> > It may be "branch cut strangeness", but if so it is very strange. The
> > integrand is clearly well-behaved, and the integral,
> > while in terms of the incomplete gamma function, seems to be off the
> usual
> > branch cut (negative real axis).
>
> Try domain:complex before calling integrate; that changes the result to
> what I think is expected.
>
>
(%i5) display2d:false;
(%o5) false
(%i6) integrate(x*cos(x^3),x);
(%o6) (gamma_incomplete(2/3,%i*x^3)+gamma_incomplete(2/3,-%i*x^3))/6
(%i7) domain:complex;
(%o7) complex
(%i8) integrate(x*cos(x^3),x);
(%o8) ((sqrt(3)*%i-1)*gamma_incomplete(2/3,%i*x^3)
+(-sqrt(3)*%i-1)*gamma_incomplete(2/3,-%i*x^3))
*(x^3)^(1/3)
/(12*x)
But the *definite* integral in both cases is wrong. Any ideas?
(%i1) display2d:false;
(%o1) false
(%i2) integrate(x*cos(x^3),x,0,1/2);
(%o2)
gamma_incomplete(2/3,%i/8)/6+gamma_incomplete(2/3,-%i/8)/6-gamma(2/3)/3
(%i3) domain:complex;
(%o3) complex
(%i4) integrate(x*cos(x^3),x,0,1/2);
(%o4)
gamma_incomplete(2/3,%i/8)/6+gamma_incomplete(2/3,-%i/8)/6-gamma(2/3)/3
I guess (emphasis on guess) that the problem originates not from
> gamma_incomplete itself but from terms of the form (-1)^(1/n) which are
> the result of simplifying or evaluating gamma_incomplete. Sorry I can't
> be more helpful.
>
I don't see any of those up here, though, and the gamma_incomplete
evaluation is correct (gives the same via W|A, Sage = Pari in my version,
mpmath, and Maxima). I think that Maxima is somehow using the "real"
antiderivative, if that makes sense - is that possible, Robert?
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