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What is Cauchy's Theorem?

Welcome back MechanicaLEi, did you know that Augustin-Louis Cauchy was a French mathematician, engineer and physicist after whom Cauchy's theorem is named? This makes us wonder, what is Cauchy's Theorem? Before we jump in check out the previous part of this series to learn about what standard transformations are? First, we need to know a little about line integral of a complex variable which has the following prerequisites, the complex plane is z equals to x plus i y. complex differential is dz equals to dx plus i dy. A curve in this plane is given by gamma of t equals to x of t plus i into y of t, defined for closed interval of a comma b. A complex function f of z is equal to u of x comma y plus i into v of x comma y. Given these prerequisites we define the complex line integral in the plane gamma as integral of of z dz in gamma equal to integral of a to b, f of gamma of t into gamma dash of t dt. A region is said to be simply-connected if any closed curve in that region can be shrunk to a point without any part of it leaving a region. For example, region c1 is simply-connected as it can be shrinked into a point and region c2 is not simply-connected as it has a white void in the middle. Cauchy's Theorem states that if f of z is analytic everywhere within a simply-connected region then: closed integral of f of z dz is equal to zero for every simple closed path c lying in the region. Hence, we first saw what line integral of a complex variable is and then went on to see what Cauchy’s theorem is? In the next episode of MechanicaLEi find out what properties of line integral are? Attributions: Doh De Oh by Kevin MacLeod is licensed under a Creative Commons Attribution licence (https://creativecommons.org/licenses/...) Source: http://incompetech.com/music/royalty-... Artist: http://incompetech.com/ Subtle Library by Fabian Measures (http://freemusicarchive.org/music/Fab...) is licensed under a Creative Commons Attribution licence ( https://creativecommons.org/licenses/...) Source: http://freemusicarchive.org/music/Fab...