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Line integrals of scalar fields over a curve ... Consider the function f(z) = 1/z, and let the contour L be the counterclockwise unit circle about 0, ...
Of these, 1 + √ 2 and −1 − √ 2 are outside the unit circle (shown in red, not to scale), whereas 1 − √ 2 and −1 + √ 2 are inside the unit circle (shown in blue). The corresponding residues are both equal to − i √ 2 / 16 , so that the value of the integral is I = 2 π i 2 ( − 2 16 i ) = π 2 4 . {\displaystyle I=2 ...
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. It generalizes the Cauchy integral theorem and Cauchy's integral formula.
The theory for the original curve can be deduced from that of the unit circle, where, because of rotational symmetry, both operators are classical singular integral operators of convolution type. The Hilbert transform satisfies the jump relations of Plemelj and Sokhotski , which express the original function as the difference between the ...
1.1 Fundamental theorem for complex line integrals. ... which traces out the unit circle. Here the following integral: =, is nonzero. The Cauchy ...
Field lines of a vector field v, around the boundary of an open curved surface with infinitesimal line element dl along boundary, and through its interior with dS the infinitesimal surface element and n the unit normal to the surface. Top: Circulation is the line integral of v around a closed loop C. Project v along dl, then sum.
A line integral (sometimes called a path integral) is an integral where the function to be integrated is evaluated along a curve. [42] Various different line integrals are in use. In the case of a closed curve it is also called a contour integral. The function to be integrated may be a scalar field or a vector field.
The set of homotopy classes of maps from a circle to a topological space form a group, which is called the first homotopy group or fundamental group of that space. The fundamental group of the circle is the group of the integers, Z; and the winding number of a complex curve is just its homotopy class.