How to override `?NumericQ`

Question

I have a function myFunc which I is to be displayed cleanly if arguments are symbolic, but which can also be numerically evaluated. I do this by defining a complicated auxiliary function pR which is evaluated only if its arguments are numerical (probably not a good idea?).

kin[a_, b_, c_] = a^2 + b^2 + c^2 - 2 a b - 2 a c - 2 b c;
pR[s_?NumericQ, m0_?NumericQ, m1_?NumericQ] := 
  1/s*Sqrt[kin[s, m0^2, m1^2]]*Log[(2 m0 m1)/(-s + m0^2 + m1^2 - Sqrt[kin[s, m0^2, m1^2]])]

The function (watered-down) I wish to define is

myFunc[n_?IntegerQ, s_, m0_, m1_] := 
  Sum[
    Binomial[n + 1, 2 idx3 + 1]*((s + m0^2 - m1^2)/(2 s))^(n - 2 idx3) 
     *(kin[s, m0^2, m1^2]/(4 s^2))^idx3, {idx3, 0, (n + 1)/2}] pR[s, m0, m1];

So for example:

myFunc[4,s,m,m]//Simplify
(*  ((m^4 - 3 m^2 s + s^2) pR[s, m, m])/s^2   *)

and the complicated mess is in pR. The problem I'm running into is:

1. I don't know how to code it so that the user can forcibly display the function pR in its entirety -- even if its arguments are symbolic, overriding the ?NumericQ.
2. I would like to be able to take derivatives (D), Limits, and perform a Taylor Series expansion on myFunc appropriately handling the auxiliary function pR. This should trigger the override in the previous point.

Any ideas? Thanks!

Answer 1

For the first question, you just use an expansion function which applies the pattern without the numeric constraint:

expandPR[exp_] := exp /. pR[s_, m0_, m1_] :> 
 1/s*Sqrt[kin[s, m0^2, m1^2]]*Log[(2 m0 m1)/(-s + m0^2 + m1^2 - Sqrt[kin[s, m0^2, m1^2]])]

myFunc[4,s,m,m]//expandPR//Simplify

((m^4 - 3 m^2 s + s^2) pR[s, m, m])/s^2

myFunc[4,s,m,m]//expandPR//Simplify

(Sqrt[s (-4 m^2 + s)] (m^4 - 3 m^2 s + s^2) Log[-((
 2 m^2)/(-2 m^2 + s + Sqrt[s (-4 m^2 + s)]))])/s^3

For the second one, you can always expand the definition and find the derivative. If that doesn't work for you, if for instance you want to express the derivative with respect to other pR functions, you can always define custom up-values for the derivative operator D and try to do some contraction over your function definition or something like that:

 pR /: D[pR[s_, m0_, m1_], d_] := D[pR[s, m0, m1] // expandPR, d]

 D[pR[x, k, m], k]

(1/(2 k m x)) Sqrt[k^4 - 2 k^2 m^2 + m^4 - 2 k^2 x - 2 m^2 x + x^2] (k^2 + m^2 - x - Sqrt[ k^4 - 2 k^2 m^2 + m^4 - 2 k^2 x - 2 m^2 x + x^2]) (-(( 2 k m (2 k - (4 k^3 - 4 k m^2 - 4 k x)/( 2 Sqrt[k^4 - 2 k^2 m^2 + m^4 - 2 k^2 x - 2 m^2 x + x^2])))/(k^2 + m^2 - x - Sqrt[ k^4 - 2 k^2 m^2 + m^4 - 2 k^2 x - 2 m^2 x + x^2])^2) + (2 m)/( k^2 + m^2 - x - Sqrt[ k^4 - 2 k^2 m^2 + m^4 - 2 k^2 x - 2 m^2 x + x^2])) + ((4 k^3 - 4 k m^2 - 4 k x) Log[(2 k m)/( k^2 + m^2 - x - Sqrt[ k^4 - 2 k^2 m^2 + m^4 - 2 k^2 x - 2 m^2 x + x^2])])/( 2 x Sqrt[k^4 - 2 k^2 m^2 + m^4 - 2 k^2 x - 2 m^2 x + x^2])