Pattern definition for replacing plus and subtract

Question

Suppose I have two functions $h_1(x)$ and $h_2(x)$, for which we know their sum and their difference: $$ h_1(x)+h_2(x) = d(x) \qquad h_1(x)-h_2(x) = i f(x) $$

I have a complicated expression, in which I'd like to replace these sums and subtractions. In principle, I could just define

h1[x_]:= 1/2 (d[x] + I f[x])
h2[x_]:= 1/2 (d[x] - I f[x])

and use Simplify. However, this is not an option as I'd like to keep the $h_i(x)$ when they appear alone (i.e. not as a sum/subtraction). So I defined a function simp as follows:

ClearAll[h1,h2,simp]
simp[a_. (b_. h1[x_] + b_. h2[x_]) + e_.] := simp[2 a b d[x] + e]
simp[a_. (b_. h1[x_] - b_. h2[x_]) + e_.] := simp[2 a b I f[x] + e]

This works for the addition, but not for the subtraction, as can be seen with the test expressions:

simp[1/2 a h1[x] + 1/2 a h2[x]]
simp[1/2 a h1[x] - 1/2 a h2[x]]

Why is this? Any idea how I can solve it?

Answer 1

The problem is that the pattern -b_. h2[x_] doesn't match the expression -1/2 a h2[x].

The reason is that putting a minus sign in front of a symbol means Times[-1, symbol] whereas a negative number is atomic (i.e. it is not represented as a positive number multiplied by -1). So a pattern like -x_ will not match a negative number:

MatchQ[-2, -x_]
(* False *)

In your case the pattern looks like Times[-1, b_. h2[x_]] and the expression looks like Times[Rational[-1,2], a h2[x], so they don't match.

One way to proceed is to use a pattern with different factors in front of the two terms and a Condition that one is the negative of the other:

simp[a_. (b_. h1[x_] + c_. h2[x_]) + e_.] /; b == -c := simp[2 a b I f[x] + e]

simp[1/2 a h1[x] - 1/2 a h2[x]]
(* simp[I a f[x]] *)