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In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane. [1][2][3] 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 ...
A contour line (also isoline, isopleth, isoquant or isarithm) of a function of two variables is a curve along which the function has a constant value, so that the curve joins points of equal value. [1][2] It is a plane section of the three-dimensional graph of the function parallel to the -plane. More generally, a contour line for a function of ...
Jordan's lemma. In complex analysis, Jordan's lemma is a result frequently used in conjunction with the residue theorem to evaluate contour integrals and improper integrals. The lemma is named after the French mathematician Camille Jordan.
The simple contour C (black), the zeros of f (blue) and the poles of f (red). Here we have ′ () =. In complex analysis, the argument principle (or Cauchy's argument principle) is a theorem relating the difference between the number of zeros and poles of a meromorphic function to a contour integral of the function's logarithmic derivative.
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 ...
v. t. e. In mathematics, the Cauchy integral theorem (also known as the Cauchy–Goursat theorem) in complex analysis, named after Augustin-Louis Cauchy (and Édouard Goursat), is an important statement about line integrals for holomorphic functions in the complex plane. Essentially, it says that if is holomorphic in a simply connected domain ...
In mathematics the estimation lemma, also known as the ML inequality, gives an upper bound for a contour integral. If f is a complex -valued, continuous function on the contour Γ and if its absolute value |f (z) | is bounded by a constant M for all z on Γ, then. where l(Γ) is the arc length of Γ. In particular, we may take the maximum.
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 function to be integrated may be a ...