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In complex analysis, the residue is a complex number which describes the behavior of line 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. Complex analysis is the branch of mathematics investigating functions of complex numbers, and is of enormous practical use in many branches of mathematics, including applied mathematics. ...
In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...
This article is about path integrals in the general mathematical sense, and not the path integral formulation of physics which was studied by Richard Feynman. ...
In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function. ...
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. ...
The residue theorem in complex analysis is a powerful tool to evaluate path integrals of meromorphic functions over closed curves and can often be used to compute real integrals as well. ...
Motivation
As an example, consider the contour integral  where C is some Jordan curve about 0. In topology, the Jordan curve theorem states that every non-self-intersecting loop in the plane divides the plane into an inside and an outside. It was proved by Oswald Veblen in 1905. ...
Let us evaluate this integral without using standard integral theorems that may be available to us. Now, the Taylor series for ez is well-known, and we substitute this series into the integrand. The integral then becomes As the degree of the Taylor series rises, it approaches the correct function. ...
 Let us bring the 1/z5 term into the series, and so, we obtain   The integral now collapses to a much simpler form. Recall that  So now the integral around C of every other term not in the form cz−1 becomes zero, and the integral is reduced to  The value 1/4! is known as the residue of ez/z5 at z = 0, and is notated as 
Calculating residues Suppose a punctured disk D = {z : 0 < |z − c| < R} 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: An annulus An annulus (from Latin anulus, little ring) is a ring-shaped geometric figure, or more generally, a term used to name a ring-shaped object. ...
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. ...
A Laurent series is defined with respect to a particular point c and a path of integration γ. The path of integration must lie in an annulus (shown here in red) inside of which f(z) is holomorphic. ...
In complex analysis, a pole of a function is a certain type of simple singularity that behaves like the singularity of f(z) = 1/zn at z = 0; a pole of a function f is a point a such that f(z) approaches infinity as z approaches a. ...
 According to the integral formula given in the Laurent series article we have: A Laurent series is defined with respect to a particular point c and a path of integration γ. The path of integration must lie in an annulus (shown here in red) inside of which f(z) is holomorphic. ...
 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.
The residue of a function f(z)=g(z)/h(z) at a simple pole c, where g and h are holomorphic functions in a neighborhood of c with h(c) = 0 and g(c) ≠ 0 is given by In complex analysis, a pole of a function is a certain type of simple singularity that behaves like the singularity of f(z) = 1/zn at z = 0; a pole of a function f is a point a such that f(z) approaches infinity as z approaches a. ...
 More generally, the residue of f around z = c, a pole of order n, can be found by the formula:  If the function f can be continued to a holomorphic function on the whole disk { z : |z − c| < R }, then Res(f, c) = 0. The converse is not generally true.
Due to its simplicity, the latter formula is more than useful in the computation of residues at first order poles.
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 the degree of the Taylor series rises, it approaches the correct function. ...
A Laurent series is defined with respect to a particular point c and a path of integration γ. The path of integration must lie in an annulus (shown here in red) inside of which f(z) is holomorphic. ...
As an example, consider calculating the residues at the singularities of the function
 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  it is apparent that the singularity at z = 0 is a removable singularity and thus the residue at z = 0 is therefore 0. In complex analysis, a removable singularity of a function is a point at which the function is not defined (a singularity) but at which the function can be defined without creating any problems. ...
The only other singularity is at z = 1. Recall
 about z = a, so, for g(z) = sin z and a = 1 we have  Introducing 1/(z − 1) gives us  So the residue of f(z) at z = 1 is sin 1.
See also The residue theorem in complex analysis is a powerful tool to evaluate path integrals of meromorphic functions over closed curves and can often be used to compute real integrals as well. ...
In mathematics, Cauchys integral formula, named after Augustin Louis Cauchy, is a central statement in complex analysis. ...
In mathematics, the Cauchy integral theorem in complex analysis, named after Augustin Louis Cauchy, is an important statement about path integrals for holomorphic functions in the complex plane. ...
In complex analysis, the evaluation of integrals of real-valued functions along intervals on the real line, is not readily found with certain integrands and methods involving only real variables. ...
In complex analysis, Moreras theorem states that if the integral of a continuous complex-valued function f of a complex variable along every simple closed curve within an open set is zero, that is, if for C any simple closed curve, then f is differentiable at every point in...
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