Difference between f1[a_, b_] := a[b]^2 and f2[a_][b_] := a[b]^2 etc.

Questions
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1

Colleagues, please - it is a difference betwen theese defintifions? It seems to me, that tey are equivalent – it is true?

In[1]:= f1[a_, b_] := a[b]*a[2 b]

In[2]:= f2[a_][b_] := a[b]*a[2 b]

In[3]:= f3[a_[b_]] := a[b]*a[2 b]

In[4]:= {f1[Func, x], f2[Func][x], f3[Func[x]]}

Out[4]= {Func[x] Func[2 x], Func[x] Func[2 x], Func[x] Func[2 x]}

Can you give me a link for study this theme? Thanks!

2

The first two accomplish the very same thing, for as long as you keep them in this form.

The third one won’t always behave the same way though. Consider

fun[x_] := x^2

f3[a_[b_]] := a[b]*a[2 b]

Then f3[fun[z]] will evaluate to f3[z^2] because fun[z] evaluates first. However, if you set SetAttributes[f3, HoldAll], then it will behave more similarly to f1 and f2:

f3[fun[z]] will now first evaluate to fun[z]*fun[2 z], then to 4 z^4.

This is not the complete story, but giving full details is not always educational. So instead I’ll leave you with this reference, which (I think) discusses all the relevant issues:

In particular, look at the section on Nonstandard Evaluation and the Hold... attributes.