Doctest handling of Maxima's one-sided symbolic limits

[rws]

Maxima gives back limit expressions in some integral computation but Sage has no idea what that is. Consequently, working with the expression leads to failure:

sage: var('a,b,t,s,k');
sage: u(t) = exp(-(t-a)^2/(2*s^2)) + exp(-(t-b)^2/(2*s^2)) ;
sage: I=integral(u(t)*exp(-I*k*t), t, -infinity, +infinity); I
-limit(1/2*sqrt(2)*sqrt(pi)*s*erf(1/2*(I*sqrt(2)*k*s^2 - sqrt(2)*a + sqrt(2)*t)/s)*e^(-1/2*k^2*s^2 - I*a*k) + 1/2*sqrt(2)*sqrt(pi)*s*erf(1/2*(I*sqrt(2)*k*s^2 - sqrt(2)*b + sqrt(2)*t)/s)*e^(-1/2*k^2*s^2 - I*b*k), t, -Infinity, plus) + limit(1/2*sqrt(2)*sqrt(pi)*s*erf(1/2*(I*sqrt(2)*k*s^2 - sqrt(2)*a + sqrt(2)*t)/s)*e^(-1/2*k^2*s^2 - I*a*k) + 1/2*sqrt(2)*sqrt(pi)*s*erf(1/2*(I*sqrt(2)*k*s^2 - sqrt(2)*b + sqrt(2)*t)/s)*e^(-1/2*k^2*s^2 - I*b*k), t, +Infinity, minus)
sage: I.simplify_full()
...
TypeError: ECL says: Error executing code in Maxima: 

Presumably a symbolic limit function would be necessary, and to connect it with what Maxima gives back.

Is there a smaller example?

This is from ​http://ask.sagemath.org/question/26004/sage-65-maxima-simplify_full-error/

[nbruin]

Funnily enough it's not the limit placeholder that seems to be the primary issue. For the most part, the generic stuff just works for that. It's the "plus" or "minus" keyword that confuses things.

sage: L=I.operands()[0].operands()[0]
limit(1/2*sqrt(2)*sqrt(pi)*s*erf(1/2*(I*sqrt(2)*k*s^2 - sqrt(2)*a + sqrt(2)*t)/s)*e^(-1/2*k^2*s^2 - I*a*k) + 1/2*sqrt(2)*sqrt(pi)*s*erf(1/2*(I*sqrt(2)*k*s^2 - sqrt(2)*b + sqrt(2)*t)/s)*e^(-1/2*k^2*s^2 - I*b*k), t, -Infinity, plus)
sage: Ls=L.operator()(*L.operands()[:-1]); Ls
limit(1/2*sqrt(2)*sqrt(pi)*s*erf(1/2*(I*sqrt(2)*k*s^2 - sqrt(2)*a + sqrt(2)*t)/s)*e^(-1/2*k^2*s^2 - I*a*k) + 1/2*sqrt(2)*sqrt(pi)*s*erf(1/2*(I*sqrt(2)*k*s^2 - sqrt(2)*b + sqrt(2)*t)/s)*e^(-1/2*k^2*s^2 - I*b*k), t, -Infinity)
sage: Ls.simplify_full()
1/2*sqrt(pi)*e^(-1/2*k^2*s^2 - I*a*k - I*b*k)*limit(sqrt(2)*s*erf(1/2*(I*sqrt(2)*k*s^2 - sqrt(2)*b + sqrt(2)*t)/s)*e^(I*a*k) + sqrt(2)*s*erf(1/2*(I*sqrt(2)*k*s^2 - sqrt(2)*a + sqrt(2)*t)/s)*e^(I*b*k), t, -Infinity)

i.e., the direct problem is that the keyword "plus" does not get translated appropriately. In fact:

sage: L.operands()[-1].is_symbol()
True
sage: L.operands()[-1]._maxima_()
_SAGE_VAR_plus

as you can see, the maxima symbol "plus" got translated to a symbolic variable, which doesn't round-trip properly. It would have been nice to recognize "plus" on the maxima side not into a variable.

For the rest:

sage: limit(L.operands()[0],t=-oo,dir='+')
limit(1/2*sqrt(2)*sqrt(pi)*s*erf(1/2*(I*sqrt(2)*k*s^2 - sqrt(2)*a + sqrt(2)*t)/s)*e^(-1/2*k^2*s^2 - I*a*k) + 1/2*sqrt(2)*sqrt(pi)*s*erf(1/2*(I*sqrt(2)*k*s^2 - sqrt(2)*b + sqrt(2)*t)/s)*e^(-1/2*k^2*s^2 - I*b*k), t, -Infinity, plus)

which gives a bit of a pointer to the sage user interface (we shouldn't use that as "placeholder", though: it punts to maxima by default, so it shouldn't be called for interpreting the result that comes back).

[kcrisman]

Conceivably related: #19203

[rws]

The limit part appears to work now, i.e. the limit is evaluated and gives sqrt(2)*sqrt(pi)*s*e^(-1/2*k^2*s^2 - I*a*k) + sqrt(2)*sqrt(pi)*s*e^(-1/2*k^2*s^2 - I*b*k). This means the simplify part is no longer in a code path. I propose to close the original issue. Is there something left?

[rws]

No, rather this ticket should add an example calling Maxima's integral that is guaranteed to return a one-sided limit.

[rws]

Replying to rws:

No, rather this ticket should add an example calling Maxima's integral that is guaranteed to return a one-sided limit.

Also, the interface to Maxima wrt plus/minus is broken with the placeholder function as well:

sage: from sage.functions.other import symbolic_limit as slimit
sage: _ = var('plus')
sage: slimit(1/x, x, 0, plus)
limit(1/x, x, 0, plus)
sage: maxima(_)
...
TypeError: Error executing code in Maxima
CODE:
	sage4 : limit((_SAGE_VAR_x)^(-1),_SAGE_VAR_x,0,_SAGE_VAR_plus)$
Maxima ERROR:
	
limit: direction must be either 'plus' or 'minus'; found: _SAGE_VAR_plus
 -- an error. To debug this try: debugmode(true);