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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. Contour integration is closely related to the calculus of residues, a method of complex analysis.
30. Apr. 2021 · The calculus of residues allows us to employ contour integration for solving definite integrals over the real domain. The trick is to convert the definite integral into a contour integral, and then solve the contour integral using the residue theorem. As an example, consider the definite integral ∫∞ − ∞ dx x2 + 1.
What exactly is the contour integral? It is nothing but a line integral. This point becomes clear if you write dz = dx + idy, Z Z. f(z)dz = (f(x + iy)dx + if(x + iy)dy). (2) C C. Therefore, it can be viewed as a line integral. with. Z A ~ · d~x, C. A~ = (f(x + iy), if(x + iy)). (3) (4) Now comes a crucial observation called Cauchy’s theorem.
Vor 4 Tagen · Contour integration is the process of calculating the values of a contour integral around a given contour in the complex plane. As a result of a truly amazing property of holomorphic functions , such integrals can be computed easily simply by summing the values of the complex residues inside the contour .
Contour integral. Consider a contour \ (C\) parametrized by \ (z (t)=x (t)+iy (t)\) for \ (a≤t≤b\). We define the integral of the complex function along \ (C\) to be the complex number. \ (\int_ {C}f (z)dz=\int_ {a}^ {b}f\left ( z\left ( t \right ) \right ) {z}'\left ( t \right )dt\). (1)
The Contour Integral of a Laurent Expansion Consider a single term a n(z−z 0)n of an expansion, integrated round a closed curve C which encircles z 0 in a positive sense (i.e., anticlockwise) once. For n ≥ 0, we can use Cauchy’s Theorem to obtain immediately I C a n(z −z 0)n dz = 0. For n < 0, first change the contour C to C ε, a ...
The first video on contour integration, part of the complex analysis lecture series. Here we introduce the concept of a contour and what it means to integrat...
- 10 Min.
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- Maths 505
