On Nov. 30, 2012, I wrote:
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>integrate (and defint) give the wrong Cauchy
>principal value result for a non-convergent
>integral:
>................
>(%i1) integrate(1/(x-1),x,0,2);
>Principal Value
>(%o1) %i*%pi
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in which the correct Cauchy principal value is 0.
Another example of an incorrect principal value
integral:
(%i2) assume(eps > 0, eps < 1);
(%o2) [eps > 0,eps < 1]
(%i20) integrate(1/((x-1)*(x-2)),x,0,3);
Principal Value
(%o20) %i*%pi-2*log(2)
/* which is wrong; as usual we let eps be a
small positive number, and calculate three
pieces: */
(%i21) i1: integrate(1/((x-1)*(x-2)),x,0,1-eps);
(%o21) log(eps+1)-log(eps)-log(2)
(%i22) i2: integrate(1/((x-1)*(x-2)),x,1+eps,2-eps);
Is 2*eps-1 positive, negative, or zero?
n;
(%o22) 2*log(eps)-2*log(1-eps)
(%i23) i3: integrate(1/((x-1)*(x-2)),x,2+eps,3);
(%o23) log(eps+1)-log(eps)-log(2)
(%i24) isum : i1+i2+i3;
(%o24) 2*log(eps+1)-2*log(1-eps)-2*log(2)
(%i25) limit(isum,eps,0,plus);
(%o25) -2*log(2)
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So the correct result is -2*log(2).
Ted Woollett