蝴蝶函数

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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^[1]：

Butterfly function Maple animation with varying parameter a

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

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

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

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

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