What is meromorphic function in complex analysis?

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 Ancient Greek meros (μέρος), meaning “part”.

How do you know if a function is meromorphic?

1. A function on a domain Ω is called meromorphic, if there exists a sequence of points p1,p2,··· with no limit point in Ω such that if we denote Ω∗ = Ω \ {p1,···} • f : Ω∗ → C is holomorphic. •
2. To see this, note that F(1/z) has either a pole or zero at z = 0. In either.
3. Pk. ( 1.
4. Pk ( 1 z − pk ) = 0.

Is an entire function meromorphic?

A function is said to be entire if it is analytic on all of C. It is said to be meromorphic if it is analytic except for isolated singularities which are poles. We develop an additive theory for mero- morphic functions, in terms of their principal part (polar part) at the poles.

What is meant by removable singularity?

A removable singularity is a singular point of a function for which it is possible to assign a complex number in such a way that becomes analytic. A more precise way of defining a removable singularity is as a singularity of a function about which the function is bounded.

Which of these functions is analytic everywhere in the complex plane?

If f(z) is analytic everywhere in the complex plane, it is called entire. Examples • 1/z is analytic except at z = 0, so the function is singular at that point. The functions zn, n a nonnegative integer, and ez are entire functions.

What does entire mean in complex analysis?

If a complex function is analytic at all finite points of the complex plane. , then it is said to be entire, sometimes also called “integral” (Knopp 1996, p. 112).

What is bounded entire function?

In complex analysis, Liouville’s theorem, named after Joseph Liouville (although the theorem was first proven by Cauchy in 1844), states that every bounded entire function must be constant. That is, every holomorphic function for which there exists a positive number such that for all in is constant.

What is Laurent theorem?

In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion cannot be applied.

How do you find the removable singularity of a function?

Definition 1. f has an isolated singularity at z = a if there is a punctured disk B(a, R)\{a} such that f is defined and analytic on this set, but not on the full disk. a is called removable singularity if there is an analytic g : B(a, R) → C such that g(z) = f(z) for 0 < |z − a| < R.

What is the meromorphic field of a complex number?

Thus, if D is connected, the meromorphic functions form a field, in fact a field extension of the complex numbers . In several complex variables, a meromorphic function is defined to be locally a quotient of two holomorphic functions. For example,

How do you find the ratio of meromorphic functions?

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. The gamma function is meromorphic in the whole complex plane.

How do you know if a function is meromorphic at z0?

“A function h (z), defined in Ω is meromorphic at z 0 in Ω if and only if in a neighborhood of z 0 ≠ z 0 it can be represented as f (z)/g (z), where f (z) and g (z) are analytic at z 0 .”

What is a meromorphic and doubly periodic function?

A meromorphic function is the ratio of two analytic functions which are analytic except for isolated singularities, called “ poles.” A doubly periodic function has two periods (ω 1 and ω 2), such that f (z + ω 1) = f (z + ω 1) = f (z) The fundamental pair of periods (demoted by omega, ω) span a parallelogram in the complex plane.

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