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
<fqgou...@colby.edu> 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
<fqgou...@colby.edu> 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
--
==================================================================
Fernando Q. Gouvea
Carter Professor of Mathematics
Colby College
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Colby College
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The painful transition from capitalism to capitalism.
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Colby College
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