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 ) = c o s ( n ∗ x ∗ a r c c o s ( x ) ) {\displaystyle K(n,x)=cos(n*x*arccos(x))} K ( n , x ) = − ( 1 / 2 ∗ I ) ∗ ( − 1 + e x p ( I ∗ ( 2 ∗ n ∗ x ∗ a r c c o s ( x ) + P i ) ) ) e x p ( ( 1 / 2 ∗ I ) ∗ ( 2 ∗ n ∗ x ∗ a r c c o s ( x ) + P i ) ) {\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))}}} K ( n , x ) = ( n x c o s 1 ( x ) + π / 2 ) K u m m e r M ( 1 , 2 , I ( 2 n x a r c c o s ( x ) + π ) ) e x p ( I ( 2 n x a r c c o s ( x ) + 2 π / 2 ) ) {\displaystyle 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 H e u n B ( 2 , 0 , 0 , 0 , 2 1 / 2 i ( 2 n x ( 1 / 2 π − x H e u n C ( 0 , 1 / 2 , 0 , 0 , 1 / 4 , x 2 x 2 − 1 ) 1 1 − x 2 ) + π ) ) H e u n C ( 0 , 1 / 2 , 0 , 0 , 1 / 4 , x 2 x 2 − 1 ) 1 1 − x 2 ( e − 1 / 2 i ( − n x π 1 − x 2 + 2 n x 2 H e u n C ( 0 , 1 / 2 , 0 , 0 , 1 / 4 , x 2 x 2 − 1 ) − π 1 − x 2 ) 1 1 − x 2 ) − 1 + 1 / 2 π ( n x + 1 ) H e u n B ( 2 , 0 , 0 , 0 , 2 1 / 2 i ( 2 n x ( 1 / 2 π − x H e u n C ( 0 , 1 / 2 , 0 , 0 , 1 / 4 , x 2 x 2 − 1 ) 1 1 − x 2 ) + π ) ) ( e − 1 / 2 i ( − n x π 1 − x 2 + 2 n x 2 H e u n C ( 0 , 1 / 2 , 0 , 0 , 1 / 4 , x 2 x 2 − 1 ) − π 1 − x 2 ) 1 1 − x 2 ) − 1 {\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}} 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 （页面存档备份，存于互联网档案馆） Weisstein, Eric W. "Carotid-Kundalini Function." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Carotid-KundaliniFunction.html （页面存档备份，存于互联网档案馆）