University Mathematics Handbook (2015)

XII. Complex Functions

Chapter 10. Poles and Zeroes of Meromorphic Functions

a.   is called a meromorphic function in domain if all of its singular points in are isolated, no necessarily simple poles.

b.  If function is meromorphic in domain with boundary and has no zeroes or poles on , then

where is the number of zeroes of on , and is the number of poles of on (a zero or a pole of order is counted times).

c.  Rouche theorem: If functions and are analytic in domain with boundary , and on , there holds , , then functions and have the same number of zeroes in .