卡咯提的-昆达利尼函数

Carotid-Kundalini function

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

$K(n,x)=cos(n*x*arccos(x))$

与其他特殊函数的关系

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

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

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

函数展开

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

帕德近似

帕德近似:

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

外部链接

• Carotid-Kundalini function Template:Wayback