Meromorphic Function

A Meromorphic Function is a complex function that is holomorphic (analytic) on an open subset of the complex plane except at a set of isolated points, where it has poles.

• Context:
  • It can (typically) be defined on a complex domain except for a discrete set of points, known as poles, where the function can take an infinite value.
  • It can (often) be expressed locally as the quotient of two holomorphic functions, where the denominator is non-zero at non-pole points.
  • It can range from simple functions, like the reciprocal function \(f(z) = \frac{1}{z}\), which has a single pole at \(z=0\), to more complex functions with multiple poles and varying orders of poles.
  • It can be described by a Laurent series expansion around each pole, where the series includes terms with negative powers.
  • It can include specific examples like the Euler–Riemann Zeta Function, which is meromorphic with a simple pole at \(s=1\).
  • It can have applications in complex analysis, number theory, and mathematical physics, where such functions often arise in the study of special functions and differential equations.
  • It can be distinguished from holomorphic functions, which are analytic everywhere on their domain, as meromorphic functions are only analytic where they do not have poles.
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• Example(s):
  • (Exponential Quotient) \(f(z) = \frac{e^z}{z-1}\), with a single pole at \(z=1\).
  • Elliptic Functions, such as: Weierstrass Elliptic Functions, with poles on a lattice in the complex plane.
  • Gamma Functions, with simple poles at all non-positive integers.
  • Zeta Functions, such as: Riemann Zeta functions (with a simple pole at (s=1) that are an infinite series function of the form [math]\displaystyle{ \zeta(s) = \sum_{n=1}^{\infty} \frac{a_n}{n^s} }[/math], where [math]\displaystyle{ a_n }[/math] are arithmetic or geometric coefficients..
  • Poincaré Seriess, with poles related to the geometry of the underlying space.
  • Bessel Functions, such as Bessel Function of the Second Kinds, with poles at non-positive integers when considered as functions of their order.
  • Hypergeometric Functions, with poles depending on the parameters involved in the function.
  • Mittag-Leffler Functions, with poles determined by the terms in their series expansion.
  • Theta Functions, such as: Jacobi Theta Function's related meromorphic functions, with poles on the upper half-plane.
  • Lambert W Functions, with branch points and poles related to their multi-valued nature.
  • ...
• Counter-Example(s):
  • Holomorphic Function, which is analytic across its entire domain without any poles.
  • Entire Function, a special case of a holomorphic function that is analytic everywhere in the complex plane.
• See: Holomorphic Function, Laurent Series, Complex Plane, Pole (Complex Analysis), Analytic Continuation, Mathematical Physics, Complex Function Theory, Pole (Complex Analysis).

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References

2024

• (Wikipedia, 2024) ⇒ https://en.wikipedia.org/wiki/Meromorphic_function Retrieved:2024-8-26.
  • In the mathematical field of complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all of D except for a set of isolated points, which are poles of the function. The term comes from the Greek meros (μέρος), meaning "part".Every meromorphic function on D can be expressed as the ratio between two holomorphic functions (with the denominator not constant 0) defined on D: any pole must coincide with a zero of the denominator.

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