I want to make the following equivalent:

http://example.com/mail/index.php
http://example.com/mail/
http://example.com/mail

The first two work OK but the last one I can't get to work. With the
following it gives 404 error in the logs but the browser says "No input file
specified." :

location ~* ^/(web)?mail/.+\.(jpg|jpeg|gif|css|png|js|ico|html)$ {
root /var/www;
}

location ~ ^/(web)?mail {
root /var/www;
index index.php;
fastcgi_index index.php;
include my_fastcgi_params;
fastcgi_pass localhost:8888;
}

I've tried all sorts of variations of this including trying to rewrite the
full uri in the last case.

The my_fastcgi_params include

fastcgi_param SCRIPT_FILENAME $document_root$fastcgi_script_name;
fastcgi_param SCRIPT_NAME $fastcgi_script_name;

What is the recommended way to configure this?

Dick