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Complex Pole

Complex pole

In complex analysis, a pole of a holomorphic function is a certain type of simple singularity that behaves like the singularity 1/zⁿ at z = 0. A pole of the function f(z) is a point z = a such that f(z) approaches infinity as z approaches a. singularity Formally, suppose U is an open subset of the complex plane C, a is an element of U and f : U − → C is a holomorphic function. If there exists a holomorphic function g : U → C and a natural number n such that :

f(z) = \frac

for all z in U − , then a is called a pole of f. If n is chosen as small as possible, then n is called the order of the pole. A pole of order 1 is called a simple pole. Equivalently, a is a pole of order n≥ 0 for a function f if there exists an open neighbourhood U of a such that f : U - → C is holomorphic and the limit :

\lim_ (z-a)^n f(z)

exists and is different from 0. The point a is a pole of order n of f if and only if all the terms the Laurent series expansion of f around a below degree −n vanishes and the term in degree −n is not zero. A pole of order 0 is a removable singularity. In this case the limit lim_z→a f(z) exists as a complex number. If the order is bigger than 0, then lim_z→a f(z) = ∞. If the first derivative of a function f has a simple pole at a, then a is a branch point of f. (The converse need not be true). A non-removable singularity that is not a pole or a branch point is called an essential singularity. A holomorphic function whose only singularities are poles is called meromorphic.

Complex analysis

Complex analysis is the branch of mathematics investigating functions of complex numbers. It is of enormous practical use in applied mathematics and in many other branches of mathematics. Complex analysis is particularly concerned with analytic functions of complex variables, known as holomorphic functions.

Complex functions

A complex function, :

w = f(z)\,

, is a function in which the independent variable,

z\,

, and the dependent variable,

w\,

, are both complex numbers: :

f\colon \mathbb\rightarrow\mathbb

. For any complex function both the independent variable and the dependent variable may be separated into real and imaginary parts: :

z = x + iy\,

and :

w = f(z) = u + iv\,

, : where

x,y,u,v \in \mathbb.

It follows that the components of the function, :

u = u(x,y)\,

and :

v = v(x,y)\,

, can be interpreted as real valued functions of the two real variables,

x\,

and

y\,

. Since the real numbers are a subset of the complex numbers, all functions of real numbers are, strictly speaking, also complex functions. In fact, the extension of real functions (exponentials, logarithms, trigonometric functions) to the complex domain is frequently used as an introduction to complex analysis. However, the study of complex functions is primarily concerned with complex-valued functions of complex variables.

Holomorphic functions

Holomorphic functions are functions which are defined in some region of the complex plane, take complex values, and are differentiable as complex functions. Complex differentiability has much stronger consequences than usual (real) differentiability. For instance, holomorphic functions are infinitely differentiable, a fact that is far from true for real differentiable functions. Most elementary functions, including the exponential function, the trigonometric functions, and all polynomial functions, are holomorphic. See also: analytic function, holomorphic sheaf and vector bundles.

Major results

One central tool in complex analysis is the path integral. The integral around a closed path of a function which is holomorphic everywhere inside the area bounded by the closed path is always zero; this is the Cauchy integral theorem. The values of a holomorphic function inside a disk can be computed by a certain path integral on the disk's boundary (Cauchy's integral formula). Path integrals in the complex plane are often used to determine complicated real integrals, and here the theory of residues among others is useful (see methods of contour integration). If a function has a pole or singularity at some point, meaning that its values "explode" and it does not have a finite value there, then one can define the function's residue at that pole, and these residues can be used to compute path integrals involving the function; this is the content of the powerful residue theorem. The remarkable behavior of holomorphic functions near essential singularities is described by the Weierstrass-Casorati theorem. Functions which have only poles but no essential singularities are called meromorphic. Laurent series are similar to Taylor series but can be used to study the behavior of functions near singularities. A bounded function which is holomorphic in the entire complex plane must be constant; this is Liouville's theorem. It can be used to provide a natural and short proof for the fundamental theorem of algebra which states that the field of complex numbers is algebraically closed. An important property of holomorphic functions is that if a function is holomorphic throughout a simply connected domain then its values are fully determined by its values on any smaller subdomain. The function on the larger domain is said to be analytically continued from its values on the smaller domain. This allows the extension of the definition of functions such as the Riemann zeta function which are initially defined in terms of infinite sums that converge only on limited domains to almost the entire complex plane. Sometimes, as in the case of the natural logarithm, it is impossible to analytically continue a holomorphic function to a non-simply connected domain in the complex plane but it is possible to extend it to a holomorphic function on a closely related surface known as a Riemann surface. All this refers to complex analysis in one variable. There is also a very rich theory of complex analysis in more than one complex dimension where the analytic properties such as power series expansion still remain true whereas most of the geometric properties of holomorphic functions in one complex dimension (such as conformality) are no longer true. The Riemann mapping theorem about the conformal relationship of certain domains in the complex plane, maybe the most important result in the one-dimensional theory, fails dramatically in higher dimensions. It is also applied in many subjects throughout engineering, particularly in power engineering.

