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Complex Integrals | Contour Integration | Complex Analysis #11

The basics of contour integration (complex integration). The methods that are used to determine contour integrals (complex Integrals) are explained and illustrated with a lot of examples with solutions. You will also learn how to parametrize the standard curves/paths and how to determine when you can ignore the path and only consider the start and end points of the path when determining the contour integral. But in summary, this video will include concepts as: ► Definition of Contour Integrals ► Definition of a Directed Smooth Curve ► Definition of a Contour ► How to Parametrize the standard curves ► A theorem about Independence of Path ► Methods to solve Contour Integrals ► Tips and Tricks for Contour Integration LINK TO COMPLEX ANALYSIS PLAYLIST    • The Complete Guide to Complex Analysi...   LINK TO CANVAS https://drive.google.com/file/d/1irTA... SUPPORT Consider subscribing, liking or leaving a comment, if you enjoyed the video or if it helped you understand the subject. It really helps me a lot. CONCEPTS FROM THE VIDEO ► Contour Integral It is like the case with normal integrals, but here the integrand is a complex valued function of a complex variable instead of a real function of a real variable. The complex function is integrated along a specific path and in general the integration will yield different results for different paths (with the same start and end points). This comes from the fact that there are many differents path from one point in the complex plane to another. 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. Furthermore if a function f(z) is continous on a directed smooth curve (gamma), and if z(t) (from t=a to t=b) is a parametrization of this curve, then: Int_(gamma) f(z) dz = Int_a^b f(z(t))*z'(t) dt ► Independence of Path There are functions that will yield the same result when integrated along different paths which have the same start and end points. If a function f(z) is continous on a contour (gamma) from z_1 to z_2, and f(z) has an antiderivative F(z) on gamma, then: Int_gamma f(z) dz = F(z_2) - F(z_1) so the only thing that matters when calculating the integral is the start and end points of gamma. ► Directed smooth curve A point set is said to be a smooth curve if it is in range of some continous value function z = z(t) on [a, b] that satisfies the following conditions: 1) z(t) has a continous derivative on [a, b] 2) z'(t) never vanishes on [a, b] 3) z(t) is one to one on [a, b] But in short this simply means that the curve should not have a any sharp corners and no intersections with itself. A directed smooth curve is a smooth curve with a direction. ► Parametrization Imagine you start at a time t_0 and start tracing the curve gamma on a graph. At any particular time, a dot is drawn and the locus of dots generated over an interval of time t_0 to t_1 constitutes the curve. You could say that by using this method we can create a function z as a function of t, such that the curve gamma is in the range of z(t) as t varies between t_0 and t_1. Then we call the function z(t) the parametrization of the curve gamma. So in short it is a way to describe the curve with the help of the function z(t). ► Contour A single point or a finite sequence of directed smooth curves, such that the terminal point of one directed smooth curve coincides with the following directed smooth curve. TIMESTAMPS 00:00 - 00:20 Definition/Theorem Contour Integrals 00:20 - 01:44 Standard Parametrizations 01:44 - 03:42 Theorem Independence of Path Examples: Compute each Integral 03:42 - 04:52 f(z) = z along a straight line 04:52 - 06:33 f(z) = z along a quarter arc of a circle 06:33 - 09:23 f(z) = z along some weird path 09:23 - 10:05 f(z) = z^bar along two connected paths 10:05 - 12:09 Notes about the most used trap in (pitfall) ERROR I missed adjusting for the "+1" term in the second integrand when I determined the primitive in the video at around 11:35. SOCIAL ► Follow me on Youtube: http://bit.ly/1NQhPJ9 ► Follow me on Twitter:   / the_mathcoach   HASHTAGS #TheMathCoach #ComplexAnalysis