Residue

The residue of a complex function at an isolated singularity is the coefficient of the (z-a)^{-1} term in its Laurent series there. It determines the function’s contour integral around that singularity.

Laurent series: A series in integer powers of a variable, allowing both positive and negative exponents. Its negative-power terms encode the local singular behavior from which a residue is read.
Complex analysis: The study of functions of complex variables and their analytic properties. Residues arise from the local structure and integration theory of complex functions.
Residue calculus: The use of complex residues to evaluate contour integrals and related expressions. It applies the residue theorem as a practical integration technique.
Removable singularity: An apparent singularity where defining the function's value restores holomorphicity. Its Laurent series has no negative powers, so its residue is zero.
Cauchy residue theorem: A theorem expressing a contour integral as 2πi times the sum of enclosed residues. It converts residues at enclosed singularities into the value of a contour integral.
Holomorphic function: A complex function that is complex differentiable throughout an open neighborhood. A function is holomorphic near a point precisely when no singularity contributes a residue there.
Jordan's lemma: A result bounding integrals over large semicircles for certain exponentially decaying functions. It helps justify closing contours when residues are used to evaluate Fourier-type integrals.
Branch point: A point around which analytic continuation can change a function's value between branches. A branch point is generally not an isolated singularity with an ordinary Laurent residue.
Contour integral: An integral evaluated along a directed curve, often in the complex plane. Residues determine contour integrals when the contour encloses isolated singularities.
Isolated singularity: A point where a function fails to be holomorphic while remaining holomorphic nearby. Residues are defined at isolated singularities, not arbitrary singular sets.

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