In ES5 and above, there is no access to that information.
In older versions of JS you can get it by using arguments.callee
.
You may have to parse out the name though, as it will probably include some extra junk. Though, in some implementations you can simply get the name using arguments.callee.name
.
Parsing:
function DisplayMyName()
{
var myName = arguments.callee.toString();
myName = myName.substr('function '.length);
myName = myName.substr(0, myName.indexOf('('));
alert(myName);
}