History

Complex analysis is one of the classical branches in mathematics with its roots in the 19th century and some even before. Important names are Euler, Gauss, Riemann, Cauchy, Weierstrass, and many more in the 20th century. Traditionally, complex analysis, in particular the theory of conformal mappings, has many applications in engineering, but it is also used throughout analytical number theory. In modern times, it became very popular through a new boost of complex dynamics and the pictures of fractals produced by iterating holomorphic functions, the most popular being the Mandelbrot set. Another important application of complex analysis today is in string theory which is a conformally invariant quantum field theory.

External links

- [http://www.math.gatech.edu/~cain/winter99/complex.html Complex Analysis -- textbook by George Cain]
- [http://www.ima.umn.edu/~arnold/502.s97/ Complex analysis course web site by Douglas N. Arnold] Category:Calculus Category:Mathematical analysis
-

Mathematical singularity

In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability. See singularity theory for general discussion of the geometric theory, which only covers some aspects. For example, the function :f(x) = 1/x on the real line has a singularity at x = 0, where it explodes to ±∞ and is not defined. The function g(x) = |x| (see absolute value) also has a singularity at x = 0, since it isn't differentiable there. Similarly, the graph defined by y² = x also has a singularity at (0,0), this time because it has a "corner" (vertical tangent) at that point. The algebraic set defined by y² = x² in the (x, y) coordinate system has a singularity (singular point) at (0, 0) because it does not admit a tangent there.

Complex analysis

In complex analysis, there are four kinds of singularity, to be described below. Suppose U is an open subset of the complex numbers C, a is an element of U, and f is a holomorphic function defined on U \ .
- The point a is a removable singularity of f if there exists a holomorphic function g defined on all of U such that f(z) = g(z) for all z in U − .
- The point a is a pole of f if there exists a holomorphic function g defined on U and a natural number n such that f(z) = g(z) / (z − a)ⁿ for all z in U − .
- The point a is an essential singularity of f if is neither a removable singularity nor a pole. The point a is an essential singularity if and only if the Laurent series has infinitely many powers of negative degree. These three types of singularities are isolated. The fourth type is branch points; they require a more verbose definition, see branch point.

From the point of view of dynamics

A finite-time singularity occurs when a kinematic variable increases towards infinity at a finite time. An example would be the bouncing motion of an inelastic ball on a plane. If idealized motion is considered, in which the same fraction of kinetic energy is lost on each bounce, the frequency of bounces becomes infinite as the ball comes to rest in a finite time. Other examples of finite-time singularities include Euler's disk and the Painlevé paradox.

Algebraic geometry and commutative algebra

In algebraic geometry and commutative algebra, a singularity is a prime ideal whose localization is not a regular local ring (alternately a scheme (mathematics) with a stalk that is not a regular local ring). For example,

y^2 - x^3 = 0

defines an isolated singular point (at the cusp)

x = y = 0

. The ring in question is given by :

C[x,y] / (y^2 - x^3) \cong C[t^2, t^3].

The maximal ideal of the localization at

(t^2, t^3)

is a height one local ring generated by two elements and thus not regular.

Open subset

In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can "wiggle" or "change" any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U; it can't be on the edge of U. As a typical example, consider the open interval (0,1) consisting of all real numbers x with 0 < x < 1. If you "wiggle" such an x a little bit (but not too much), then the wiggled version will still be a number between 0 and 1. Therefore, the interval (0,1) is open. However, the interval (0,1] consisting of all numbers x with 0 < x ≤ 1 is not open; if you take x = 1 and wiggle a tiny bit in the positive direction, you will be outside of (0,1]. Note that whether a given set U is open depends on the surrounding space, the "wiggle room". For instance, the set of rational numbers between 0 and 1 (exclusive) is open in the rational numbers, but it is not open in the real numbers. Note also that "open" is not the opposite of "closed". First, there are sets which are both open and closed (called clopen sets); in R and other connected spaces, only the empty set and the whole space are clopen, while the set of all rational numbers smaller than √2 is clopen in the rationals. Also, there are sets which are neither open nor closed, such as (0,1] in R.

