4
$\begingroup$

I have a symbolic function (which is in fact an operator) f[expr, x] where the second argument is a local variable for f (kind of like how Integrate[expr, x] is set up).

How do I make analysis functions like D act on f[expr, x] similar to how D acts on Inactive[Integrate]:

D[Inactive[Integrate][Sin[p x^2], x], p]
Inactive[Integrate][x^2 Cos[p x^2], x]   
D[f[Sin[p x^2], x], p]
x^2 Cos[p x^2] Derivative[1, 0][f][Sin[p x^2], x] (*Wrong*)

This output is wrong because x^2 Cos[p x^2] has incorrectly been pulled out of f.

Edit: It should work if f appears arbitrarily deep inside expr. Another test case that needs to work:

D[a * f[Sin[p x^2], x], p]

with the desired result:

a*f[x^2*Cos[p*x^2], x] (*Desired*)
$\endgroup$
  • 3
    $\begingroup$ You can overload D on f with e.g. f /: D[_f, x_] := someOperation[f, x] -- do you mean something else? $\endgroup$ – Mr.Wizard Jun 17 '16 at 15:14
  • 2
    $\begingroup$ I should have written f /: D[body_f, x_] := someOperation[body, x] $\endgroup$ – Mr.Wizard Jun 17 '16 at 15:24
7
$\begingroup$

I think this is good case for TagSetDelayed. Rather than redefining D, you associate f with the desired upvalue.

f /: D[f[exp_, x_], p_] := f[D[exp, p], x]

D[f[Sin[p x^2], x], p]
(* f[x^2 Cos[p x^2], x] *)
| improve this answer | |
$\endgroup$
  • 1
    $\begingroup$ This is nice; but if D doesn't directly wrap f, it doesn't seem to work: for example D[a * f[Sin[p x^2], x], p]. Can you suggest a workaround? $\endgroup$ – QuantumDot Jun 17 '16 at 15:36
  • $\begingroup$ @Quantum, is f supposed to be a linear operator, or not? It should not be too hard to modify chuy's answer to incorporate a linearity property, if so. $\endgroup$ – J. M.'s technical difficulties Jun 17 '16 at 16:12
  • 1
    $\begingroup$ @QuantumDot You can use NonConstants -> {f} option. This will prevent D from using built in chain rule on expressions with head f and result will contain D[f[...], ...], so no matter how deep f is inside differentiated expression its D-overriding UpValues will be used. You'll also need to change p_ to p__, so that D with more than two arguments (including options) will be matched. $\endgroup$ – jkuczm Jun 17 '16 at 22:08
  • $\begingroup$ @jkuczm NonConstants solved my problem. Please post as answer and I will accept. $\endgroup$ – QuantumDot Jun 18 '16 at 5:56
1
$\begingroup$

Using jkuczm comment together with chuy's answer, I have a satisfactory solution.

Consider the following example expression (f is a symbolic operator introduced in OP's question):

expr = Cos[ f[Sin[p x^2], x] ]
  1. When using D, set option NonConstants->{f} to prevent it from operating on f. This works for any expression where f appears arbitrarily deep. (jkuczm)

    D[expr, p, NonConstants -> {f}]
    
    (*  -D[f[Sin[p x^2], x], p, NonConstants -> {f}] Sin[f[Sin[p x^2], x]] *)
    
  2. Then, define a rule specifying how D should act on f (chuy):

    f /: D[f[exp_, x_], p_, NonConstants -> {f}] := 
      f[D[exp, p, NonConstants -> {f}], x]
    

Now we have:

D[expr, p, NonConstants -> {f}]

(* -f[x^2 Cos[p x^2], x] Sin[f[Sin[p x^2], x]] *)

as desired.

| improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.