[Maxima] Cauchy principal value integral

Javed Alam jalam1001 at gmail.com
Thu Apr 9 17:15:00 CDT 2009


Hi
I have some examples of residues, poles and complex integrals here along
with the theory.

http://www.eng.ysu.edu/~jalam/engr6924s07/sessions/session8/session8.htm
<http://www.eng.ysu.edu/~jalam/engr6924s07/sessions/session9/session9.htm>

http://www.eng.ysu.edu/~jalam/engr6924s07/sessions/session9/session9.htm

Javed

On Thu, Apr 9, 2009 at 4:39 PM, Edwin Woollett <woollett at charter.net> wrote:

>
>  On 4-8-09 Edwin Woollett wrote :
>  ------------------------------------
> > It is straightforward to use Maxima's
> > limit function to work out the principal
> > value integral from its basic definition:
> ---------------------------------------
>
> It is quicker to use integrate in its definite
> integral mode to get the definition of a principal value
> integral, but you will need to answer questions
> or else prep with an assume statement:
>
>
> (%i1) display2d : false$
> (%i2) assume ( eps > 0, eps < 1 )$
>
> /* example 1:    1/x integrated over [-1, 2 ] */
>
> (%i3) integrate ( 1/x, x, -1, -eps ) + integrate ( 1/x, x, eps, 2 );
> (%o3) log(2)   /* success */
>
> /*  the deftness of ldefint  */
>
> (%i4) ldefint(1/x,x,-1,2);
> (%o4) log(2)
>
> /* the ineptness of integrate  */
>
> (%i5) integrate(1/x,x,-1,2);
> Principal Value
> (%o5) log(2)+2*%i*%pi
>
> /* example 2:   1/(x^2 - 1) over [0, 2]  */
>
> (%i6) integrate(1/(x^2-1),x,0,1-eps) +
>               integrate(1/(x^2-1),x,1+eps,2);
> (%o6) log(eps+2)/2-log(2-eps)/2-log(3)/2
> (%i7) limit(%,eps,0,plus);
> (%o7) -log(3)/2   /* success  */
>
> /*  the deftness of ldefint  */
>
> (%i8) ldefint(1/(x^2-1),x,0,2);
> (%o8) -log(3)/2
>
> /* the ineptness of integrate  */
>
> (%i9) integrate(1/(x^2-1),x,0,2);
> Principal Value
> (%o9) log(%i)-log(3)/2-log(-1)/2
> -----------------------------------------
>
> These two principal value integrals  illustrate
>  the deftness of ldefint, and the ineptness of
> integrate for principle value integrals. I suspect
> that integrate is not using ratsimp and logcontract
> correctly (see below for a possible wrong
> path) and that ldefint is doing the job right.
>
>
> Here is a fresh start with Maxima and a possible path
> which leads integrate to the wrong answer:
>
> (%i1) display2d:false$
>
> /*  indefinite integral for example 2  */
>
> (%i2) ix : integrate(1/(x^2-1),x);
> (%o2) log(x-1)/2-log(x+1)/2
> (%i3) assume(eps>0, eps<1)$
>
> (%i4) i1 : subst(x=1 - eps,ix) - subst( x = 0,ix );
> (%o4) log(-eps)/2-log(2-eps)/2-log(-1)/2
>
> (%i5) i2 : subst(x=2,ix) - subst(x = 1 + eps,ix);
> (%o5) log(eps+2)/2-log(eps)/2-log(3)/2
>
> (%i6) ip : i1 + i2;
> (%o6) log(eps+2)/2-log(eps)/2+log(-eps)/2-log(2-eps)/2-log(3)/2-log(-1)/2
>
> we now take the limit without getting the log args
> inside one log, and get an incorrect answer which reproduces
> what integrate did above.
>
> (%i7) limit(ip,eps,0,plus);
> (%o7) log(%i)-log(3)/2-log(-1)/2
>
> if we just do logcontract, still incorrect.
>
> (%i8) logcontract(ip);
> (%o8) log(eps+2)/2-log(eps)/2+log(-eps)/2-log(2-eps)/2-log(3)/2-log(-1)/2
> (%i9) limit(%,eps,0,plus);
> (%o9) log(%i)-log(3)/2-log(-1)/2
>
> correct way: ratsimp, then logcontract:
>
> (%i10) ratsimp(ip);
> (%o10) (log(eps+2)-log(eps)+log(-eps)-log(2-eps)-log(3)-log(-1))/2
> (%i11) logcontract(%);
> (%o11) -log(-(3*eps-6)/(eps+2))/2
> (%i12) limit(%,eps,0,plus);
> (%o12) -log(3)/2
> success.
> ==============
>
>
> On 4-8-09 Michel Talon  wrote:
> ----------------------------------------------
> > This integral doesn't exist, in the mathematical sense, period. You can
> > find
> > any real answer you want if you approach 0 in an unsymmetric way. The
> > principal part prescription (approach 0 from right and left in a
> symmetric
> > way) is a totally ad-hoc prescription which has no justification.
>
> ----------------------------------------
> An integral with a "pole" on the contour does not exist in the
>  strict sense, but for a simple pole on the real axis, one defines the
>  Cauchy principal value of g(x) at a singular point c as
>  limit( integrate(g(x),x,a,c - eps) + integrate(g(x),x,c + eps,b )
> ,eps,0,plus).
>
>  If this limit exists, this definition provides a unique and well defined
> result.
>
> This result, well defined if it exists, does not imply that the integral is
> "proper".
>
>
>  Michel is quite right that one does not need to resort to the work
>  of defining a principle value integral if a well defined  result involving
>  independent modes of approach using two parameters eps and epsprime via
>  the definition; (again see Gradshteyn $ Ryzhik 7th, p. 252 )
>
>  limit(  limit( integrate(g(x),x,a,c - eps) +
>                    integrate(g(x),x,c + epsprime, b ) ,eps,0,plus),
>           epsprime,0,plus )
>
>  is found.
>
> Nevertherless, the principle value integral meaning is clear and well
> defined.
>
> Ted Woollett
>
>
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