Simplification of exponential and trigonometric expression

Re:

Is there any way to ask Maxima to simplify q(x,a,phi,r,theta,psi) into the
nicer form of p(x,a,phi,r,theta,psi)?


I don't see any "automagic" way to do this.  I think you'll have to pick
apart the expression, transform the parts with a variety of functions
(factor, trigexpand, trigreduce, etc.) and put them back together.

Over the years, I've thought that we ought to support this sort of
transformation within Maxima, letting you say "focus on part(...) of this
expression, work on it for a while, and substitute it back in".  Of course
you can do this easily enough by hand, something like this:

            subpart: part(expr,2,1,2);
           ..... various transformations...
           newsubpart: <<transformed version>>;
           substpart(newsubpart,expr,2,1,2);

But I think we can support this better.

                  -s

On Sat, Sep 29, 2012 at 9:42 PM, Michele Dall'Arno <
michele.dallarno at gmail.com> wrote:

> Thank you very much for your help (and for introducing me to the function
> scanmap).
>
> I made a couple of mistakes (now fixed) in the normalizations of q:
>
> define(Wvac(x,p), 2/(%pi) * exp(-2*(x^2+p^2)))$
> define(Wsq(x,p), Wvac(exp(-r)*(cos(theta/2)*x + sin(theta/2)*p),
> exp(+r)*(-sin(theta/2)*x + cos(theta/2)*p)))$
>
> define(Wsqd(x,p), Wsq(x-a*cos(phi),p-a*sin(phi))**)$
> define(W(x,p), Wsqd(cos(psi)*x-sin(psi)*p,**sin(psi)*x+cos(psi)*p))$
> assume(A<0)$
> F : integrate(2/(%pi)*exp(A*p^2+B***p+C), p, minf, inf)$
> l : makelist(coeff(expand(log(%pi/**2*W(x,p))),p,n),n,0,2)$
>
> define(q(x,a,phi,r,theta,psi), (trigreduce(trigexpand(subst([**A = l[3],
> B = l[2], C = l[1]], F)))));
>
> and of p:
>
> xavg(a,psi,phi) := a*cos(psi-phi)$
> sigma2(r,theta,psi) := %e^(2*r)*cos(psi-theta/2)^2+%**
> e^(-2*r)*sin(psi-theta/2)^2$
> define(p(x,a,phi,r,theta,psi), sqrt(2/(%pi*sigma2(r,theta,**psi))) *
> exp(-2*(x-xavg(a,psi,phi))^2/**sigma2(r,theta,psi)));
>
> so that now
>
>
> qq: q(x,a,phi,r,theta,psi)$
> pp: p(x,a,phi,r,theta,psi)$
> qq1: scanmap(factor,trigreduce(qq))**;
> pp1: scanmap(factor,trigreduce(pp))**;
>
> qq1 and pp1 are equivalent.
>
> Is there any way to ask Maxima to simplify q(x,a,phi,r,theta,psi) into the
> nicer form of p(x,a,phi,r,theta,psi)?
>
> It may sound pointless as we already know the two functions are the same,
> but I would really like to learn how to do smart simplification in Maxima.
>
> Thanks a lot,
>
> Michele
>
>
>
> On 09/30/2012 12:51 AM, Stavros Macrakis wrote:
>
>> First of all, your functions don't seem to be equal:
>>
>> q(0,0,0,0,0,0) => 1/sqrt(%pi)
>> p(0,0,0,0,0,0) => sqrt(2)/sqrt(%pi)
>>
>> Let's put them in similar forms to see the differences:
>>
>> qq: q(x,a,phi,r,theta,psi)$
>> pp: p(x,a,phi,r,theta,psi)$
>>
>> qq1: scanmap(factor,trigreduce(qq))**;
>> pp1: scanmap(factor,trigreduce(pp))**;
>>
>> You can also look at radcan(qq1) and radcan(pp1).
>>
>> Looking over the two expressions, I see quite a few differences, though
>> the
>> structure is the same.  Quite a few coefficients are different (factor of
>> -1, factor of 2) and it seems unlikely (at a quick glance) that these
>> cancel.
>>
>>                    -s
>>
>>
>> On Sat, Sep 29, 2012 at 4:19 AM, Michele Dall'Arno <
>> michele.dallarno at gmail.com> wrote:
>>
>>  Hello,
>>>
>>> I know that the function q(x,a,phi,r,theta,psi) defined as
>>>
>>> define(Wvac(x,p), 1/(%pi) * exp(-(x^2+p^2)))$
>>> define(Wsq(x,p), Wvac(exp(-r)*(cos(theta/2)*x + sin(theta/2)*p),
>>> exp(r)*(-sin(theta/2)*x + cos(theta/2)*p)))$
>>> define(Wsqd(x,p), Wsq(x-a*cos(phi),p-a*sin(phi))****)$
>>> define(W(x,p), Wsqd(cos(psi)*x-sin(psi)*p,****sin(psi)*x+cos(psi)*p))$
>>> assume(A<0)$
>>> F : integrate(1/(%pi)*exp(A*p^2+B*****p+C), p, minf, inf)$
>>> l : makelist(coeff(expand(log(%pi*****W(x,p))),p,n),n,0,2)$
>>> define(q(x,a,phi,r,theta,psi), (trigreduce(trigexpand(subst([****A =
>>> l[3],
>>>
>>> B = l[2], C = l[1]], F)))));
>>>
>>> is equivalent to the function p(x,a,phi,r,theta,psi) defined as
>>>
>>> xavg(a,psi,phi) := a*cos(psi-phi)$
>>> sigma2(r,theta,psi) := %e^((-2)*r)*cos(psi-theta/2)^****
>>> 2+%e^(2*r)*sin(psi-theta/2)^2$
>>> define(p(x,a,phi,r,theta,psi), sqrt(2/(%pi*sigma2(r,theta,****psi))) *
>>> exp(-2*(x-xavg(a,psi,phi))^2/****sigma2(r,theta,psi)));
>>>
>>>
>>> but I am not able to simplify q to the form of p. Can you help me?
>>>
>>> Thank you,
>>>
>>> Michele
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>>> >
>>>
>>>
>>
>