Calling a javascript function recursively

I can create a recursive function in a variable like so:

/* Count down to 0 recursively.
 */
var functionHolder = function (counter) {
    output(counter);
    if (counter > 0) {
        functionHolder(counter-1);
    }
}

With this, functionHolder(3); would output 3 2 1 0. Let's say I did the following:

var copyFunction = functionHolder;

copyFunction(3); would output 3 2 1 0 as above. If I then changed functionHolder as follows:

functionHolder = function(whatever) {
    output("Stop counting!");

Then functionHolder(3); would give Stop counting!, as expected.

copyFunction(3); now gives 3 Stop counting! as it refers to functionHolder, not the function (which it itself points to). This could be desirable in some circumstances, but is there a way to write the function so that it calls itself rather than the variable that holds it?

That is, is it possible to change only the line functionHolder(counter-1); so that going through all these steps still gives 3 2 1 0 when we call copyFunction(3);? I tried this(counter-1); but that gives me the error this is not a function.

Using Named Function Expressions:

You can give a function expression a name that is actually private and is only visible from inside of the function ifself:

var factorial = function myself (n) {
    if (n <= 1) {
        return 1;
    }
    return n * myself(n-1);
}
typeof myself === 'undefined'

Here myself is visible only inside of the function itself.

You can use this private name to call the function recursively.

See 13. Function Definition of the ECMAScript 5 spec:

The Identifier in a FunctionExpression can be referenced from inside the FunctionExpression's FunctionBody to allow the function to call itself recursively. However, unlike in a FunctionDeclaration, the Identifier in a FunctionExpression cannot be referenced from and does not affect the scope enclosing the FunctionExpression.

Please note that Internet Explorer up to version 8 doesn't behave correctly as the name is actually visible in the enclosing variable environment, and it references a duplicate of the actual function (see patrick dw's comment below).

Using arguments.callee:

Alternatively you could use arguments.callee to refer to the current function:

var factorial = function (n) {
    if (n <= 1) {
        return 1;
    }
    return n * arguments.callee(n-1);
}

The 5th edition of ECMAScript forbids use of arguments.callee() in strict mode, however:

(From MDN): In normal code arguments.callee refers to the enclosing function. This use case is weak: simply name the enclosing function! Moreover, arguments.callee substantially hinders optimizations like inlining functions, because it must be made possible to provide a reference to the un-inlined function if arguments.callee is accessed. arguments.callee for strict mode functions is a non-deletable property which throws when set or retrieved.

You can access the function itself using arguments.callee ^[MDN]:

if (counter>0) {
    arguments.callee(counter-1);
}

This will break in strict mode, however.

You can use the Y-combinator: (Wikipedia)

// ES5 syntax
var Y = function Y(a) {
  return (function (a) {
    return a(a);
  })(function (b) {
    return a(function (a) {
      return b(b)(a);
    });
  });
};

// ES6 syntax
const Y = a=>(a=>a(a))(b=>a(a=>b(b)(a)));

// If the function accepts more than one parameter:
const Y = a=>(a=>a(a))(b=>a((...a)=>b(b)(...a)));

And you can use it as this:

// ES5
var fn = Y(function(fn) {
  return function(counter) {
    console.log(counter);
    if (counter > 0) {
      fn(counter - 1);
    }
  }
});

// ES6
const fn = Y(fn => counter => {
  console.log(counter);
  if (counter > 0) {
    fn(counter - 1);
  }
});