蝴蝶函数维基百科,自由的 encyclopedia 提示:此条目页的主题不是蝶形线。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 h d ( x , y ) = ( x 2 − y 2 ) s i n ( x + y a ) x 2 + y 2 {\displaystyle hd(x,y)={\frac {(x^{2}-y^{2})sin({\frac {x+y}{a}})}{x^{2}+y^{2}}}} = ( x 2 − y 2 ) ∗ ( x + y ) ∗ H e u n B ( 2 , 0 , 0 , 0 , ( 2 ) ∗ s q r t ( I ∗ ( x + y ) / a ) ) ( a ∗ e x p ( I ∗ ( 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}))}}} = − ( 1 / 2 ∗ I ) ∗ ( x 2 − y 2 ) ∗ W h i t t a k e r M ( 0 , 1 / 2 , ( 2 ∗ 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})}}} = − ( 1 / 2 ∗ I ) ∗ ( x 2 − y 2 ) ∗ ( Γ ( 1 , − ( 2 ∗ I ) ∗ ( x + y ) / a ) − 1 ) ( e x p ( 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}))}}} = ( 1 / 2 ) ∗ ( x 2 − y 2 ) ∗ ( x + y ) ∗ ( π ) ∗ ( 2 ) ∗ B e s s e l J ( 1 / 2 , ( x + y ) / a ) ( a ∗ ( ( 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}))}}}
提示:此条目页的主题不是蝶形线。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 h d ( x , y ) = ( x 2 − y 2 ) s i n ( x + y a ) x 2 + y 2 {\displaystyle hd(x,y)={\frac {(x^{2}-y^{2})sin({\frac {x+y}{a}})}{x^{2}+y^{2}}}} = ( x 2 − y 2 ) ∗ ( x + y ) ∗ H e u n B ( 2 , 0 , 0 , 0 , ( 2 ) ∗ s q r t ( I ∗ ( x + y ) / a ) ) ( a ∗ e x p ( I ∗ ( 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}))}}} = − ( 1 / 2 ∗ I ) ∗ ( x 2 − y 2 ) ∗ W h i t t a k e r M ( 0 , 1 / 2 , ( 2 ∗ 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})}}} = − ( 1 / 2 ∗ I ) ∗ ( x 2 − y 2 ) ∗ ( Γ ( 1 , − ( 2 ∗ I ) ∗ ( x + y ) / a ) − 1 ) ( e x p ( 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}))}}} = ( 1 / 2 ) ∗ ( x 2 − y 2 ) ∗ ( x + y ) ∗ ( π ) ∗ ( 2 ) ∗ B e s s e l J ( 1 / 2 , ( x + y ) / a ) ( a ∗ ( ( 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}))}}}