I derive in detail the specific form of 2D Newton-Raphson for a simple root-finding problem on the complex plane. Try it out and create your own fractals!Video. Notes.
Category Archives: Complex numbers
Green’s function for diffusion in 1D. Part 2.
I finalize the discussion started in Part 1 with an example showing the behavior for a specific source, which is a simple Gaussian in space and in time. I comment on how to make sure the desired initial conditions are satisfied by adding a solution to the homogeneous equation. Video. Notes.
Green’s function for diffusion in 1D. Part 1.
I solve for the Green’s function of the diffusion operator on the whole real line. To that end, I use Fourier transforms in space and time and contour integration (more on that later!).Video. Notes.
The quantum free particle
Using background on complex numbers and Fourier modes, here is a first discussion of the most essential quantum mechanical system. Video. Notes.
Euler’s formula
A quick look at the derivation of Euler’s formula and its uses and connections to trigonometric and hyperbolic sine and cosine functions of complex variables. Video. Notes.
Complex numbers basics
Definition and basic properties of complex numbers. Addition, multiplication, division, conjugation. Polar representation. Trigonometric identities. Differentiation & Cauchy-Riemann conditions. Simple examples. Video. Notes.