Cauchy principal value and the "normal" definition.

Question

Suppose that $\int^{\infty}_{-\infty}f(x)\, dx$ exist. How to prove that $\lim_{b\to\infty}\int^{b}_{-b}f(x)\, dx$ also exist, and $\int^{\infty}_{-\infty}f(x)\, dx=\lim_{b\to\infty}\int^{b}_{-b}f(x)\, dx$

How do you define the LHS? Is this lebesgue integral or riemmann integral? — Git Gud, Jun 09 '13 at 10:24

score 2 · Accepted Answer · answered Jun 09 '13 at 10:30

2

$\int_{-\infty}^{\infty}f(x)\, dx$ exist mean that both $$ \int_{-\infty}^{0}f(x)\, dx,\,\int_{0}^{\infty}f(x)\, dx $$

exist

This means that the two limits

$$ \lim_{b\to\infty}\int_{-b}^{0}\, f(x)\, dx,\,\lim_{b\to\infty}\int_{0}^{b}\, f(x)\, dx $$

exist.

Since the two limits exists you can add them up and get the desired result

answered Jun 09 '13 at 10:30

Belgi

23,150

I did that, but it seem to me too simple. Thanks! – 17SI.34SA Jun 09 '13 at 10:47

Cauchy principal value and the "normal" definition.

1 Answers1