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What is the notion of a principal value of an integral when the singularity appears at one endpoint? Namely,

$ PV \int_a^b f(t) dt = ? $,

where the integral is convergent in the upper limit, but divergent in the lower. Thank you for any ideas and/or references.

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The usual Cauchy "principal value" gives you a finite result due to cancellation. The analogue in this case would have to be something that converges as an improper Riemann integral, i.e. $\lim_{x \to a+} \int_x^b f(t)\ dt$ converges, although the Lebesgue integral might not converge. –  Robert Israel May 20 '12 at 21:37
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2 Answers

up vote 3 down vote accepted

Let me give an example: you want to define a distribution on $\mathbb R$ which coincides with $1/t$ on $(0,+\infty)$ and vanishes on $(-\infty,0)$. Let us take $$ T=\frac{d}{dt}(H(t)\ln t),\quad H=1_{\mathbb R_+}, $$ with a distributional derivative. You can easily generalize to a 1D situation with a finite number of poles of finite type. For instance, to define a distribution on $\mathbb R$ which coincides with $1/t^2$ on $(0,+\infty)$ and vanishes on $(-\infty,0)$, you take $$ -\frac{dT}{dt}. $$

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This method coincides with the so-called Hadamard finite-part prescription, which generalizes the Cauchy principal value prescription. A standard reference for this is Gelfand-Shilov, Generalized Functions, volume 1. –  Igor Khavkine May 20 '12 at 22:45
    
Thank you for all the answers. The Hadamard prescription seems to be best suited for my purposes. –  Michał Eckstein May 21 '12 at 9:55
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This is an old question but since it concerns the non-symmetric case and this has not been addressed by the above answers, the following solution might be of interest. The portuguese mathematician Sebastiao e Silve gave an elementary definition of definite integals of distributions which makes, for example, functions such as $\frac 1 {x^2} \sin\frac 1x$ integrable on $[0,\infty[$. The first ingredient is the simple fact that each distribution on the line (or a subinterval) has a primitive. The definite integral is then defined as in elementary calculus, using the concepts of the value of a distribution at a point, respectively limits of a distribution at a point (including one-sided limits as required in this question). Of course, these need not exist in the general case and so there are restrictions required for the existence of definite integrals, as one would expect. The precise definitions and examples can be found in Campos Ferreira's book on Distributions (Pitman) which is essentially based on courses Sebastiao e Silva gave in the 60's. As a sample, for a distribution $f$ defined near infinity, write $\lim_{s \to \infty} f(s)= \lambda$ if there is an integer $p$ and a continuous function $F$ defined on a neighbourhood of infinity so that $f=D^p F$ (derivative in distributional sense) and $\frac {F(s)}{s^p} \to\frac{\lambda}{p!}$ as $s \to \infty$.

We remark that in addition to this application of these concepts, there are many situations where they are, despite their simplicity, of some consequence. For example, the comment that one often reads in the literature, that distributions don't have values at points is unnecessarily pessimistic. Most distributions which are of practical value do have values at most points---simple example, the Dirac $\delta$ function, which , of course, has values at all points, apart from its singularity. This is a very simple example, but there are much more subtle ones where the assignment of a value is not, a priori, obvious.

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