Sagemath :Replace an expression in a formula by a function define previously

Question

I have defined a function F by $F(x)=(f(x)+h)/g(x)$ where $f, g$ are functions and $h$ a variable.

Suppose that I write $f(x)+h$, how can i ask to sagemath to use F in the result: $F(x)*g(x)$

Thanks

Answer 1

This challenge resists a bit and requires perseverance.

Setting: three functions, one variable, one relation.

sage: f = function('f')
sage: g = function('g')
sage: F = function('F')
sage: h = SR.var('h')
sage: relation = F(x) == (f(x) + h)*g(x)

Warm-up: substituting a subexpression by a chosen replacement in an expression works:

sage: a = (f(x) + h)*g(x)
sage: a
(h + f(x))*g(x)
sage: a.subs({f(x) + h: F(x)})
F(x)*g(x)

Sage will however not go out of its lengths and rearrange things to force a substitution we have in mind but don't describe precisely:

sage: b = f(x) + h
sage: b
h + f(x)
sage: b.subs({(f(x) + h)/g(x): F(x)})
h + f(x)

There just was no actual occurrence of the given subexpression to substitute, so nothing happened.

If we tell it exactly what to substitute, it will:

sage: b.subs({f(x) + h: F(x)*g(x)})
F(x)*g(x)

If we are willing to prepare the ground "by hand", we can divide by g(x) to make the desired subexpression appear, remultiply immediately to cancel out the division, but hold the multiplication so the cancellation does not actually get simplified, and then substitute:

sage: bb = (b / g(x)).mul(g(x), hold=True)
sage: bb
((h + f(x))/g(x))*g(x)
sage: bb.subs({(f(x) + h)/g(x): F(x)})
F(x)*g(x)

Now that we are warmed up, let us restate our goal: get rid of any occurrence of f(x) in an expression, applying appropriate substitutions, derived from the relation in the question.

Recall the relation that we want to use for replacements.

sage: relation = F(x) == (f(x) + h)*g(x)

Introduce an auxiliary variable, say z, and substitute it for the expression to remove in the relation, and solve to get our replacement dictionary:

sage: z = SR.var('z')
sage: fz = {f(x): z}
sage: rep = solve(relation.subs(fz), z, solution_dict=True)
sage: rep
[{z: -(h*g(x) - F(x))/g(x)}]

The dictionary came in a list, let's unwrap it out of that list.

sage: rep = rep[0]
sage: rep
{z: -(h*g(x) - F(x))/g(x)}

Now we can use that to substitute, first replacing f(x) by z, then z by the result of solving for z in the modified relation:

sage: b
h + f(x)
sage: b.subs(fz).subs(rep)
h - (h*g(x) - F(x))/g(x)

Almost there... try simplifying...

sage: b.subs(fz).subs(rep).simplify()
h - (h*g(x) - F(x))/g(x)

Didn't work... try harder...

sage: b.subs(fz).subs(rep).simplify_full()
F(x)/g(x)

We did it!