4.4 Meromorphic functions and their properties

Meromorphic functions are like complex-valued functions with a few quirks. They're smooth sailing everywhere except for some isolated bumps called singularities. These functions are super important in complex analysis because they pop up all over the place.

Rational functions are a special type of meromorphic function. They're just fractions of polynomials, but they're incredibly useful. Understanding their zeros and poles helps us solve all sorts of problems in math and science.

Meromorphic Functions and Rational Functions

Definition and Properties of Meromorphic Functions

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• Meromorphic function defined as complex-valued function analytic on entire complex plane except at isolated singularities
• Isolated singularities include removable singularities, poles, and essential singularities
• Meromorphic functions can be expressed as ratio of two holomorphic functions
• Possess Laurent series expansions around their singularities
• Examples include rational functions ($\frac{P(z)}{Q(z)}$), trigonometric functions ($\tan z$), and exponential functions ($e^{\frac{1}{z}}$)

Rational Functions as Special Meromorphic Functions

• Rational function expressed as quotient of two polynomials $R(z) = \frac{P(z)}{Q(z)}$
• P(z) and Q(z) represent polynomials with no common factors
• Degree of numerator P(z) can be less than, equal to, or greater than degree of denominator Q(z)
• Rational functions always meromorphic on entire complex plane
• Examples include $\frac{z^2 + 1}{z - 2}$ and $\frac{3z + 1}{z^2 - 4}$

Zeros and Poles of Meromorphic Functions

• Zeros occur where function equals zero, represented by roots of numerator polynomial
• Poles occur where denominator equals zero, represented by roots of denominator polynomial
• Order of zero or pole determined by multiplicity of corresponding root
• Simple zero or pole has order 1, double zero or pole has order 2, etc.
• Residue of function at a pole calculated using limit formula or coefficient of z^(-1) in Laurent series
• Examples: $f(z) = \frac{z^2 - 1}{z - 2}$ has zeros at z = ±1 and a simple pole at z = 2

Theorems on Meromorphic Functions

Fundamental Theorems in Complex Analysis

• Liouville's theorem states bounded entire function must be constant
  • Applies to functions analytic and bounded on entire complex plane
  • Proves fundamental theorem of algebra as corollary
  • Used in proofs of other important results in complex analysis
• Fundamental theorem of algebra asserts every non-constant polynomial has at least one complex root
  • Implies polynomial of degree n has exactly n complex roots (counting multiplicity)
  • Proves algebraic closure of complex numbers
  • Connects algebra and complex analysis

Principles for Counting Zeros and Poles

• Argument principle relates number of zeros and poles inside a contour to contour integral
  • Formula: $N - P = \frac{1}{2\pi i} \oint_C \frac{f'(z)}{f(z)} dz$
  • N represents number of zeros, P represents number of poles (counting multiplicity)
  • Contour C must not pass through any zeros or poles of f(z)
  • Used in various applications, including proof of Rouché's theorem
• Rouché's theorem provides method for determining number of zeros of sum of two functions
  • If |f(z)| > |g(z)| on closed contour C, then f(z) and f(z) + g(z) have same number of zeros inside C
  • Useful for locating zeros of polynomials and other analytic functions
  • Applications in control theory and signal processing

Advanced Theorems and Applications

• Mittag-Leffler theorem allows construction of meromorphic functions with prescribed poles and principal parts
  • Generalizes partial fraction decomposition to infinite series
  • States any meromorphic function can be expressed as sum of its principal parts plus an entire function
  • Used in constructing functions with specific singularity structures
  • Applications in complex analysis, number theory, and mathematical physics
• Residue theorem connects contour integrals to sum of residues at poles
  • Formula: $\oint_C f(z) dz = 2\pi i \sum_{k=1}^n \text{Res}(f, a_k)$
  • Simplifies calculation of complex integrals
  • Used in evaluation of real integrals through contour integration techniques