How to correctly specify variables in a function? [duplicate]

Question

There is an expression

(Sqrt[\[Pi]] (a b^2 d^3 + a (b^2 d^3 + 2 a (b + b d^3))) r[1, 1] r[1, 4])/(2 c^2 (b + b d^3)^(3/2)) - ( Sqrt[\[Pi]] (a b^2 d^4 + a (b^2 d^4 + 2 c (b d + b d^3))) r[5, 2] r[ 1, 4])/(2 a^2 (b d + b c^3)^(3/2))

where a, b, c, d, r[1, 1], r[1, 4], r[5, 2] are variables.

I would like to write this expression as a function f where a, b, c, d, r[1, 1], r[1, 4], r[5, 2] are variables. It can be indicated that a, b, c and d are variables as follows f[a_,b_,c_,d_]:= the expression, but how can I specify that r[1, 1], r[1, 4] and r[5, 2] are also variables f[a_,b_,c_,d_,...]:= the expression?

Answer 1

I'm pretty sure I'm misunderstanding something, so this isn't so much an answer as an attempt to force more clarity.

Here is a function:

theFunction[arg1_, arg2_, arg3_, arg4_, arg5_, arg6_, arg7_] := 
  ((arg1*arg2^2*arg4^3 + arg1*(arg2^2*arg4^3 + 2*arg1*(arg2 + arg2*arg4^3)))*arg5*arg6*Sqrt[Pi])/
   (2*arg3^2*(arg2 + arg2*arg4^3)^(3/2)) - 
  ((arg1*arg2^2*arg4^4 + arg1*(arg2^2*arg4^4 + 2*arg3*(arg2*arg4 + arg2*arg4^3)))*arg6*arg7*Sqrt[Pi])/
   (2*arg1^2*(arg2*arg3^3 + arg2*arg4)^(3/2))

Applying the function as follows will reproduce your original expression:

theFunction[a, b, c, d, r[1, 1], r[1, 4], r[5, 2]]

And so, in the specific example you gave in one of your comments:

theFunction[1, 2, 1, 1, 2, 3, 5]
(* -9*Sqrt[Pi] *)

Is that what you're after?

Answer 2

Using r essentially as a private keyword is dangerous, if r is not protected. For instance, setting r = 7 will generally mess up any code that assumes r is an undefined symbol. You can Protect it, which will protect your code from most bugs, but Protect can be overridden. (There is also Locked, if you want, then the following won't work.)

That aside, assuming r is safe but not protected:

ClearAll[f];
f[a_, b_, c_, d_, r11_, r14_, r52_] := Block[{r},
   (*Unprotect[r];*)  (* uncomment if r is protected *)
   r[1, 1] = r11; r[1, 4] = r14; r[5, 2] = r52;
   (*Protect[r];*)    (* uncomment if r is protected *)
   (* OP's expression *)
   (Sqrt[\[Pi]] (a b^2 d^3 + a (b^2 d^3 + 2 a (b + b d^3))) r[1, 1] r[
        1, 4])/(2 c^2 (b + b d^3)^(3/2)) - (Sqrt[\[Pi]] (a b^2 d^4 + 
         a (b^2 d^4 + 2 c (b d + b d^3))) r[5, 2] r[1, 
        4])/(2 a^2 (b d + b c^3)^(3/2))
   ];

f[1, 2, 1, 1, 2, 3, 5]
(*  -9 Sqrt[\[Pi]]  *)

Again, I will point out that we're assuming it is safe to treat Global`r as a reserved keyword. If this is combined with other code that uses r in a different way, you may find you have bugs. I offer this answer in the spirit of showing the difficulties of using a tool (Mma) in ways it wasn't really designed for. It would be better practice to rewrite your r variables as r11, r14, r52, which is essentially what @lericr's answer recommends.