Definitions

The concept of open sets can be formalized in various degrees of generality.

Function-analytic

A point set in Rⁿ is called open when every point P of the set is an inner point.

Euclidean space

A subset U of the Euclidean n-space Rⁿ is called open if, given any point x in U, there exists a real number ε > 0 such that, given any point y in Rⁿ whose Euclidean distance from x is smaller than ε, y also belongs to U. (Equivalently, U is open if every point in U has a neighbourhood contained in U) Intuitively, ε measures the size of the allowed "wiggles". An example of an open set in R² (on a plane) would be all the points within a circle radius r, which satisfy the equation

r>\sqrt

. Because the distance of any point p in this set from the edge of the set is greater than zero:

r-\sqrt>0

, we can set ε to half of this distance, which means ε is also greater than zero, and all the points that are within a distance of ε to p are also in the set, thus satisfying the conditions for an open set.

Metric spaces

A subset U of a metric space (M,d) is called open if, given any point x in U, there exists a real number ε > 0 such that, given any point y in M with d(x,y) < ε, y also belongs to U. (Equivalently, U is open if every point in U has a neighbourhood contained in U) This generalises the Euclidean space example, since Euclidean space with the Euclidean distance is a metric space.

Topological spaces

In topological spaces, the concept of openness is taken to be fundamental. One starts with an arbitrary set X and a family of subsets of X satisfying certain properties that every "reasonable" notion of openness is supposed to have. (Specifically: the union of open sets is open, the finite intersection of open sets is open, and in particular the empty set and X itself are open.) Such a family T of subsets is called a topology on X, and the members of the family are called the open sets of the topological space (X,T). Note that infinite intersections of open sets need not be open. Sets that can be constructed as the intersection of countably many open sets are denoted G_δ sets. The topological definition of open sets generalises the metric space definition: If you start with a metric space and define open sets as before, then the family of all open sets will form a topology on the metric space. Every metric space is hence in a natural way a topological space. (There are however topological spaces which are not metric spaces.)

Uses

Every subset A of a topological space X contains a (possibly empty) open set; the largest such open set is called the interior of A. It can be constructed by taking the union of all the open sets contained in A. Given topological spaces X and Y, a function f from X to Y is continuous if the preimage of every open set in Y is open in X. The map f is called open if the image of every open set in X is open in Y. An open set on the real line has the characteristic property that it is a countable union of disjoint open intervals.

Manifolds

A manifold is called open if it is a manifold without boundary and if it is not compact. This notion differs somewhat from the openness discussed above. Category:general topology ja:開集合

Holomorphic function

Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. This is a much stronger condition than real differentiability and implies that the function is infinitely often differentiable and can be described by its Taylor series. The term analytic function is often used interchangeably with "holomorphic function", although note that the former term has several other meanings. A function that is holomorphic on the whole complex plane is called an entire function. The phrase "holomorphic at a point a" means not just differentiable at a, but differentiable everywhere within some open disk centered at a in the complex plane. Biholomorphic means a holomorphic function with a holomorphic inverse function.

Definition

If U is an open subset of C and f : U → C is a function, we say that f is complex differentiable at the point z₀ of U if the limit :

f'(z_0) = \lim_

exists. The limit here is taken over all sequences of complex numbers approaching z₀, and for all such sequences the difference quotient has to approach the same number f '(z₀). Intuitively, if f is complex differentiable at z₀ and we approach the point z₀ from the direction r, then the images will approach the point f(z₀) from the direction f '(z₀) r, where the last product is the multiplication of complex numbers. This concept of differentiability shares several properties with real differentiability: it is linear and obeys the product, quotient and chain rules. If f is complex differentiable at every point z₀ in U, we say that f is holomorphic on U. We say that f is holomorphic in the point z₀ if it is holomorphic on some neighborhood of z₀. We say that f is holomorphic on some non-open set A if it is holomorphic in an open set containing A. An equivalent definition is the following. A complex function f(x + iy) = u + iv is holomorphic if and only if it satisfies the Cauchy-Riemann equations and u and v have continuous first partial derivatives with respect to x and y.

Examples

All polynomial functions in z with complex coefficients are holomorphic on C, and so are the trigonometric functions of z and the exponential function. (The trigonometric functions are in fact closely related to and can be defined via the exponential function using Euler's formula). The principal branch of the logarithm function is holomorphic on the set C - . The square root function can be defined as :

\sqrt = e^

and is therefore holomorphic wherever the logarithm ln(z) is. The function 1/z is holomorphic on . Typical examples of functions which are not holomorphic are complex conjugation and taking the real part.

