Complex Analysis, Video #25 (Complex Arithmetic, Part 25)
Main Topics: numerically check 5th roots from previous video. Define an open disk and visualize it using RegionPlot on Mathematica.
Do a quick numerical check of the 5th roots of -1 + sqrt(3)*I from the previous video (use "N" on Mathematica). Use the percent sign % to refer back to the preceeding output (as a list) and raise every number in the list to the 5th power. Definition of an "open disk" of radius positive radius "rho", centered at a given complex number z0. Use set-builder notation to define, as well as the modulus of the difference z - z0, which represents the distance between the points representing z and z0 in the complex plane. Also write in terms of rectangular coordinates x and y by using the distance formula (Pythagorean Theorem). To be an open disk, we must use a strict inequality so that we do NOT include the boundary points of the disk. Use RegionPlot to graph a disk of radius 2 centered at the point 3 + 4*i. RegionPlot graphs it as if the boundary circle is included, but it is not. We can use Show, Graphics, Dashed, and Circle to emphasize that the boundary is not included.