Corless & Davenport provide a whole bestiarium of rules. This is a small part of the most simple cases, which I sampled more or less for 'all day use' as reference. They are based on the 'unwinding number' (which is a sheet number of according Riemannian surfaces). It turns out, that Maple can 'proof' the identities, if one does not use the definition, but uses the version given in the help pages (= version 2 in the following).
I could not convince Maple to proof the equivalence of the 3 definitions (though the primary definition should be true to the statement in the help (i.e. K1 = K2, see below), but that certainly has to be a hard-coded fact). In Complex Analysis this certainly all is seen to be basic. But I had fun and trouble with it in Maple, which also showed some astonishing weakness in that field (for my taste, may be due to my handling).
Note that in the references the sign for 'unwinding number' has been changed at some time to the definition now being used in Maple. So check that first before using any of the formulae.
complex_rules.mws (Worksheet, 50kB)
complex_rules.mws.pdf (view as pdf)
Edited 07 Jan 2010: this is an update