蝴蝶函数

Butterfly function Maple animation with varying parameter a

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：

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

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

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

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

$=\frac{(1/2)*(x^2-y^2)*(x+y)*\sqrt{(}\pi )*\sqrt{(}2)*BesselJ(1/2,(x+y)/a)}{(a*\sqrt{(}(x+y)/a)*(x^2+y^2))}$

级数展开

$-sin(y/a)-cos(y/a)*x/a+((1/2)*y^2*sin(y/a)/a^2+2*sin(y/a))*x^2/y^2+((1/6)*y^2*cos(y/a)/a^3+2*cos(y/a)/a)*x^3/y^2+(-(1/24)*y^2*sin(y/a)/a^4-(1/2)*sin(y/a)/a^2-(1/2)*sin(y/a)*(y^2+4*a^2)/(a^2*y^2))*x^4/y^2+O(x^5)$

渐近展开

$-sin((x+y)/a)+2*x^2*sin((x+y)/a)/y^2-2*sin((x+y)/a)*x^4/y^4+2*sin((x+y)/a)*x^6/y^6-2*x^8*sin((x+y)/a)/y^8+2*sin((x+y)/a)*x^{1}0/y^{1}0-2*sin((x+y)/a)*x^{1}2/y^{1}2+O(1/y^{1}4)$