查看“︁蝴蝶函数”︁的源代码

{{not|蝶形線}}
[[File:Butterfly function plot.gif|thumb|300px|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<ref>Weisstein, Eric W. "Butterfly Function." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ButterflyFunction.html {{Wayback|url=http://mathworld.wolfram.com/ButterflyFunction.html |date=20200319005351 }}</ref>：

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

<math>=\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))}</math>

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

<math>=\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))}  </math>

<math>=\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))}</math>
==级数展开==
<math>  {-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)}     </math>
==渐近展开==
<math> {-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^10/y^10-2*sin((x+y)/a)*x^12/y^12+O(1/y^14)}  </math>

==参考文献==
<references/>

[[Category:特殊函数]]