I was showing my students a famous calculus example of an integral that
can be computed in one order of the variables but not in the other.
Knowing that SageMath can compute anything, the students suggested
trying the integral the "wrong" way.
The "right" way is
sage: integrate(integrate(sin(x^2),y,0,x),x,0,1)
-1/2*cos(1) + 1/2
The "wrong" way is
sage: integrate(integrate(sin(x^2),x,y,1),y,0,1)
-1/16*(-1)^(3/4)*((sqrt(2) + 4*(-1)^(1/4))*e^I - sqrt(-I)*((I +
1)*sqrt(2)*(-1)^(1/4)*e^(2*I) - (I + 1)*sqrt(2)*(-1)^(1/4)*e^I) +
I*sqrt(2)*e^I - 2*(-1)^(1/4)*e^(2*I) - (I + 1)*sqrt(2) -
2*(-1)^(1/4))*e^(-I)
Is there any way to get Sage to check that these are equal?
The obvious thing does not seem to work:
sage: -1/16*(-1)^(3/4)*((sqrt(2) + 4*(-1)^(1/4))*e^I - sqrt(-I)*((I +
1)*sqrt(2) ....: *(-1)^(1/4)*e^(2*I) - (I + 1)*sqrt(2)*(-1)^(1/4)*e^I) +
I*sqrt(2)*e^I - 2* ....: (-1)^(1/4)*e^(2*I) - (I + 1)*sqrt(2) -
2*(-1)^(1/4))*e^(-I) == -1/2*cos(1) ....: +1/2
-1/16*(-1)^(3/4)*((sqrt(2) + 4*(-1)^(1/4))*e^I - sqrt(-I)*((I +
1)*sqrt(2)*(-1)^(1/4)*e^(2*I) - (I + 1)*sqrt(2)*(-1)^(1/4)*e^I) +
I*sqrt(2)*e^I - 2*(-1)^(1/4)*e^(2*I) - (I + 1)*sqrt(2) -
2*(-1)^(1/4))*e^(-I) == -1/2*cos(1) + 1/2
Thanks,
Fernando
--
==================================================================
Fernando Q. Gouvea
Carter Professor of Mathematics
Colby College
Mayflower Hill 5836
Waterville, ME 04901
fqgou...@colby.edu http://www.colby.edu/~fqgouvea
I have had a perfectly wonderful evening, but this wasn't it.
--Groucho Marx
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