Properties

Because complex differentiation is linear and obeys the product, quotient, and chain rules, the sums, products and compositions of holomorphic functions are holomorphic, and the quotient of two holomorphic functions is holomorphic wherever the denominator is non-zero. Every holomorphic function is infinitely often complex differentiable at every point. It coincides with its own Taylor series and the Taylor series converges on every open disk that lies completely inside the domain U. The Taylor series may converge on a larger disk; for instance, the Taylor series for the logarithm converges on every disk that does not contain 0, even in the vicinity of the negative real line. See holomorphic functions are analytic for a proof. If one identifies C with R², then the holomorphic functions coincide with those functions of two real variables which solve the Cauchy-Riemann equations, a set of two partial differential equations. Close to points with non-zero derivative, holomorphic functions are conformal in the sense that they preserve angles and the shape (but not size) of small figures. Cauchy's integral formula states that every holomorphic function inside a disk is completely determined by its values on the disk's boundary. From an algebraic point of view the set of holomorphic functions on an open set is a commutative ring and a complex vector space. In fact, it is a locally convex topological vector space, with the seminorms being the suprema on compact subsets.

Several variables

A complex analytic function of several complex variables is defined to be analytic and holomorphic at a point if it is locally expandable (within a polydisk, a cartesian product of disks, centered at that point) as a convergent power series in the variables. This condition is stronger than the Cauchy-Riemann equations; in fact it can be stated as follows: A function of several complex variables is holomorphic if and only if it satisfies the Cauchy-Riemann equations and is locally square-integrable.

Extension to functional analysis

The concept of a holomorphic function can be extended to the infinite-dimensional spaces of functional analysis. The article on the Fréchet derivative reviews the concept of a holomorphic function on a Banach space.

Terminology

Today, many mathematicians prefer the term "holomorphic function" to "analytic function", as the latter is a more general concept. This is also because an important result in complex analysis is that every holomorphic function is complex analytic, a fact that does not follow directly from the definitions. The term "analytic" is however also in wide use. The word "holomorphic" derives from the Greek holos meaning "whole" and morphe meaning "form" or "appearance".

Laurent series

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. The Laurent series was named after and first published by Pierre Alphonse Laurent in 1843. Karl Weierstrass discovered it first in 1841 but did not publish it. The Laurent series for a complex function f(z) about a point c is given by: :

f(z)=\sum_^\infty a_n(z-c)^n

where the a_n are constants, defined by a path integral which is a generalization of Cauchy's integral formula: :

a_n=\frac \oint_\gamma \frac.\,

The path of integration γ is counterclockwise around a closed, rectifiable path containing no self-intersections, enclosing c and lying in an annulus A in which f(z) is holomorphic. The expansion for f(z) will be valid anywhere inside this annulus. The annulus is shown in red in the diagram on the right, along with an example of a suitable path of integration labelled γ. In practice, this formula is rarely used because the integrals are difficult to evaluate; instead, one typically pieces together the Laurent series by combining known Taylor expansions. The numbers a_n and c are most commonly taken to be complex numbers, although there are other possibilities, as described below.

Convergent Laurent series

Laurent series with complex coefficients are an important tool in complex analysis, especially to investigate the behavior of functions near singularities. singularities Consider for instance the function f(x) = e^−1/x² with f(0) = 0. As a real function, it is infinitely often differentiable everywhere; as a complex function however it is not differentiable at x = 0. By plugging −1/x² into the series for the exponential function, we obtain its Laurent series which converges and is equal to f(x) for all complex numbers x except at the singularity x=0. The graph opposite shows e^−1/x² in black and its Laurent approximations :

\sum_^n(-1)^j\,

for n = 1, 2, 3, 4, 5, 6, 7 and 50. As n → ∞, the approximation becomes exact for all (complex) numbers x except at the singularity x = 0. More generally, Laurent series can be used to express holomorphic functions defined on an annulus, much as power series are used to express holomorphic functions defined on a disc. Suppose ∑_{−∞ < n < ∞} a_n(z − c)ⁿ is a given Laurent series with complex coefficients a_n and a complex center c. Then there exists a unique inner radius r and outer radius R such that:
- The Laurent series converges on the open annulus A := . To say that the Laurent series converges, we mean that both the positive degree power series and the negative degree power series converge. Furthermore, this convergence will be uniform on compact sets. Finally, the convergent series defines a holomorphic function f(z) on the open annulus.
- Outside the annulus, the Laurent series diverges. That is, at each point of the exterior of A, the positive degree power series or the negative degree power series diverges.
- On the boundary of the annulus, one cannot make a general statement, except to say that there is at least one point on the inner boundary and one point on the outer boundary such that f(z) cannot be holomorphically continued to those points. It is possible that r may be zero or R may be infinite; at the other extreme, it's not necessarily true that r is less than R. These radii can be computed as follows: :

