changing a symbol into a function variable

Question

Struggled for a while on this problem and hopefully you can help... I have a mathematical expression that currently has a symbol where I used to have a function variable. How do I make the symbol back into a variable, so that my expression becomes a function again? My function became an expression when I saved it in a matrix, and so now I'm trying to make the different matrix elements back into functions after having constructed the matrix. Thanks in advance.

INCLUDING SOME CODE (my first time contributing on this website...)

Imagine a For loop sum (I've removed a substantial amount of code that's irrelevant for now)

Pstatesalpha[λ_]=0;
RankMatrix={};
elecdipole (*some vector of numbers delem*)
SumPstatesalpha[λ_, Alpharank_]  (*a predefined function of λ, Alpharank*)

For[Alpharank = 1, Alpharank <= 3, Alpharank++,(
   For[jsum = 1, jsum <=Length[elecdipole], jsum++,(
Pstatesalpha[λ_] = (delem*a0)^2*SumPstatesalpha[λ,Alpharank] + Pstatesalpha[λ];
   )]
AppendTo[RankMatrix, {Alpharank,Pstatesalpha[λ]}]
)]

Note that I've saved Pstatesalpha[λ] as a function into RankMatrix. When I want to access (for instance)

RankMatrix[[1,2]]

Mathematica returns an expression that includes $\lambda$, but I can't use it as a function anymore.

Answer 1

Second try.

In the comments below Spawn's answer you said that you are trying to construct a table of functions. Does this do what you want?:

Pstatesalpha[λ_] = 0;

RankMatrix = {};

For[jsum = 1, jsum <= 3, jsum++,
 Pstatesalpha[λ_] = (delem*a0)^2*SumPstatesalpha[λ] + Pstatesalpha[λ];
 AppendTo[RankMatrix, {Alpharank, Function @@ {λ, Pstatesalpha[λ]}}]
]

The key detail is appending Function @@ {λ, Pstatesalpha[λ]} which will build a function with the evaluated expression Pstatesalpha[λ] as the body and the parameter λ.

Now:

RankMatrix[[1, 2]][17]

a0^2 delem^2 SumPstatesalpha[17]

---

Since the above was Accepted (thanks) let me explain a bit better.

I use the method fn @@ {args} to evaluate args and pass them to a function fn (using Apply) that otherwise holds its arguments (has a Hold attribute).

In the code above, copied from the question, there is a global symbol λ which if assigned a value will break the code. One method to avoid this is to use a Formal Symbol, entered Esc$xEsc where x is any a-z letter. These symbols exist specifically for cases like this. Another method is to use Slot as in the syntax # + 2 & (the FullForm of # is Slot[1]). That might look like this (in context):

AppendTo[RankMatrix, {Alpharank, Evaluate[Pstatesalpha[#]] &}]

Evaluate is used instead of Apply and List, but many methods are possible:

Evaluate[Pstatesalpha[#]] &

Function @@ {Pstatesalpha[#]}

With[{body = Pstatesalpha[#]}, body &]

Pstatesalpha[#] /. x_ :> (x &)

With each of these Pstatesalpha[#] is evaluated outside of Function and then inserted into it.

Answer 2

Ok if I understand correctly you want to create a list of functions for example

RankMatrix={}

For[jsum = 1, jsum <= 3, jsum++,
       Pstatesalpha[jsum][x_] = (delem*a0)^2*SumPstatesalpha[x] + Pstatesalpha[x];
       AppendTo[RankMatrix, {Alpharank,Pstatesalpha[jsum]}]
    ]

Now on RankMatrix you have a matrix including a set of functions and to access one of them : RankMatrix[[1,2]][λ]

In Mathematica we often don't use the procedural methods employed in the usual programming languages, like for, do etc. I give you an equivalent solution using Mathematica's command for matrix creation:

RankMatrix= Array[Function[Pstatesalpha[#][x_] = (delem*a0)^2*SumPstatesalpha[x] + 
                  Pstatesalpha[x];{Alpharank, Pstatesalpha[#]}], 
                  3]