I'm trying to understand how to use symbolic expressions as arguments. The following example represents my problem:
Q[f_, s0_] := Q[f, s0] = D[f[s0], s0]
Q[Q[f, s0], s0]
where Q is simply first derivative of function f. As a result of the second line, I expect to see second derivative of f. But I'm getting
f'[s0]'[s0] + f''[s0][s0]
Could you help me please find a way to obtain a correct answer?
Thank you in advance!
You are effectively calling Q with two different argument patterns (Q[foo, t] in the first call in the nested calls and Q[bar[t],t] in the outer call) but your Q is defined only for the first argument pattern. If you add the definition for the second signature, you get the desired result:
ClearAll[Q]
Q[f_, s0_] := Q[f, s0] = D[f[s0], s0]
Q[f_[s0_], s0_] := Q[f, s0]
Q[Q[f, s0], s0]
(f^′′)[s0]
Q[Q[f[s0], s0], s0]
(f^′′)[s0]
I suggest you using Derivative:
Q[f_] := Derivative[1][f]
Q[f_, s0_] := Derivative[1][f][s0]
Q[f, s0]
f'[s0]
Q[Q[f], s0]
f''[s0]
BTW, the number 1 in the first [...] after Derivative can be other integers, so that the nested application can actually be lifted.
Try this:
Q[function_,variable_,n_? Positive]/;Element[n,Integers]:=Nest[D[#,variable]&,function[variable],n]
Example:
Q[f,s0,1]
Q[f,s0,2]
(*f'[s0]*)
(*f''[s0]*)
Q[f_, s0_] := Q[f, s0] = D[f, s0]in a fresh kernel, and then evaluateQ[Q[f, s0], s0]again. $\endgroup$