Definition of complex conjugate?

Question

I came across the following way of defining complex conjugate in mathematica [a]:

SuperStar[y_] := y /. Complex[u_, v_] -> Complex[u, -v]

I am able to run this program, but I don't understand the way of defining it.

For example, let us define a function f as

f[x_] := x^2 + 1

Then I define (instead of *, I am using n):

f_^n := f /. t_ -> t + 1

Evaluating this results in following output:

"Tag Power in f_^n is Protected. "

However, for * in the superscript, it works fine.

I want to understand how they have defined the function? Does it use pattern matching? What is special with "*" in superscript position?

[a]: Patric Tam: A physicist guide's to mathematica

Answer 1

It looks like if you define a function such as

$g\_^+ := /.g \rightarrow -g$

where the + is entered using a superscript, then $5^+$ gets you -5. So it seems that it doesn't like numbers or variables, but other kinds of symbols are okay. I think when you do f_^n, it replaces it with Power[f_, n] := f/.t_ -> t+1. Power is protected so that you don't accidentally change it and all of a sudden break your ability to calculate powers properly.

It looks to me like it is using pattern matching to find any instance where you have some thing with a superscript star and replaces it. Similarly with my + example, you can do $(a b c)^+$ and get $-abc$.

It looks like you're mostly interested in understanding function definitions and pattern matching, so you probably already know this but I thought I'd mention it just in case, MMA does have a built-in conjugate symbol that can be entered as Esc conj Esc.