方程共轭复数解在实平面上的几何意义

Geometric Meaning of Complex Roots of Equation in Real Plane

研究了实系数代数方程的复数解发生变化时实平面上y=F(x)几何图形的变化。结果表明,当复数解虚部的平方较小时,函数在实部的附近形成极值或拐点,且虚部的平方越小,极值或拐点越明显;当虚部的平方较大时,上述极值或拐点将逐渐不明显;当虚部的平方趋于无穷时,函数值与复数解虚部的平方成正比而与复数解的实部无关,上述极值或拐点将消失。所给结论可用于预测含复数解函数曲线的形状,也可用于改变函数曲线的形状。提出了影像方程的概念,从而可以研究一般方程复数解在实平面上的几何意义。

The relationship between the complex roots of an algebra equation F （x） = 0 and the figure of function y =F （x） in a real plane was revealed. The results show that an extreme value or an inflection point will be formed gradually when the value of the square of imaginary number of the roots becomes small gradually. When the value of the square becomes large gradually, the extreme value or the inflection point will gradually become vague or even disappear. When the value of the square becomes large enough, the values of F （x） is only directly proportion to the square of imaginary number of the roots and is independent of the real part of the complex roots. The figure of function y = F （x） containing complex roots can be either predicted or altered according to the conclusions. A concept, shadow equation, was put forward to search the real geometric meaning of complex roots for general equation.