Mathematics for Civil Engineers by Yang Xin-She

Author:Yang, Xin-She
Language: eng
Format: epub
Publisher: Dunedin Academic Press Limited
Published: 2018-06-14T16:00:00+00:00

That is to say that the integral over any curve between two points is independent of the path (see Fig. 6.8), as we have demonstrated in the previous example. This property becomes very useful for evaluation of integrals.

Let us introduce Cauchy’s integral formula for complex integration without proof. The integral of f(z)/(z − z0) over any simply closed path Γ enclosing a point z0 in the domain Ω is

which is the basic form of the Cauchy integral formula. In addition, the same integral value can be obtained if the integration is carried out along a circle D enclosed z0. Therefore, we can choose any integration path as convenient and necessary.

As we can see that the integrals here are along a closed contour, such integrals are also called contour integrals.

Example 6.6

Let us look at a simple example to calculate the contour integral of

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