Mathematica defines a function as list

Question

This might be a very silly question but i'm defining a function as:

\[Chi][AR_] := ((AR^(2) - 1)/(AR^(2) + 1));

which is clearly a scalar function of AR; however when I do:

Dimensions[\[Chi][AR]]

It tells me this is a vector of two elements. Inded i tried to check what those elements where and i got:

In[21]:= \[Chi][AR][[2]]

Out[21]= 1/(1 + AR^2)

while the first element is:

In[22]:= \[Chi][AR][[1]]

Out[22]= -1 + AR^2

this is really annoying, anybody does why it's happening? (Maybe it's a silly question the solution is obvious but i'm too much asleep to notice it...)

thanks in advance

Answer 1

Dimensions>>Generalizations and Extensions says:

Dimensions works with any head,not just List.

So,

Dimensions[foo[x,y]]
(* {2} *)

and

foo[[1]]
(* x *)

foo[[2]]
(* y *)

In your case, checking the FullForm of χ[z]

FullForm[χ[z]]

gives

Times[Plus[-1,Power[z,2]],Power[Plus[1,Power[z,2]],-1]]

Thus,

χ[z][[0]]
(* Times *)

χ[z][[1]]
(* -1 + z^2 *)

χ[z][[2]]
(* 1/(1 + z^2) *)