Complex Integration and Finding Zeros of the Zeta Function

In this video we examine the other half of complex calculus: integration. We explain how the idea of a complex line integral arises naturally from real definite integrals via Riemann sums, and we examine some of the properties of this new sort of integral. In particular, we consider some complications that arise when trying to apply the fundamental theorem of calculus to complex functions. We then bring these ideas to the central focus of this series: the zeta function and the Riemann hypothesis. By the end of the video, we will be able to use complex integrals to approximate the location of the zeroes of the zeta function (or those of any other complex function for that matter)! -- Please subscribe for more content like this! If you'd like to leave a tip, click the Thanks button under this video. We appreciate any support towards making more videos! Also, check out our second channel @zetamathdoespuzzles , where we stream sudoku and other puzzle content three times a week. Animations in this video were produced using Manim Community Edition. For more information, go to https://manim.community