## #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
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
2009-10-04

• priority: 5 --> 6

• 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
2009-10-04

• status: open --> closed