[Maxima] Cauchy principal value integral

Stanislav Maslovski stanislav.maslovski at gmail.com
Wed Apr 8 15:12:02 CDT 2009

```On Wed, Apr 08, 2009 at 04:16:18PM +0200, Michel Talon wrote:
> Edwin Woollett wrote:
>
> >
> >
> > The correct value of integrate ( 1/x,  x,  -1, 2 )
> >  is log(2) = 0.693147..  when considered as a
> >  principal value integral, since we can write this
>
> "Correct" depnds on the point of view.

I think that in mathematics there is only one commonly accepted and
used definition for the pricipal value of an improper integral, and
that is the Cauchy principal value.

BTW, I tried to check the above example in Maxima 5.13.0 and it asks
me about signs of of (x-1) and (x+2) before giving an (wrong) answer.
This also does not make any sense.

> >
> > (%i2) integrate(1/x,x,-1,2);
> > Principal Value
> > (%o2) log(2)+2*%i*%pi
> >
> > We know the integral must be a real number, so
> > we know integrate(..) cannot be trusted here.
> > Integrate appears to be closing the complex plane
> > contour for a case where Jordan's lemma is not
> > satisfied?
> ....
> > ----------------------------------
> > Perhaps integrate should check the integrand
> > against conditions of Jordan's lemma before
> > returning an answer which is wrong.
> >
>
> In physics it is traditional, and *very* important to write
>
> 1/x = PP 1/x + %i*%pi*Dirac_delta(x)

By PP 1/x you mean something that gives zero when integrated over a
segment symmetric with respect to the point x = 0? Then I would say
that it is not related, first of all because with this definition
one will get after integration log(2) + %i*%pi, i.e. something
different from what Maxima gets.

Secondly, this definition for sure can be useful in physics when you
consider integrals of some physical quantities like polarizabilities or
dielectric functions. But Maxima is a mathematical package. It should
not depend, for instance, on the type of the time dependence that one
uses in physical problems (exp(+i omega t) or exp(-i omega t), this
will change the sign of your second term).

I think that the real reason for the imaginary part in the answer must
be something really stupid like that at some point(s) in the integration
Maxima takes logarithm(s) of negative number(s).

> Where the real part is indeed a principal part, and there is an imaginary
> part given by a delta "function". This result is the basis of causality
> analysis, Kramers-Kronig relations, etc. and is indeed related to
> application of Cauchy relation for complex integrals.
>
> It may well be that the maxima integration routine takes this in view.
>
> --
> Michel Talon
>
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--
Stanislav
```