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In complex analysis (a branch of mathematics), a pole is a certain type of singularity of a function, nearby which the function behaves relatively regularly, in contrast to essential singularities, such as 0 for the logarithm function, and branch points, such as 0 for the complex square root function.

A function f of a complex variable z is meromorphic in the neighbourhood of a point *z*_{0} if either f or its reciprocal function 1/*f* is holomorphic in some neighbourhood of *z*_{0} (that is, if f or 1/*f* is complex differentiable in a neighbourhood of *z*_{0}).

A zero of a meromorphic function f is a complex number z such that *f*(*z*) = 0. A **pole** of f is a **zero** of 1/*f*.

This induces a duality between *zeros* and *poles*, that is obtained by replacing the function f by its reciprocal 1/*f*. This duality is fundamental for the study of meromorphic functions. For example, if a function is meromorphic on the whole complex plane, including the point at infinity, then the sum of the multiplicities of its poles equals the sum of the multiplicities of its zeros.

A function of a complex variable z is holomorphic in an open domain U if it is differentiable with respect to z at every point of U. Equivalently, it is holomorphic if it is analytic, that is, if its Taylor series exists at every point of U, and converges to the function in some neighbourhood of the point. A function is meromorphic in U if every point of U has a neighbourhood such that either f or 1/*f* is holomorphic in it.

A ** zero ** of a meromorphic function f is a complex number z such that *f*(*z*) = 0. A **pole** of f is a zero of 1/*f*.

If f is a function that is meromorphic in a neighbourhood of a point of the complex plane, then there exists an integer n such that

is holomorphic and nonzero in a neighbourhood of (this is a consequence of the analytic property). If *n* > 0, then is a *pole* of **order** (or multiplicity) n of f. If *n* < 0, then is a zero of order of f. *Simple zero* and *simple pole* are terms used for zeroes and poles of order *Degree* is sometimes used synonymously to order.

This characterization of zeros and poles implies that zeros and poles are isolated, that is, every zero or pole has a neighbourhood that does not contain any other zero and pole.

Because of the *order* of zeros and poles being defined as a non-negative number n and the symmetry between them, it is often useful to consider a pole of order n as a zero of order –*n* and a zero of order n as a pole of order –*n*. In this case a point that is neither a pole nor a zero is viewed as a pole (or zero) of order 0.

A meromorphic function may have infinitely many zeros and poles. This is the case for the gamma function (see the image in the infobox), which is meromorphic in the whole complex plane, and has a simple pole at every non-positive integer. The Riemann zeta function is also meromorphic in the whole complex plane, with a single pole of order 1 at *z* = 1. Its zeros in the left halfplane are all the negative even integers, and the Riemann hypothesis is the conjecture that all other zeros are along Re(*z*) = 1/2.

In a neighbourhood of a point a nonzero meromorphic function f is the sum of a Laurent series with at most finite *principal part* (the terms with negative index values):

where n is an integer, and Again, if *n* > 0 (the sum starts with , the principal part has n terms), one has a pole of order n, and if *n* ≤ 0 (the sum starts with , there is no principal part), one has a zero of order .

A function is *meromorphic at infinity* if it is meromorphic in some neighbourhood of infinity (that is outside some disk), and there is an integer n such that

exists and is a nonzero complex number.

In this case, the point at infinity is a pole of order n if *n* > 0, and a zero of order if *n* < 0.

For example, a polynomial of degree n has a pole of degree n at infinity.

The complex plane extended by a point at infinity is called the Riemann sphere.

If f is a function that is meromorphic on the whole Riemann sphere, then it has a finite number of zeros and poles, and the sum of the orders of its poles equals the sum of the orders of its zeros.

Every rational function is meromorphic on the whole Riemann sphere, and, in this case, the sum of orders of the zeros or of the poles is the maximum of the degrees of the numerator and the denominator.

- The function

- is meromorphic on the whole Riemann sphere. It has a pole of order 1 or simple pole at and a simple zero at infinity.

- The function

- is meromorphic on the whole Riemann sphere. It has a pole of order 2 at and a pole of order 3 at . It has a simple zero at and a quadruple zero at infinity.

- The function

- is meromorphic in the whole complex plane, but not at infinity. It has poles of order 1 at . This can be seen by writing the Taylor series of around the origin.

- The function

- has a single pole at infinity of order 1, and a single zero at the origin.

All above examples except for the third are rational functions. For a general discussion of zeros and poles of such functions, see Pole–zero plot § Continuous-time systems.

The concept of zeros and poles extends naturally to functions on a *complex curve*, that is complex analytic manifold of dimension one (over the complex numbers). The simplest examples of such curves are the complex plane and the Riemann surface. This extension is done by transferring structures and properties through charts, which are analytic isomorphisms.

More precisely, let f be a function from a complex curve M to the complex numbers. This function is holomorphic (resp. meromorphic) in a neighbourhood of a point z of M if there is a chart such that is holomorphic (resp. meromorphic) in a neighbourhood of Then, z is a pole or a zero of order n if the same is true for

If the curve is compact, and the function f is meromorphic on the whole curve, then the number of zeros and poles is finite, and the sum of the orders of the poles equals the sum of the orders of the zeros. This is one of the basic facts that are involved in Riemann–Roch theorem.