r = \limsup_ |a_|^

= \limsup_ |a_n|^

We take R to be infinite when this latter lim sup is zero. Conversely, if we start with an annulus of the form A = and a holomorphic function f(z) defined on A, then there always exists a unique Laurent series with center c which converges (at least) on A and represents the function f(z). As an example, let :

f(z) =

This function has singularities at z = 1 and z = 2i, where the denominator of the expression is zero and the expression is therefore undefined. A Taylor series about z = 0 (which yields a power series) will only converge in a disc of radius 1, since it "hits" the singularity at 1. However, there are three possible Laurent expansions about z = 0, depending on the region z is in.
- One is defined on the disc where |z| < 1; it is the same as the Taylor series, :

f(z) = \frac \sum_^\infty \left(\frac-1\right)z^k

.
- Another one is defined on the annulus where 1 < |z| < 2, caught between the two singularities, :

f(z) = \frac \left(\sum_^\infty \frac + \sum_^\infty \fracz^k\right)

.
- The third one is defined on the infinite annulus where 2 < |z| < ∞, :

f(z) = \frac \sum_^\infty \frac

. The case r = 0, i.e. a holomorphic function f(z) which may be undefined at a single point c, is especially important.
The coefficient a_-1 of the Laurent expansion of such a function is called the residue of f(z) at the singularity c; it plays a prominent role in the residue theorem. For an example of this, consider :

f(z) =  + e^

This function is holomorphic everywhere except at z = 0. To determine the Laurent expansion about c = 0, we use our knowledge of the Taylor series of the exponential function: :

f(z) = \cdots + \left (  \right ) z^ + \left (  \right ) z^ + 2z^ + 2 + \left (  \right ) z + \left (  \right ) z^2 + \left (  \right ) z^3 + \cdots

and we find that the residue is 2.

Formal Laurent series

Formal Laurent series are Laurent series that are used without regard for their convergence. The coefficients a_k may then be taken from any commutative ring K. In this context, one only considers Laurent series where all but finitely many of the negative-degree coefficients are zero. Furthermore, the center c is taken to be zero. Two such formal Laurent series are equal if and only if their coefficient sequences are equal. The set of all formal Laurent series in the variable x over the coefficient ring K is denoted by K((x)). Two such formal Laurent series may be added by adding the coefficients, and because of the finiteness of the negative-degree coefficients, they may also be multiplied using convolution of the coefficient sequences. With these two operations, K((x)) becomes a commutative ring. If K is a field, then the formal power series over K form an integral domain Kx. The field of quotients of this integral domain can be identified with K((x)). Category:Complex analysis Category:Mathematical series ja:ローラン展開

Limit (mathematics)

In mathematics, the concept of a "limit" is used to describe the behavior of a function, as its argument gets "close" to either some point, or infinity; or the behavior of a sequence's elements, as their index approaches infinity. Limits are used in calculus and other branches of mathematical analysis to define derivatives and continuity. The concept of the "limit of a function" is further generalized to the concept of topological net, while the limit of a sequence is closely related to limit and direct limit in category theory.

Limit of a function

Main article: limit of a function

Limit of a function at a point

Suppose f(x) is a real function and c is a real number. The expression: :

\lim_f(x) = L

means that f(x) can be made to be as close to L as desired by making x sufficiently close to c. In that case, we say that "the limit of f(x), as x approaches c, is L". Note that this statement can be true even if f(c)

\neq

L. Indeed, the function f(x) need not even be defined at c. Two examples help illustrate this. Consider

f(x) = \frac

as x approaches 2. In this case, f(x) is defined at 2 and equals its limit of 0.4: As x approaches 3, f(x) approaches 0.3 and hence we have

\lim_f(x)=0.3

. In the case where

f(c) = \lim_ f(x)

