1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

328 CHAPTER 8 • RESIDUE THEORY

Definition 8 .3: Meromorphic function
A function f is said to be m eromorphic in a domain D provided the only
singularities of f are isolated poles and removable singularities.

We make three important observations relating to this definition.

• Analytic functions are a special case of meromorphic functions.
  
  
  • Rational functions f (z) = Gt~l' where P (z) and Q (z) are polynomials, are
    meromorphic in t he entire complex plane.
  
  
  • By definition, meromorphic functions have no essential singularities.
  
  
  
  

Suppose that f is analytic at each point on a simple closed contour C and f

is meromorphic in the domain that is the interior of C. We assert without proof

that Theorem 7 .13 can be extended to meromorphic functions so that f has at

most a finite number of zeros that lie ins ide C. Since the function g (z) = JfzJ

is also meromorphic, it can have only a finite number of zeros inside C, and so

f can have a t most a finite number of poles that lie inside C.

Theorem 8.8, known as the argument principle, is useful in determining the

number of zeros and poles that a function bas.

(8-34)