By the way, the problem *might* be larger than that. In Maxima, one can do :
(%i1) display2d:false;
(%o1) false
(%i2) define(h(x),integrate(f(t),t,g1(x),g2(x)));
defint: lower limit of integration must be real; found g1(x)
-- an error. To debug this try: debugmode(true);
(%i3) declare(g1,real,g2,real);
(%o3) done
(%i4) define(h(x),integrate(f(t),t,g1(x),g2(x)));
(%o4) h(x):='integrate(f(t),t,g1(x),g2(x))
(%i5) diff(h(x),x);
(%o5) f(g2(x))*'diff(g2(x),x,1)-f(g1(x))*'diff(g1(x),x,1)
which is both correct and useful.
As far as I know, one cannot declare a function as real-valued in Sage :
sage: g1=function("g1",x,domain="real")
---------------------------------------------------------------------------
TypeError Traceback (most recent call last)
<ipython-input-43-ab6717b797c7> in <module>()
----> 1 g1=function("g1",x,domain="real")
/usr/local/sage-5.10/local/lib/python2.7/site-packages/sage/calculus/var.so
in sage.calculus.var.function (sage/calculus/var.c:1253)()
/usr/local/sage-5.10/local/lib/python2.7/site-packages/sage/calculus/var.so
in sage.calculus.var.function (sage/calculus/var.c:1299)()
/usr/local/sage-5.10/local/lib/python2.7/site-packages/sage/symbolic/function_factory.pyc
in function(s, *args, **kwds)
310 names = [s]
311
--> 312 funcs = [function_factory(name, **kwds) for name in names]
313
314 if len(args) > 0:
TypeError: function_factory() got an unexpected keyword argument 'domain'
But also :
sage: g1=function("g1",x)
sage: assume(g1,"real")
---------------------------------------------------------------------------
TypeError Traceback (most recent call last)
<ipython-input-45-bd761ce74d13> in <module>()
----> 1 assume(g1,"real")
/usr/local/sage-5.10/local/lib/python2.7/site-packages/sage/symbolic/assumptions.pyc
in assume(*args)
410 else:
411 try:
--> 412 x.assume()
413 except KeyError:
414 raise TypeError, "assume not defined for objects of
type '%s'"%type(x)
/usr/local/sage-5.10/local/lib/python2.7/site-packages/sage/symbolic/expression.so
in sage.symbolic.expression.Expression.assume
(sage/symbolic/expression.cpp:8750)()
TypeError: self (=g1(x)) must be a relational expression
but also :
sage: assume(g1(x),"real")
---------------------------------------------------------------------------
TypeError Traceback (most recent call last)
<ipython-input-46-0d7aa50473b1> in <module>()
----> 1 assume(g1(x),"real")
/usr/local/sage-5.10/local/lib/python2.7/site-packages/sage/symbolic/assumptions.pyc
in assume(*args)
410 else:
411 try:
--> 412 x.assume()
413 except KeyError:
414 raise TypeError, "assume not defined for objects of
type '%s'"%type(x)
/usr/local/sage-5.10/local/lib/python2.7/site-packages/sage/symbolic/expression.so
in sage.symbolic.expression.Expression.assume
(sage/symbolic/expression.cpp:8750)()
TypeError: self (=g1(x)) must be a relational expression
hence :
sage: g2(x)=function("g2",x)
sage: h(x)=integrate(f(t),t,g1(x), g2(x))
---------------------------------------------------------------------------
RuntimeError Traceback (most recent call last)
<ipython-input-48-fdf29d67d922> in <module>()
----> 1 __tmp__=var("x"); h = symbolic_expression(integrate(f(t),t,g1(x),
g2(x))).function(x)
/usr/local/sage-5.10/local/lib/python2.7/site-packages/sage/misc/functional.pyc
in integral(x, *args, **kwds)
738 """
739 if hasattr(x, 'integral'):
--> 740 return x.integral(*args, **kwds)
741 else:
742 from sage.symbolic.ring import SR
/usr/local/sage-5.10/local/lib/python2.7/site-packages/sage/symbolic/expression.so
in sage.symbolic.expression.Expression.integral
(sage/symbolic/expression.cpp:39592)()
/usr/local/sage-5.10/local/lib/python2.7/site-packages/sage/symbolic/integration/integral.pyc
in integrate(expression, v, a, b, algorithm)
686 return indefinite_integral(expression, v)
687 else:
--> 688 return definite_integral(expression, v, a, b)
689
690 integral= integrate
/usr/local/sage-5.10/local/lib/python2.7/site-packages/sage/symbolic/function.so
in sage.symbolic.function.Function.__call__
(sage/symbolic/function.cpp:5114)()
/usr/local/sage-5.10/local/lib/python2.7/site-packages/sage/symbolic/integration/integral.pyc
in _eval_(self, f, x, a, b)
171 for integrator in self.integrators:
172 try:
--> 173 return integrator(*args)
174 except NotImplementedError:
175 pass
/usr/local/sage-5.10/local/lib/python2.7/site-packages/sage/symbolic/integration/external.pyc
in maxima_integrator(expression, v, a, b)
19 result = maxima.sr_integral(expression,v)
20 else:
---> 21 result = maxima.sr_integral(expression, v, a, b)
22 return result._sage_()
23
/usr/local/sage-5.10/local/lib/python2.7/site-packages/sage/interfaces/maxima_lib.pyc
in sr_integral(self, *args)
745 raise ValueError, "Computation failed since Maxima
requested additional constraints; using the 'assume' command before
integral evaluation *may* help (example of legal syntax is 'assume(" +
s[4:k] +">0)', see `assume?` for more details)\n" + s
746 else:
--> 747 raise error
748
749 def sr_sum(self,*args):
RuntimeError: ECL says: Error executing code in Maxima: defint: lower limit
of integration must be real; found g1(x)
Should this problem be added to
Trac#14976<http://trac.sagemath.org/ticket/14976>or reported elsewhere ?
Emmanuel
Charpentier
[ Snip... ]
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