, f is said to be continuous at x = c. But it is not always the case. Consider :

g(x)=\left\

Branch point

In complex analysis, a branch point may be thought of informally as a point z₀ at which a "multiple-valued function" changes values when one winds once around z₀. Examples: - 0 is a branch point of the square root function. Suppose w = √z, and z starts at 4 and moves along a circle of radius 4 centered at 0. The dependent variable w changes while depending on z in a continuous manner. When z has made one full circle, going from 4 back to 4 again, w will have made one half-circle, going from the positive square root of 4, i.e., from 2, to the negative square root of 4, i.e., −2. - 0 is also a branch point of the natural logarithm. Since e⁰ is the same as e^2πi, both 0 and 2πi are among the multiple values of Log(1). As z moves along a circle of radius 1 centered at 0, w = Log(z) goes from 0 to 2πi. - In trigonometry, since tan(π/4) and tan (5π/4) are both equal to 1, the two numbers π/4 and 5π4 are among the multiple values of arctan(1). The imaginary units i and −i are branch points of the arctangent function. That this is so may be seen by observing that the derivative (d/dz) arctan(z) = 1/(1 + z²) has simple poles at those two points, since the denominator is zero at those points. - If the derivative f′ of a function f has a simple pole at a point a, then has f has a branch point at a. (The converse is false, since the square-root function is a counterexample.) In order to work with single-valued functions, it is customary to construct branch cuts in the complex plane, namely arcs out of branch points in the complement of which there is a well-defined branch of the function in question. An example for :

F(z) = \sqrt \sqrt

is to make a branch cut along the interval [0,1] on the real axis, connecting the two branch points of the function. The same idea can be applied to the function √z; but in that case one has to perceive that the point at infinity is the appropriate 'other' branch point to connect to from 0, for example along the whole negative real axis. See also principal branch. The branch cut device may appear arbitrary (it is); but it is very useful, for example in the theory of special functions. An invariant explanation of the branch phenomenon is developed in Riemann surface theory (of which it is historically the origin), and more generally in the ramification and monodromy theory of algebraic functions and differential equations. Category:Complex analysis

Essential singularity

In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits extreme behavior. Formally, consider an open subset U of the complex plane C, an element a of U, and a holomorphic function f defined on U - . The point a is called an essential singularity for f if it is a singularity which is neither a pole nor a removable singularity. For example, the function f(z) = exp(1/z) has an essential singularity at a = 0. The point a is an essential singularity if and only if the limit :

\lim_f(z)

does not exist as a complex number nor equals infinity. This is the case if and only if the Laurent series of f at the point a has infinitely many negative degree terms (the principal part is an infinite sum). The behavior of holomorphic functions near essential singularities is described by the Weierstrass-Casorati theorem and by the considerably stronger Picard's great theorem. The latter says that in every neighborhood of an essential singularity a, the function f takes on every complex value, except possibly one, infinitely often. Category:Complex analysis

Zero (complex analysis)

In complex analysis, a zero of a holomorphic function f is a complex number a such that f(a) = 0.

Multiplicity of a zero

A complex number a is a simple zero of f, or a zero of multiplicity 1 of f, if f can be written as :

f(z)=(z-a)g(z)\,

where g is a holomorphic function g such that g(a) is not zero. Generally, the multiplicity of the zero of f at a is the positive integer n for which there is a holomorphic function g such that :

f(z)=(z-a)^ng(z)\  \mbox\ g(a)\neq 0.\,

Existence of zeroes

The fundamental theorem of algebra says that every nonconstant polynomial with complex coefficients has at least one zero in the complex plane. This is in contrast to the situation with real zeroes: some polynomial functions with real coefficients have no real zeroes (but since real numbers are complex numbers, they still have complex zeroes). An example is f(x) = x² + 1.

Residue (complex analysis)

In complex analysis, the residue is a complex number which describes the behavior of path integrals of a meromorphic function around a singularity. Residues can be computed quite easily and, once known, allow the determination of more complicated path integrals via the residue theorem.

Motivation

As an example, consider the contour integral :

\oint_C \,dz

where C is some Jordan curve about 0. Let us evaluate this integral without using standard integral theorems that may be available to us. Now, the Taylor series for e^z is well-known, and we substitute this series into the integrand. The integral then becomes: :

\oint_C \left(1+z+ +  +  +  +  + \ldots\right)\,dz

Let us bring the 1/z⁵ term into the series, and so, we obtain :

\oint_C ++ +  +  +  +  + \ldots\,dz

\oint_C ++ +  +  +  +  + \ldots\,dz

The integral now collapses to a much simpler form. Recall :

\oint_C  \,dz=0,\quad a \in \mathbb, \ a \ne 1

So now the integral around C of every other term not in the form cz⁻¹ becomes zero, and the integral is reduced to :

\oint_C  \,dz=\oint_C\,dz=(2\pi i)

The value 1/4! is known as the residue of e^z/z⁵ at z=0, and is notated as :

\mathrm_0 ,\ \mathrm\ \mathrm_ ,\ \mathrm\ \mathrm(f,0)

Calculating residues

Suppose a punctured disk D = in the complex plane is given and f is a holomorphic function defined (at least) on D. The residue Res(f, c) of f at c is the coefficient a₋₁ of (z − c)⁻¹ in the Laurent series expansion of f around c. At a simple pole, the residue is given by: :

\operatorname(f,c)=\lim_(z-c)f(z).

