蝴蝶函数

蝴蝶函数（Butterfly function）因其图形似蝴蝶而得名，蝴蝶函数由下列公式给出：

hd(x,y)=\frac{(x^2-y^2)sin(\frac{x+y}{a})}{x^2+y^2}

=\frac{(x^2-y^2)(x+y)HeunB(2, 0, 0, 0, \sqrt(2)sqrt(I(x+y)/a))}{(aexp(I(x+y)/a)*(x^2+y^2))}

=-\frac{(1/2I)(x^2-y^2)WhittakerM(0, 1/2, (2I)*(x+y)/a)}{(x^2+y^2)}

=\frac{-(1/2I)(x^2-y^2)(\Gamma(1, -(2I)(x+y)/a)-1)}{(exp(I(x+y)/a)*(x^2+y^2))}

=\frac{(1/2)(x^2-y^2)(x+y)\sqrt(\pi)\sqrt(2)BesselJ(1/2, (x+y)/a)}{(a\sqrt((x+y)/a)*(x^2+y^2))}
级数展开
{-sin(y/a)-cos(y/a)x/a+((1/2)y^2sin(y/a)/a^2+2sin(y/a))x^2/y^2+((1/6)y^2cos(y/a)/a^3+2cos(y/a)/a)x^3/y^2+(-(1/24)y^2sin(y/a)/a^4-(1/2)sin(y/a)/a^2-(1/2)sin(y/a)(y^2+4a^2)/(a^2y^2))*x^4/y^2+O(x^5)}
渐近展开
{-sin((x+y)/a)+2x^2sin((x+y)/a)/y^2-2sin((x+y)/a)x^4/y^4+2sin((x+y)/a)x^6/y^6-2x^8sin((x+y)/a)/y^8+2sin((x+y)/a)x^10/y^10-2sin((x+y)/a)x^12/y^12+O(1/y^14)}

参考文献