(10-18)            .  

Corollary 10.1 (The Implicit Formula with a point at Infinity).  In equation (10-18) the point at infinity can be introduced as one of the prescribed points in either the z plane or the  w plane.

Case 1.  If  ,  then we can write    and substitute this expression into Equation (10-18) to obtain    which can be rewritten as   and simplifies to obtain  
    
            .  

Case 2.  If  ,  then we can write   and substitute this expression into Equation (10-18) to obtain   which can be rewritten as    and simplifies to obtain

(10-21)        .  

      Equation (10-21) is sometimes used to map the crescent-shaped region that lies between the tangent circles onto an infinite strip.

Proof.

Proof of Corollary 10.1 is in the book.

Complex Analysis for Mathematics and Engineering