2 n x arccos ( x ) + π ) ) 4 n x arccos ( x ) + 2 π {\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 }}}
K ( n , x ) ≈ 1 − ( 1 / 8 ) ∗ n 2 ∗ π 2 ∗ x 2 + ( 1 / 2 ) ∗ n 2 ∗ π ∗ x 3 + ( ( 1 / 384 ) ∗ n 4 ∗ π 4 − ( 1 / 2 ) ∗ n 2 ) ∗ x 4 + ( − ( 1 / 48 ) ∗ n 4 ∗ π 3 + ( 1 / 12 ) ∗ n 2 ∗ π ) ∗ x 5 + O ( x 6 ) {\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:
K ( n , x ) ≈ { 1800.0 + ( − 36.4 n 4 + 516.0 ) x + ( − 46.3 n 4 − 1830.0 n 2 − 71.0 ) x 2 + ( 1820.0 n 2 + 37.4 n 6 − 44.3 n 4 + 81.9 ) x 3 1800.0 + ( − 36.4 n 4 + 516.0 ) x + ( − 46.3 n 4 + 368.0 n 2 − 71.0 ) x 2 + ( − 7.48 n 6 − 44.3 n 4 − 363.0 n 2 + 81.9 ) x 3 } {\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\}}
Carotid-Kundalini function