**Complex analysis**, traditionally known as the **theory of functions of a complex variable**, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including the branches of hydrodynamics, thermodynamics, and particularly quantum mechanics. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering.

In mathematics, a **holomorphic function** is a complex-valued function of one or more complex variables that is, at every point of its domain, complex differentiable in a neighborhood of the point. The existence of a complex derivative in a neighbourhood is a very strong condition, for it implies that any holomorphic function is actually infinitely differentiable and equal, locally, to its own Taylor series (*analytic*). Holomorphic functions are the central objects of study in complex analysis.

In complex analysis, a branch of mathematics, the **Casorati–Weierstrass theorem** describes the behaviour of holomorphic functions near their essential singularities. It is named for Karl Theodor Wilhelm Weierstrass and Felice Casorati. In Russian literature it is called Sokhotski's theorem.

In complex analysis, an **elliptic function** is a meromorphic function that is periodic in two directions. Just as a periodic function of a real variable is defined by its values on an interval, an elliptic function is determined by its values on a fundamental parallelogram, which then repeat in a lattice. Such a doubly periodic function cannot be holomorphic, as it would then be a bounded entire function, and by Liouville's theorem every such function must be constant. In fact, an elliptic function must have at least two poles in a fundamental parallelogram, as it is easy to show using the periodicity that a contour integral around its boundary must vanish, implying that the residues of all simple poles must cancel.

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".

In mathematics, particularly in complex analysis, a **Riemann surface** is a one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together.

In mathematics, **complex geometry** is the study of complex manifolds, complex algebraic varieties, and functions of several complex variables. Application of transcendental methods to algebraic geometry falls in this category, together with more geometric aspects of complex analysis.

In mathematics, the **complex plane** or ** z-plane** is a geometric representation of the complex numbers established by the

The **Riemann–Roch theorem** is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It relates the complex analysis of a connected compact Riemann surface with the surface's purely topological genus *g*, in a way that can be carried over into purely algebraic settings.

In mathematics, a **modular form** is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections with number theory. Modular forms appear in other areas, such as algebraic topology, sphere packing, and string theory.

In complex analysis, a branch of mathematics, an **isolated singularity** is one that has no other singularities close to it. In other words, a complex number *z _{0}* is an isolated singularity of a function

In the mathematical field of complex analysis, **Nevanlinna theory** is part of the theory of meromorphic functions. It was devised in 1925, by Rolf Nevanlinna. Hermann Weyl has called it "one of the few great mathematical events of century." The theory describes the asymptotic distribution of solutions of the equation *f*(*z*) = *a*, as *a* varies. A fundamental tool is the Nevanlinna characteristic *T*(*r*, *f*) which measures the rate of growth of a meromorphic function.

In the mathematical field of complex analysis, a **branch point** of a multi-valued function is a point such that the function is discontinuous when going around an arbitrarily small circuit around this point. Multi-valued functions are rigorously studied using Riemann surfaces, and the formal definition of branch points employs this concept.

In complex analysis, **Picard's great theorem** and **Picard's little theorem** are related theorems about the range of an analytic function. They are named after Émile Picard.

In complex analysis, the **argument principle** relates the difference between the number of zeros and poles of a meromorphic function to a contour integral of the function's logarithmic derivative.

In mathematics, with special application to complex analysis, a *normal family* is a pre-compact subset of the space of continuous functions. Informally, this means that the functions in the family are not widely spread out, but rather stick together in a somewhat "clustered" manner. Sometimes, if each function in a normal family *F* satisfies a particular property , then the property also holds for each limit point of the set *F*.

In mathematics, and particularly in the field of complex analysis, the **Weierstrass factorization theorem** asserts that every entire function can be represented as a product involving its zeroes. The theorem may be viewed as an extension of the fundamental theorem of algebra, which asserts that every polynomial may be factored into linear factors, one for each root.

In algebraic geometry, **divisors** are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors. Both are ultimately derived from the notion of divisibility in the integers and algebraic number fields.

In mathematics, the **Riemann sphere**, named after Bernhard Riemann, is a model of the **extended complex plane**, the complex plane plus a point at infinity. This extended plane represents the **extended complex numbers**, that is, the complex numbers plus a value ∞ for infinity. With the Riemann model, the point "∞" is near to very large numbers, just as the point "0" is near to very small numbers.

In mathematics, a **planar Riemann surface** is a Riemann surface sharing the topological properties of a connected open subset of the Riemann sphere. They are characterized by the topological property that the complement of every closed Jordan curve in the Riemann surface has two connected components. An equivalent characterization is the differential geometric property that every closed differential 1-form of compact support is exact. Every simply connected Riemann surface is planar. The class of planar Riemann surfaces was studied by Koebe who proved in 1910 as a generalization of the uniformization theorem that every such surface is conformally equivalent to either the Riemann sphere or the complex plane with slits parallel to the real axis removed.

- Conway, John B. (1986).
*Functions of One Complex Variable I*. Springer. ISBN 0-387-90328-3. - Conway, John B. (1995).
*Functions of One Complex Variable II*. Springer. ISBN 0-387-94460-5. - Henrici, Peter (1974).
*Applied and Computational Complex Analysis 1*. John Wiley & Sons.

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