Need:

All pages in the new website will be generated using WordPress, but the homepage, because of its sophistication, will be a php page developed independently of WordPress.



Issue:

When a user enters http://mysite.com into a browser I would like them to default to the page that was developed independently of WordPress (for example, index3.php). Index3.php will be homepage with links to all other pages which are WordPress pages. All WordPress pages use the index.php page in the root WordPress directory which is the same as the web root (http://mysite.com). Would I be able to redirect for http://mysite.com/ to http://mysite.com/index3.php but all other WordPress pages, for example http://library.gc.cuny.edu/wordpress/test-circ/ (uses index.php) will still be able to use the http://mysite.com/index.php without being redirected?

mod-rewrite is part of the Apache project. Have you enabled mod-rewrite and tried it. You can do most things like you referenced with it. The syntax can be a bit thick if you are not familiar with them. There are lots of examples on Apache and the net though.

I have done several things similar to your request with it.