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：

Butterfly function Maple animation with varying parameter a

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

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

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

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

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

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

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