#1698 integrate(exp(-x^(%i)),x,0,1); => Is %i an integer?

closed
nobody
6
2009-10-04
2009-06-25
Robert Dodier
No

integrate(exp(-x^(%i)),x,0,1); => Is %i an integer?

Discussion

  • Barton Willis
    Barton Willis
    2009-06-25

    Answering yes gives two more questions and a spurious value for the integral

    (%i98) integrate(exp(-x^(%i)),x,0,1);
    Is %i an integer?no;
    Is yx positive or negative?pos;
    Is yx-1 positive, negative, or zero?pos;

    (%o98) %i*gamma_incomplete(-%i,1)-true

     
  • Dan Gildea
    Dan Gildea
    2009-10-04

    • priority: 5 --> 6
     
  • Dieter Kaiser
    Dieter Kaiser
    2009-10-04

    With revision 1.56 of compar.lisp %i has the following properties:

    (%i12) featurep(%i,imaginary);
    (%o12) true
    (%i13) featurep(%i,complex);
    (%o13) true
    (%i14) featurep(%i,noninteger);
    (%o14) true

    The integral of this bug report no longer ask a question about %i. We get:

    (%i15) integrate(exp(-x^%i),x,0,1);
    (%o15) %i*(%i*('limit(%i*gamma_incomplete(-%i,-log(x+1))/2
    -%i*gamma_incomplete(%i,-log(x+1))/2,x,0,minus)
    +%i*gamma_incomplete(%i,1)/2-%i*gamma_incomplete(-%i,1)/2)
    +'limit(-gamma_incomplete(%i,-log(x+1))/2
    -gamma_incomplete(-%i,-log(x+1))/2,x,0,minus)
    +gamma_incomplete(%i,1)/2+gamma_incomplete(-%i,1)/2)

    The above integral can not be evaluated completely because the limit of gamma_incomplete(-%i,x) as x->0 is not known.

    The indefinite integral is correct:

    (%i16) integrate(exp(-x^%i),x);
    (%o16) %i*gamma_incomplete(-%i,x^%i)*x*(x^%i)^%i

    Closing this bug report as fixed.

    Dieter Kaiser

     
  • Dieter Kaiser
    Dieter Kaiser
    2009-10-04

    • status: open --> closed