On 4-8-09 Michel Talon wrote
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"Correct" depnds on the point of view.
In physics it is traditional, and *very* important to write
1/x = PP 1/x + %i*%pi*Dirac_delta(x)
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.
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1. I am using "principal value integral" in the usual mathematics sense,
Let pri ( f(x), x, c, a, b ) stand for:
limit ( integrate ( f(x)/(x-c), x, a, c - eps) +
integrate ( f(x)/(x-c), x, c + eps, b ) ,
eps, 0, plus) ),
which ( if the limit exists ) defines the principal value of the
improper
integral: integrate ( f(x)/(x-c), x, a, b ). See p.252,
sec.3.05, Gradshteyn and
Ryzhik, Table of integrals, Series and Products, 7th ed. (Jeffrey &
Zwillinger).
2. The symbolic (operator) expression you have written down above actually
should be written:
limit ( 1/(x - %i*eps), eps, 0, plus ) = P( 1/x ) + %i*%pi*delta(x)
which only has meaning when both sides are multiplied by f(x)*dx and
integrated over
an interval which includes x = 0, in which case it means
limit( integrate ( f(x)/(x - %i*eps), x, a, b), eps, 0, plus ) = pri (
f(x), x, 0, a, b) + %i*%pi*f(0)
provided a < 0 < b and f(x) is a smooth function near x = 0.
(Refs: a. Paul Roman, Advanced Quantum Theory, Addison-Wesley, 1965, p.
718
b. R. Shankar, Principles of Quantum Mechanics, 2nd ed, Plenum
Press, 1994,
page 662.
3. Since integrate ( 1/x, x, -1, 2 ) involves the integration of a real
function
along the real axis, the result must be a real number, and there is
no physics magic which can turn a real number into an imaginary number.
Ted Woollett