Sheng Li (gmachine1729) wrote,
Sheng Li
gmachine1729

Principal values (of integrals)

Originally published at 狗和留美者不得入内. You can comment here or there.

I’ve been looking through my Gamelin’s Complex Analysis quite a bit lately. I’ve solved some exercises, which I’ve written up in private. I was just going over the section on principal values, which had a very neat calculation. I’ll give a sketch of that one here.

Take an integral \int_a^b f(x)dx such that on some x_0 \in (a,b) there is a singularity, such as \int_{-1}^1 \frac{1}{x}dx. The principal value of that is defined as

PV \int_a^b f(x)dx = \lim_{\epsilon \to 0}\left(\int_a^{x_0 - \epsilon} + \int_{x_0 + \epsilon}^b\right)f(x)dx.

The example the book presented was

PV\int_{-\infty}^{\infty} \frac{1}{x^3 - 1} = -\frac{\pi}{\sqrt{3}}.

Its calculation invokes both the residue theorem and the fractional residue theorem. Our integrand, complexly viewed, has a singularity at e^{2\pi i / 3}, with residue \frac{1}{3z^2}|_{z = e^{2\pi i / 3}} = \frac{e^{2\pi i / 3}}{3}, which one can arrive at with so called Rule 4 in the book, or more from first principles, l’Hopital’s rule. That is the residue to calculate if we had the half-disk in the half plane, arbitrarily large. However, with our pole at 1 we must indent it there. The integral along the arc obviously vanishes. The infinitesimal arc spawned by the indentation, the integral along which, can be calculated by the fractional residue theorem, with any -\pi, the minus accounting for the clockwise direction. This time the residue is at 1, with \frac{1}{3z^2}|_{z = 1} = \frac{1}{3}. So that integral, no matter how small \epsilon is, is -\frac{\pi}{3}i. 2\pi i times the first residue we calculated minus that, which is extra with respect to the integral, the principal value thereof, that we wish to calculate, yields -\frac{\pi}{\sqrt{3}} for the desired answer.

Let’s generalize. Complex analysis provides the machinery to compute integrals not to be integrated easily by real means, or something like that. Canonical is having the value on an arc go to naught as the arc becomes arbitrarily large, and equating the integral with a constant times the sum of the residues inside. We’ve done that here. Well, it turns out that if the integral has an integrand that explodes somewhere on the domain of integration, we can make a dent there, and minus out the integral along its corresponding arc.

Tags: analysis, complex analysis
Subscribe

  • 中国少数民族的语系分类

    汉藏语系 汉族:汉语族,12.2 亿 壮族:壮侗语族,1690 万,主要在广西 彝族:彝语支,870 万,主要在西南 土家族:840 万,主要在湖南,湖北,贵州 藏族:藏缅语族,630 万,主要在西藏 侗族 布衣族 白族 哈尼族 黎族 傣族…

  • 中国那帮英语培训机构基本都是垃圾,他们的创办人及工作者应当被处置

    刚跟朋友说了国庆放假,我在健身房,碰到了一个俄罗斯姑娘和个三十出头的中国女人一起。我跟她用俄语稍微聊了几句,也加了她和她的中国同事微信。然后,我得知她们是在培训班教小学生英语的。俄罗斯姑娘跟我说她在俄罗斯的 Перм 上大学,专业是德语,现在在中国已经待了一年了。…

  • 拼音英语字符比对服务

    我和朋友做了一个汉字转拼音的服务。什么给了我这个启发?是他有一次想让我在微信上用英语,可是我不想,因为觉得中文更快。然后自己做了个 小试验。 >>>…

  • Post a new comment

    Error

    Anonymous comments are disabled in this journal

    default userpic

    Your reply will be screened

  • 0 comments