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greeting students and welcome back to another lesson on complex variables in this video Im going to define branch points and branch cuts and build a foundation thatll allow us to perform complex integration of functions with branches to build this foundation Ill start by discussing the natural log of a complex number Z suppose I have a function W equals F of Z which equals lon of Z if you remember my very first video on complex variables youll recall that I can write any complex number as its real part plus I times its imaginary part in the case of W I can write it as u plus IV where u and v are real numbers I can also write a complex number Z in the complex plane as a polar representation with Z being the distance from the origin R times the exponential of I times the angle relative to the positive real axis theta in that case the natural log of Z will become the natural log of R plus I times theta note that R and theta are both positive real numbers if I now rearrange this equat