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第二百四十四章 柯西主值(微积分)(2 / 2)

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求积分容易,关键是需要给他找到一个合理性,这个合理性是什么?

就是连续性大致存在,而在无穷大点处也有连续不断接近的性质。

只要这样,就可以求积分。

存在的合法性,就是可以不断的接近,这种不断的接近就是一种连续性,妙哉!

在求无穷大区间的积分的时候,只需要让其变成定积分的形式,先求出积分的式子,之后让取点积分区间那个值成为一种接近无限的值。

还可以在无穷大的点哪里,取左右分开求积分那种形式,在无穷大点处也带入定值,让最后的那个积分公式取无穷来计算即可。

这种值就是柯西主值。

柯西主值是在微积分中,实数线上的某类瑕积分,为纪念柯西而得此名。

瑕积分(improper integral)是高等数学中微积分的一种,是被积函数带有瑕点的广义积分。

在物理学中有Kramers–Kronig定理,就是说响应和耗散分别是一个函数的实部和虚部,他们之间由一个柯西主值积分相联系。

实验上一般测量响应或者耗散的其中一个,然后按Kramers–Kronig定理积分取柯西主值就可以得到另一个。

这里的积分是不能收敛的,如果不取柯西主值,物理学家就无法进行下一步。

喜欢数学心请大家收藏:数学心本站更新速度全网最快。

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