Cauchy Principal value

**dannyboycurtis** · December 8th 2009, 03:04 PM

Im a little confused on evaluating an improper integral from negative infinity to infinity.
If I know that $\int^\infty_{-\infty}f$ converges to A, how do I show that the Cauchy Principle value,
$\lim_{a\to \infty}\int^a_{-a}f=A$ ??
I know $\int^\infty_{-\infty}fdx=\lim_{a\to -\infty}\int^0_{a}fdx + \lim_{b\to\infty}\int^b_{0}fdx=-\infty+\infty$
How does this imply that the CPV is A???

**Jose27** · December 8th 2009, 05:09 PM

Originally Posted by dannyboycurtis

Im a little confused on evaluating an improper integral from negative infinity to infinity.
If I know that $\int^\infty_{-\infty}f$ converges to A, how do I show that the Cauchy Principle value,
$\lim_{a\to \infty}\int^a_{-a}f=A$ ??
I know $\int^\infty_{-\infty}fdx=\lim_{a\to -\infty}\int^0_{a}fdx + \lim_{b\to\infty}\int^b_{0}fdx=-\infty+\infty$
How does this imply that the CPV is A???

This doesn't make sense to me, if you know that your first integral converges then by definition your last two integrals must exist and be finite.

Edit: That being said $\lim_{a\rightarrow \infty} \int_{-a}^{a} f = \lim_{a\rightarrow \infty} \int_{-a}^{0} f + \lim_{a\rightarrow \infty} \int_{0}^{a} f =A$

**dannyboycurtis** · December 8th 2009, 05:42 PM

how do I show that $\lim_{a\rightarrow \infty} \int_{-a}^{0} f + \lim_{a\rightarrow \infty} \int_{0}^{a} f =A$ ?

Thread: Cauchy Principal value

LinkBack

Thread Tools

Display

Cauchy Principal value

Similar Math Help Forum Discussions

Principal Ideal

Product of non principal ideals is principal.

[SOLVED] Subsequence of a Cauchy Sequence is Cauchy

First Principal derivatives

Principal branch

Search Tags