I have always believed that mathematics is about thinking rather than memorizing. The trigonometric identities were among the things we were told to memorize at school, and not only I struggled with that but I also actively rebelled against this approach. For me, mathematics is fundamentally about having a minimal but sufficient set of definitions and axioms, understanding them deeply, and then deriving everything else from these foundations. During one of my Complex Analysis lectures in my undergraduate Applied Mathematics studies, I was introduced to Euler's formula. When I discovered how it could be used to derive some of the most important trigonometric identities, I was more than relieved. I could finally derive the identities easily whenever I needed them, instead of relying on rote memorization, or panicking about my failure to memorize them.

In this post we are going to explore the so called orthogonal functions, and some of their properties. We are also going to show that these orthogonal functions are closely related to the least-squares approximation method. This alternative to the least-squares method can be helpful in certain cases when the least-squares produces a hard to solve linear system.

Chebyshev polynomials are a sequence of orthogonal polynomials that play a central role in numerical analysis, approximation theory, and applied mathematics. They are named after the Russian mathematician Pafnuty Chebyshev and come in two primary types: Chebyshev polynomials of the first kind (\(T_n(x)\)) and Chebyshev polynomials of the second kind (\(U_n(x)\)). In this post we are going to focus on the Chebyshev polynomials of the first kind.

In this post we are going to explore the Fourier method for solving the 1D wave equation. The method is more known under the name of the method of separation of variables. For the 1D wave equation we are going to show the application of the method to a fixed string. We are also going to attempt to outline some of the physical interpretations of the fixed string.