Maple does not simplify if variables have "assume"s on them

unruh

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Aug 10, 2012, 2:53:06 PM8/10/12

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I am having trouble with Maple not simplifying expressions with
"assumes" on them.

For example, the answer to one of my integrations was
N12:=1/4 I (-epsilon~^2 -4IOmega~ +epsilon~^2)

where the ~ are the expressions with "assume"
(assume(epsilon>0);assume(Omega>0))
The correct answer should be just Omega~, but there is no way I can get
that answer no matter how many "simplify" or "expand" I put around the
original expression of N12.
Ie, maple seems to think that one of those epsilon~ is a different
variable from the other one.

If I copy that output to another input line, then maple simplifies the
expression.

Axel Vogt

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Aug 10, 2012, 3:10:32 PM8/10/12

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It would help if you provide your full code (as far as needed),
otherwise it is hard to understand.

PS: you may wish to provide it with all operational signs
(like multiplication etc), for copy & paste.

It seems a bit that this may be the reason, since Maple
would simplify it to what you expect.

unruh

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Aug 10, 2012, 4:28:57 PM8/10/12

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OK. Here it is.

> Gamma:=epsilon^2/2:assume(epsilon>0); assume(Omega>0); assume(t,real):
> phi1:=proc(t,x); -epsilon/2*exp((I*Omega+Gamma)*(t-x)); end proc:
> phi2:=proc(t,x); epsilon/2*exp((I*Omega-Gamma)*(t+x)); end proc:
> pi1:=proc(t,x); diff (phi1(t,x),t);end proc:
> pi2:=proc(t,x); diff (phi2(t,x),t);end proc:
> q1:=proc(t); exp((I*Omega+Gamma)*t);end proc:q2:=proc(t);
> exp((I*Omega-Gamma)*t);end proc:
> p1:=proc(t); -diff(q1(t),t);end proc:
> p2:=proc(t);-diff(q2(t),t);end proc:
> N12:=simplify(I/2*
( 2*int(conjugate(phi1(t,x))*pi2(t,x)-conjugate(pi1(t,x))*phi2(t,x),x=0..infinity)
+conjugate(q1(t))*p2(t)-conjugate(p1(t))*q2(t)
+epsilon*(conjugate(phi1(t,0))*q2(t)-conjugate(q1(t))*phi2(t,0)) ));

2 2
N12 := 1/4 I (-epsilon~ - 4 I Omega~ + epsilon~ )

Peter Pein

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Aug 10, 2012, 5:26:22 PM8/10/12

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Hi,

as it is almost midnight here, I do not have the patience to try it
myself - sorry.

I would guess that it makes sense to put all assumptions to the very
beginning of your code and to define Gamma later.

.. just a shot in the dark.

Good luck, Peter

Axel Vogt

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Aug 11, 2012, 6:30:56 AM8/11/12

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On 10.08.2012 22:28, unruh wrote:
> On 2012-08-10, Axel Vogt <&nor...@axelvogt.de> wrote:
>> On 10.08.2012 20:53, unruh wrote:
>>> I am having trouble with Maple not simplifying expressions with
>>> "assumes" on them.
>>>
>>> For example, the answer to one of my integrations was
>>> N12:=1/4 I (-epsilon~^2 -4IOmega~ +epsilon~^2)
>>>

...

>> It seems a bit that this may be the reason, since Maple
>> would simplify it to what you expect.
>
> OK. Here it is.

...

> 2 2
> N12 := 1/4 I (-epsilon~ - 4 I Omega~ + epsilon~ )

There are several issues (I do not use semicolon after writing
'proc' and always want to see, which variables are local or
global, t::real is a correct usage while you also use it on
differentiation).

Suggestion (and that differs from your result):

restart;
Gamma:=epsilon^2/2:
-epsilon/2*exp((I*Omega+Gamma)*(t-x)): phi1:=unapply(%, t,x);
epsilon/2*exp((I*Omega-Gamma)*(t+x)): phi2:=unapply(%, t,x);
diff (phi1(t,x),t): pi1:=unapply(%, t,x);
diff (phi2(t,x),t): pi2:=unapply(%, t,x);
exp((I*Omega+Gamma)*t): q1:=unapply(%, t);
exp((I*Omega-Gamma)*t): q2:=unapply(%, t);
-diff(q1(t),t): p1:=unapply(%, t) ;
-diff(q2(t),t): p2:=unapply(%, t);

tmp0:='I/2*
( 2*Int(conjugate(phi1(t,x))*pi2(t,x)-conjugate(pi1(t,x))*phi2(t,x),x=0..infinity)
+conjugate(q1(t))*p2(t)-conjugate(p1(t))*q2(t)
+epsilon*(conjugate(phi1(t,0))*q2(t)-conjugate(q1(t))*phi2(t,0)) )';
simplify(%) assuming (0 < epsilon, 0<Omega, t::real):
simplify(%) assuming (0 < epsilon, 0<Omega, t::real): # twice, yes
tmp1:=%;

tmp:=value(tmp1) assuming (0 < epsilon, 0<Omega, t::real);
2
tmp := 1/4 I epsilon + Omega