According to the integral formula given in the Laurent series article we have: :

\operatorname(f,c) = 
 \int_\gamma f(z)\,dz

where γ traces out a circle around c in a counterclockwise manner. We may choose the path γ to be a circle in radius ε around c where ε is as small as we desire. To calculate the residue of a function around z = c, a pole of order n, one may use the following formula: :

\mathrm(f,c) = \frac \cdot \lim_ \frac\left( f(z)\cdot (z-c)^ \right)

If the function f can be continued to a holomorphic function on the whole disk , then Res(f, c) = 0. The converse is not generally true.

Series methods

If parts or all of a function can be expanded into a Taylor series or Laurent series, which may be possible if the parts or the whole of the function has a standard series expansion, calculating the residue is significantly simpler than by other methods. As an example, consider calculating the residues at the singularities of the function :

f(z)=

which may be used to calculate certain contour integrals. This function appears to have a singularity at z=0, but if one factorizes the denominator and thus writes the function as :

f(z)=

it is apparent that the singularity at z=0 is a removable singularity and thus the residue at z=0 is therefore 0. The only other singularity is at z=1. Recall :

g(z) = g(a) + g'(a)(z-a) +  + + \cdots

about z=a, so, for g(z)=sin z and a=1 we have : $\sin = \sin + \cos(z-1)+ + +\cdots$ Introducing 1/(z-1) gives us : $= + + + +\cdots$ So the residue of f(z) at z=1 is sin 1.
External links

- [http://www.exampleproblems.com/wiki/index.php/Complex_Variables#Residues Exampleproblems.com] Category:Complex analysis ja:留数

Electronic filter

Electronic filters are electronic circuits which perform signal processing functions. Electronic filters can be:
- passive or active
- analog or digital
- discrete-time (sampled) or continuous-time
- linear or non-linear The most common types of electronic filters are linear filters, regardless of other aspects of their design. See the article on linear filters for details on their design and analysis. Most filters are a type of resonant system.

History

The oldest forms of electronic filters are passive analog linear filters, constructed using only resistors and capacitors or resistors and inductors. These are known as RC and RL single pole filters respectively. More complex mutipole LC filters have also been in existence for many years and the operation of such filters is well understood with many books having been written about them. Hybrid filters have also been made, typically involving combinations of analog amplifiers with mechanical resonators or delay lines. Other devices such as CCD delay lines have also been used as discrete-time filters.With the availability of digital signal processing, active digital filters have become common.

Classification by technology

Passive filters

Single pole types

The simplest electronic implementations of linear filters are based on combinations of resistors, inductors and capacitors. These filters exist in so-called RC, RL, LC and RLC varieties. All these types are collectively known as passive filters, because they do not depend upon an external power supply. Inductors block high-frequency signals and conduct low-frequency signals, while capacitors do the reverse. A filter in which the signal passes through an inductor, or in which a capacitor provides a path to earth, therefore presents less attenuation to low-frequency signals than high-frequency signals and is a low-pass filter. If the signal passes through a capacitor, or has a path to ground through an inductor, then the filter presents less attenuation to high-frequency signals than low-frequency signals and is a high-pass filter. Resistors on their own have no frequency-selective properties, but are added to inductors and capacitors to determine the time-constants of the circuit, and therefore the frequencies to which it responds. At very high frequencies (above about 100 megahertz), sometimes the inductors consist of single loops or strips of sheet metal, and the capacitors consist of adjacent strips of metal. These are called stubs.

Multipole types

Second order filters are measured by their quality or "Q" factor. A filter is said to have a high Q if it selects or rejects a narrow range of frequencies compared with its centre frequency. Q is defined as center frequency/3dB bandwidth.

Active filters

Active filters are implemented using a combination of passive and active (amplifying) components. Operational amplifiers are frequently used in active filter designs. These can have high Q, and achieve resonance without the use of inductors. However, their upper frequency limit is limited by the bandwidth of the amplifiers used.

