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Defining a function within a module

• To: mathgroup at smc.vnet.net
• Subject: [mg22462] Defining a function within a module
• From: "David Park" <djmp at earthlink.net>
• Date: Wed, 8 Mar 2000 02:21:59 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

Dear MathGroup,

I wish to define and use a function within a module, using an expression passed to
the module. This is a simpler example, where I simply return the function. If I use
the mkfun routine, described in Section 2.6.5 of the Mathematica Book, everything
works. Here x is a global variable without a value, and expr is an expression
containing x.

foo1[expr_] :=
  Module[{mkfun},
    mkfun[x_, body_] := Function[x, body];
    mkfun[x, expr]
    ]

foo1[x^2]
Function[x, x^2]

But, since mkfun simply substitutes x and body into the unevaluated Function, the
following should also work.

foo2[expr_] :=
  Module[{g},
    g = Hold[Function[x, body]] /. body -> expr;
    ReleaseHold[g]
    ]

foo2[x^2]
Function[x, x^2]

It does, but body is a global variable and so the routine will fail if body has a
value. Therefore, make body internal to the module.

foo3[expr_] :=
  Module[{g, body},
    g = Hold[Function[x, body]] /. body -> expr;
    ReleaseHold[g]
    ]

foo3[x^2]
Function[x$, x^2]

This does not work because the variable name has been changed. Why was it changed
even though it was within a Hold? Why did making body a Module variable rather than a
global variable cause it to be changed? But in the following routine, we can change
x$ back to x.

foo4[expr_] :=
  Module[{g, body},
    g = Hold[Function[x, body]] /. body -> expr;
    ReleaseHold[g] /. x$ -> x
    ]

foo4[x^2]
Function[x, x^2]

In fact, we don't even need the Hold, since it didn't work anyway.

foo5[expr_] :=
  Module[{},
    Function[x, expr] /. x$ -> x
    ]

And here is an even simpler method for obtaining our function.

foo6[expr_] :=
  Module[{g},
    g[x, expr] /. g -> Function
    ]

foo6[x^2]
Function[x, x^2]

I wish I had a better understanding of when Function does and does not change the
variable name.

David Park
djmp at earthlink.net
http://home.earthlink.net/~djmp/

• Follow-Ups:
  • Re: Defining a function within a module
    • From: Hartmut Wolf <hwolf@debis.com>