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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.
24. Mai 2024 · 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 .
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.
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.
Contour integration is also known as path integration or complex line integration. Contour integrals arose in the study of holomorphic and meromorphic functions in complex analysis, but they are now used in a wide range of applications, including the computation of inverse Laplace transforms and Z transforms, definite integrals and sums, and ...
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.
- 14,2K
- Maths 505
Contour integral Definition If f(z) is a continuous function on E ˆC, and (t) : [a;b] !E is a smooth (or piecewise smooth) curve, we define Z f(z)dz = Z b a f((t)) 0(t)dt Intuition: think of plugging in z = (t); dz = d (t) dt dt. By above corollary: Z f0(z)dz = f((b)) f((a))
