Diracδ函数的数学表示

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A Mathematical Representation of Dirac δ Function

摘要: Dirac提出的δ函数δ(x)满足:∫_-∞^+∞δ(x)dx=1;当x≠0时δ(x)=0,在标准分析中,这两个条件是互相矛盾的.本文认为Diracδ函数应属于微观的,把实数系R无限缩小成核子a(0),先在a(0)上定义Diracδ函数δ(x),使它在a(0)上的积分等于1,然后把δ(x)的定义域从a(0)扩张到无穷区间(-∞,+∞),使它符合上述Dirac提出的条件.举出几个满足这个定义的δ函数的例子.最后,从这个定义推出一些δ函数δ(x)的重要性质.

Abstract: Dirac's definition of a δ function δ(x)in the real number system R is the idealization of a function satisfying the following conditions These conditions in standard analysis are inconsistent since the integral of any function which has a value of zero with the exception of that which at one point.

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