You can add a class to show it open when the document loads if thats what you mean? Why not just do it with CSS?
$(document).ready(function(){
$('#advanced-search-btn').addClass('open');
});
In regards to you want to remove the JS function, why not just delete this also?
/** ADVANCED SERACH */
if ($.cookie('ts_advanced_search') == 'open') {
$('#advanced-search').show();
$('#advanced-search-btn').addClass('open');
}
$('#advanced-search-btn').click(function () {
if ($("#advanced-search").is(":visible")) {
$.cookie('ts_advanced_search', 'closed',{ expires: 60, path: '/' });
$("#advanced-search div").animate(
{
opacity: "0"
},
150,
function(){
$('#advanced-search-btn').removeClass('open');
$("#advanced-search").slideUp(150);
}
);
}
else {
$("#advanced-search").slideDown(150, function(){
$.cookie('ts_advanced_search', 'open',{ expires: 60, path: '/' });
$("#advanced-search div").animate(
{
opacity: "1"
},
150
);
$('#advanced-search-btn').addClass('open');
});
}
});