 Welcome back MechanicalEI. Did you know that Augustine Lois 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 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 iy. Complex differential is dz equals to dx plus i into 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 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 0 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.