Hi All,
In Maxima (embedded in SageMath) you can use:
expand(trigrat(integrate(integrate(sin(x^2),x,y,1),y,0,1)));
in order to get exactly the same result in both cases.
Daniel
En miércoles, 8 de diciembre de 2021 23:02:00 GMT+2, Fernando Q. Gouvea
<[email protected]> escribió:
I see. So the difference between this and, say, 1+1==2 (which returns True) is
that 1+1 and 2 are numbers, not symbolic things.
Fernando
On 12/8/2021 3:37 PM, William Stein wrote:
On Wed, Dec 8, 2021 at 12:22 PM Fernando Q. Gouvea <[email protected]> wrote:
Thank you, that works. What is strange is that this does not:
sage: right=integrate(integrate(sin(x^2),y,0,x),x,0,1)
sage: wrong=integrate(integrate(sin(x^2),x,y,1),y,0,1)
sage: real(wrong)==right
-1/2*cos(1) + 1/2 == -1/2*cos(1) + 1/2
Is Sage seeing a difference there that I don't?
I think I don't understand the difference between real(wrong)==right and
bool(real(wrong)==right).
In Sage "[symbol thing] == [symbolic thing]" is a constructor for a symbolic
equation.
sage: SR(2) == SR(2) 2 == 2 sage: parent(SR(2) == SR(2)) Symbolic Ring sage:
bool(SR(2) == SR(2)) True # only because Sage can *prove* they are equal --
it's false if it can't prove they are equal, even if they are equal...
https://cocalc.com/wstein/support/gouvea-equals
Fernando
On 12/8/2021 1:23 PM, William Stein wrote:
You can compare the real and imaginary parts directly.
https://cocalc.com/wstein/support/2021-12-08-gouvea
sage: bool(wrong.real() == right) True sage: wrong.imag() 0
On Wed, Dec 8, 2021 at 10:07 AM Fernando Q. Gouvea <[email protected]> wrote:
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
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Colby College
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[email protected] http://www.colby.edu/~fqgouvea
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Colby College
Mayflower Hill 5836
Waterville, ME 04901
[email protected] http://www.colby.edu/~fqgouvea
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