I Notation like G(r1, r2 ; z)

In what context is this ? Can you link to the original source ?

 

I think ( though not 100% sure ) that the semicolon is used in these examples to separate variables from parameters; for example, in G(r1,r2;z), the z is a complex parameter which you choose to be of a specific value ( and which doesn't vary within the function ), whereas r1 and r2 are actual variables in the Green's function. If we were talking about computer programming, this would be analogous to the difference between constants and variables.

Let's wait for one of the experts here to confirm or correct this, though.