Digital filters

Operational amplifier Digital signal processing allows the inexpensive construction of a wide variety of filters. The signal is sampled and an analog to digital converter turns the signal into a stream of numbers. A computer program running on a CPU or a specialized DSP, less often a hardware implementation of the algorithm, calculates an output number stream. This output is converted to a signal by passing it through a digital to analog converter. There are problems with noise introduced by the conversions, but these can be controlled and limited for many useful filters. Due to the sampling involved, the input signal must be of limited frequency content or aliasing will occur. See also: Digital filter.

Other filter technologies

Quartz filters and piezoelectrics

In the late 1930s, engineers realized that small mechanical systems made of rigid materials such as quartz would acoustically resonate at radio frequencies, i.e. from audible frequencies (sound) up to several hundred megahertz. Some early resonators were made of steel, but quartz quickly became favored. The biggest advantage of quartz is that it is piezoelectric. This means that quartz resonators can directly convert their own mechanical motion into electrical signals. Quartz also has a very low coefficient of thermal expansion which means that quartz resonators can produce stable frequencies over a wide temperature range. Quartz crystal filters have much higher quality factors than LCR filters. When higher stabilities are required, the crystals and their driving circuits may be mounted in a "crystal oven" to control the temperature. For very narrow band filters, sometimes several crystals are operated in series. Engineers realized that a large number of crystals could be collapsed into a single component, by mounting comb-shaped evaporations of metal on a quartz crystal. In this scheme, a "tapped delay line" reinforces the desired frequencies as the sound waves flow across the surface of the quartz crystal. The tapped delay line has become a general scheme of making high-Q filters in many different ways.

SAW filters

SAW (surface acoustic wave) filters are electromechanical devices commonly used in radio frequency applications. Electrical signals are converted to a mechanical wave in a piezoelectric crystal; this wave is delayed as it propagates across the crystal, before being converted back to an electrical signal by further electrodes. The delayed outputs are recombined to produce a direct analog implementation of a finite impulse response filter. This hybrid filtering technique is also found in an analog sampled filter.

Garnet filters

Another method of filtering, at microwave frequencies from 800MHz to about 5GHz, is to use a synthetic single crystal yttrium iron garnet sphere made of a chemical combination of yttrium and iron (YIGF, or yttrium iron garnet filter). The garnet sits on a strip of metal driven by a transistor, and a small loop antenna touches the top of the sphere. An electromagnet changes the frequency that the garnet will pass. The advantage of this method is that the garnet can be tuned over a very wide frequency by varying the strength of the magnetic field.

Atomic filters

For even higher frequencies and greater precision, the vibrations of atoms must be used. Atomic clocks use caesium masers as ultra-high Q filters to stabilize their primary oscillators. Another method, used at high, fixed frequencies with very weak radio signals, is to use a ruby maser tapped delay line.

External links and references

- Catalog of passive filter types and component values. The Bible for practical electronic filter design.
-
- [http://www.national.com/an/AN/AN-779.pdf National Semiconductor AN-779] application note describing analog filter theory
- [http://www.vias.org/feee/filters_02.html Fundamentals of Electrical Engineering and Electronics] - Detailed explanation of all types of filters Category:Filter theory Category:Electronic circuits

Category:Complex analysis

Complex analysis is the branch of mathematics investigating holomorphic functions, i.e. functions which are defined in some region of the complex plane, take complex values, and are differentiable as complex functions. Complex differentiability has much stronger consequences than usual (real) differentiability. For instance, every holomorphic function is representable as power series in every open disc in its domain of definition, and is therefore analytic. In particular, holomorphic functions are infinitely differentiable, a fact that is far from true for real differentiable functions. Most elementary functions, such as all polynomials, the exponential function, and the trigonometric functions, are holomorphic. See also : holomorphic sheaves and vector bundles. Category:Mathematical analysis Category:Calculus Category:Multivariate calculus th:Category:การวิเคราะห์เชิงซ้อน

Wyndham Vale, Victoria

Wyndham Vale is a suburb in Melbourne, Victoria. It is in the City of Wyndham in the Western Suburbs. Wyndham Vale is just north west of Werribee and starts off at the Junction of McGraths Rd and Ballan Rd. Approximately 8km to the east is Hoppers Crossing. Wyndham Vale does not have a major shopping center and most residents have to travel in to Werribee or The Plaza in Hoppers Crossing to do their shopping. A shopping center is planned in Wyndham Vale which will be built in the new estate suburb of Manor Lakes although considered the end of the Melbourne Metropolitain area Wyndham Vale is not the furthest suburb of Wyndham. This is Little River which is considered the start of the Geelong Metropolitain area.

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