CAJ | 학술논문

为研究以3x+1推广函数T(x)为代表的复指数函数的分形性质,构造了一个3x+1近似推广函数B(x).通过对B(x)的复解析分析,得出了B(x)在实轴和复平面上不动点的性质与求解方法,证明了0点为唯一的吸引不动点;说明了B(x)在复平面上迭代的对称性,求出了函数迭代在整数点处的敛散区域.最后通过绘制函数的分形图像对求得的B(x)分形性质加以验证,同时提出了B(x)收敛域和发散域逐层嵌套的猜想.
위연구이3x+1추엄함수T(x)위대표적복지수함수적분형성질,구조료일개3x+1근사추엄함수B(x).통과대B(x)적복해석분석,득출료B(x)재실축화복평면상불동점적성질여구해방법,증명료0점위유일적흡인불동점;설명료B(x)재복평면상질대적대칭성,구출료함수질대재정수점처적렴산구역.최후통과회제함수적분형도상대구득적B(x)분형성질가이험증,동시제출료B(x)수렴역화발산역축층감투적시상.
In order to study the fractal character of complex exponential functions represented by generalized 3x+1 function T(x),a generalized approximate 3x+1 function B(x) is constructed. With complex analytical analysis for B(x), the fixed points of B(x) at real axis and C-plane are found. It is proved that zero is the only attract fixed point and the iteration of B(x) on the complex plane is symmetrical. And the domains of the constringency and the divergence of B(x) are given. Finally the fractal images are drawn to validate the fractal character of B(x) and a conjecture, which the constringency and the divergence of B(x) is nested, is put forward.