using jquery $.ajax to call a PHP function

Use $.ajax to call a server context (or URL, or whatever) to invoke a particular 'action'. What you want is something like:

$.ajax({ url: '/my/site',
         data: {action: 'test'},
         type: 'post',
         success: function(output) {
                      alert(output);
                  }
});

On the server side, the action POST parameter should be read and the corresponding value should point to the method to invoke, e.g.:

if(isset($_POST['action']) && !empty($_POST['action'])) {
    $action = $_POST['action'];
    switch($action) {
        case 'test' : test();break;
        case 'blah' : blah();break;
        // ...etc...
    }
}

I believe that's a simple incarnation of the Command pattern.

I developed a jQuery plugin that allows you to call any core PHP function or even user defined PHP functions as methods of the plugin: jquery.php

After including jquery and jquery.php in the head of our document and placing request_handler.php on our server we would start using the plugin in the manner described below.

For ease of use reference the function in a simple manner:

    var P = $.fn.php;

Then initialize the plugin:

P('init', 
{
    // The path to our function request handler is absolutely required
    'path': 'http://www.YourDomain.com/jqueryphp/request_handler.php',

    // Synchronous requests are required for method chaining functionality
    'async': false,

    // List any user defined functions in the manner prescribed here
            // There must be user defined functions with these same names in your PHP
    'userFunctions': {

        languageFunctions: 'someFunc1 someFunc2'
    }
});             

And now some usage scenarios:

// Suspend callback mode so we don't work with the DOM
P.callback(false);

// Both .end() and .data return data to variables
var strLenA = P.strlen('some string').end();
var strLenB = P.strlen('another string').end();
var totalStrLen = strLenA + strLenB;
console.log( totalStrLen ); // 25

// .data Returns data in an array
var data1 = P.crypt("Some Crypt String").data();
console.log( data1 ); // ["$1$Tk1b01rk$shTKSqDslatUSRV3WdlnI/"]

Demonstrating PHP function chaining:

var data1 = P.strtoupper("u,p,p,e,r,c,a,s,e").strstr([], "C,A,S,E").explode(",", [], 2).data();
var data2 = P.strtoupper("u,p,p,e,r,c,a,s,e").strstr([], "C,A,S,E").explode(",", [], 2).end();
console.log( data1, data2 );

Demonstrating sending a JSON block of PHP pseudo-code:

var data1 = 
        P.block({
    $str: "Let's use PHP's file_get_contents()!",
    $opts: 
    [
        {
            http: {
                method: "GET",
                header: "Accept-language: en\r\n" +
                        "Cookie: foo=bar\r\n"
            }
        }
    ],
    $context: 
    {
        stream_context_create: ['$opts']
    },
    $contents: 
    {
        file_get_contents: ['http://www.github.com/', false, '$context']
    },
    $html: 
    {
        htmlentities: ['$contents']
    }
}).data();
    console.log( data1 );

The backend configuration provides a whitelist so you can restrict which functions can be called. There are a few other patterns for working with PHP described by the plugin as well.