Electrostatics and Green's Functions: Mathematical Methods

This lecture covers mathematical methods for physicists, focusing on electrostatics and the use of Green's functions. The instructor begins by addressing a previous question about units in the context of forces acting on a spring. The discussion transitions to the concept of delta functions and their role in representing forces. The instructor explains how to derive the electrostatic potential inside a cylinder with different potentials on its halves, emphasizing the importance of Laplace's equation. The lecture highlights the relationship between analytic functions and harmonic functions, demonstrating how the real part of an analytic function can represent the electrostatic potential. The Poisson kernel is introduced as a method to compute potentials from boundary conditions. The instructor illustrates the derivation of the potential using complex analysis, integrating over the unit disk and discussing the implications of branch cuts in logarithmic functions. The lecture concludes with a discussion on the behavior of electric fields near discontinuities in potential, reinforcing the connection between mathematical methods and physical applications.