You can simply help by rating individual sections of the book that you feel were inappropriately rated! This page was last edited on 16 November 2016, at 14:18. This all formula of integration pdf is about the line integral in the complex plane.
One use for contour integrals is the evaluation of integrals along the real line that are not readily found by using only real variable methods. One method can be used, or a combination of these methods, or various limiting processes, for the purpose of finding these integrals or sums. This definition of a curve coincides with the intuitive notion of a curve, but includes a parametrization by a continuous function from a closed interval. This more precise definition allows us to consider what properties a curve must have for it to be useful for integration.
In the following subsections we narrow down the set of curves that we can integrate to only include ones that can be built up out of a finite number of continuous curves that can be given a direction. These requirements correspond to requiring that we consider only curves that can be traced, such as by a pen, in a sequence of even, steady strokes, which only stop to start a new piece of the curve, all without picking up the pen. Contours are often defined in terms of directed smooth curves. These provide a precise definition of a “piece” of a smooth curve, of which a contour is made.
A smooth curve that is not closed is often referred to as a smooth arc. It is most useful to consider curves independent of the specific parametrization. Note that not all orderings of the points are the natural ordering of a smooth curve. In fact, a given smooth curve has only two such orderings. Also, a single closed curve can have any point as its endpoint, while a smooth arc has only two choices for its endpoints. Contours are the class of curves on which we define contour integration. This includes all directed smooth curves.