Residue (mathematics): Difference between revisions

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In complex analysis, the residue of a function f holomorphic in an open set $\Omega$ with possible exception of a point $z_{0}\in \Omega$ where the function may admit an isolated singularity, is a particular number describing behaviour of f around $z_{0}$ .

More precisely, if a function f is holomorphic in a neighbourhood of $z_{0}$ (but not necessarily at $z_{0}$ itself), with either a removable singularity or a pole at $z_{0}$ , then it can be represented as a Laurent series around this point, that is

f(z)=\sum _{n=-N}^{\infty }c_{n}(z-z_{0})^{n}

with some $N\in \mathbb {N} \cup \{\infty \}$ and coefficients $c_{n}\in \mathbb {C} .$

The coefficient $c_{-1}$ is the residue of f at $z_{0}$ , denoted as $\mathrm {Res} (f,z_{0})$ or ${\underset {z=z_{0}}{\mathrm {Res} }}f(z).$

Although the choice of the coefficient may look arbitrary, it turns out that it is well motivated by the particularly important role played by this number in the theory of complex functions. For example, the residue allows to evaluate path integrals of the function f via the residue theorem. This technique finds many applications in real analysis as well.

@@ Line 1: / Line 1: @@
-In complex analysis, the '''residue'''  of a complex function ''f''  [[holomorphic]] in a neighbourhood $\Omega$  of a point <math>z_0\in\mathbb{C}<math> is a particular number characterising behaviour of ''f'' around this point.
+{{subpages}}
+In complex analysis, the '''residue'''  of a function ''f''  [[holomorphic function|holomorphic]] in an open set <math>\Omega</math> with possible exception of a point <math>z_0\in\Omega</math> where the function may admit an [[isolated singularity]], is a particular number describing behaviour of ''f'' around <math>z_0</math>.
+More precisely, if a function ''f'' is holomorphic in a neighbourhood of <math>z_0</math> (but not necessarily at <math>z_0</math> itself), with either a [[removable singularity]] or a [[pole (complex analysis)|pole]] at <math>z_0</math>, then it can be represented as a [[Laurent series]] around this point, that is
-More formally, if a function ''f'' is holomorphic in a neighbourhood of <math>z_0</math>
-then it can be represented as the Laurent series around this point, that is
 :<math>f(z) = \sum_{n=-N}^\infty c_n (z-z_0)^n</math>
-with some <math>N\in \mathbb{N}</math> and coefficients <math>c_n\in \mathbb{C}.</math>
+with some <math>N\in \mathbb{N}\cup\{\infty\}</math> and coefficients <math>c_n\in \mathbb{C}.</math>
-The coefficient <math>c_{-1}<math> is the residue of ''f'' at <math>z_0<math>, denoted as
-<math>\mathrm{Res}(f,z_0) or <math>\mathrm{Res}_{z_0}f.</math>
-Although the choice of the coefficient may look arbitrary, it turns out that it is well motivated by the particularly important role played by this number in the theory of complex function.
+The coefficient <math>c_{-1}</math> is the '''residue''' of ''f'' at <math>z_0</math>, denoted as
-For example, the residue allows to evaluate [[path integral]]s of the function ''f'' via the [[residue theorem]]. This technique finds many application in real analysis as well.
+<math>\mathrm{Res}(f,z_0)</math> or <math>\underset{z=z_0}{\mathrm{Res}}f(z).</math>
-[[Category:CZ Live]]
+Although the choice of the coefficient may look arbitrary, it turns out that it is well motivated by the particularly important role played by this number in the theory of complex functions.
-[[Category:Mathematics Workgroup]]
+For example, the residue allows to evaluate [[path integral]]s of the function ''f'' via the [[residue theorem]]. This technique finds many applications in real analysis as well.
-[[Category:Stub Articles]]

Residue (mathematics): Difference between revisions

Latest revision as of 16:56, 12 November 2008

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