Zeros and poles

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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. Zeros and poles_sentence_0

A function f of a complex variable z is meromorphic in the neighbourhood of a point z0 if either f or its reciprocal function 1/f is holomorphic in some neighbourhood of z0 (that is, if f or 1/f is complex differentiable in a neighbourhood of z0). Zeros and poles_sentence_1

A zero of a meromorphic function f is a complex number z such that f(z) = 0. Zeros and poles_sentence_2

A pole of f is a zero of 1/f . Zeros and poles_sentence_3

This induces a duality between zeros and poles, that is obtained by replacing the function f by its reciprocal 1/f . Zeros and poles_sentence_4

This duality is fundamental for the study of meromorphic functions. Zeros and poles_sentence_5

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. Zeros and poles_sentence_6

Definitions Zeros and poles_section_0

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. Zeros and poles_sentence_7

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. Zeros and poles_sentence_8

A zero of a meromorphic function f is a complex number z such that f(z) = 0. Zeros and poles_sentence_9

A pole of f is a zero of 1/f. Zeros and poles_sentence_10

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. Zeros and poles_sentence_11

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. Zeros and poles_sentence_12

In this case a point that is neither a pole nor a zero is viewed as a pole (or zero) of order 0. Zeros and poles_sentence_13

A meromorphic function may have infinitely many zeros and poles. Zeros and poles_sentence_14

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. Zeros and poles_sentence_15

The Riemann zeta function is also meromorphic in the whole complex plane, with a single pole of order 1 at z = 1. Zeros and poles_sentence_16

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. Zeros and poles_sentence_17

At infinity Zeros and poles_section_1

exists and is a nonzero complex number. Zeros and poles_sentence_18

For example, a polynomial of degree n has a pole of degree n at infinity. Zeros and poles_sentence_19

The complex plane extended by a point at infinity is called the Riemann sphere. Zeros and poles_sentence_20

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. Zeros and poles_sentence_21

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. Zeros and poles_sentence_22

Examples Zeros and poles_section_2

Zeros and poles_unordered_list_0

  • The functionZeros and poles_item_0_0

Zeros and poles_unordered_list_1

  • The functionZeros and poles_item_1_1

Zeros and poles_unordered_list_2

  • The functionZeros and poles_item_2_2

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  • The functionZeros and poles_item_3_3

All above examples except for the third are rational functions. Zeros and poles_sentence_23

For a general discussion of zeros and poles of such functions, see Pole–zero plot § Continuous-time systems. Zeros and poles_sentence_24

Function on a curve Zeros and poles_section_3

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). Zeros and poles_sentence_25

The simplest examples of such curves are the complex plane and the Riemann surface. Zeros and poles_sentence_26

This extension is done by transferring structures and properties through charts, which are analytic isomorphisms. Zeros and poles_sentence_27

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. Zeros and poles_sentence_28

This is one of the basic facts that are involved in Riemann–Roch theorem. Zeros and poles_sentence_29

See also Zeros and poles_section_4

Zeros and poles_unordered_list_4


Credits to the contents of this page go to the authors of the corresponding Wikipedia page: en.wikipedia.org/wiki/Zeros and poles.