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Contour integration is closely related to the calculus of residues, [4] a method of complex analysis. One use for contour integrals is the evaluation of integrals along the real line that are not readily found by using only real variable methods. [5] Contour integration methods include:
Then, the residue at the point c is calculated as: (,) = = = = using the results from contour integral of a monomial for counter clockwise contour integral around a point c. Hence, if a Laurent series representation of a function exists around c, then its residue around c is known by the coefficient of the ( z − c ) − 1 {\displaystyle ...
Naturally the analogues of contour integrals will be harder to handle; when n = 2 an integral surrounding a point should be over a three-dimensional manifold (since we are in four real dimensions), while iterating contour (line) integrals over two separate complex variables should come to a double integral over a two-dimensional surface.
The integral here is a complex contour integral which is path-independent because is holomorphic on the whole complex plane . In many applications, the function argument is a real number, in which case the function value is also real.
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. [1] The terms path integral, curve integral, and curvilinear integral are also used; contour integral is used as well, although that is typically reserved for line integrals in the complex plane.
The path C is the concatenation of the paths C 1 and C 2.. Jordan's lemma yields a simple way to calculate the integral along the real axis of functions f(z) = e i a z g(z) holomorphic on the upper half-plane and continuous on the closed upper half-plane, except possibly at a finite number of non-real points z 1, z 2, …, z n.
In complex analysis, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as well.
Theorem (Cauchy) — If is a rectifiable Jordan curve in and if : is a continuous mapping holomorphic throughout the inner region of , then =, the integral being a complex contour integral. Proof We regard the complex plane as R 2 {\displaystyle \mathbb {R} ^{2}} .