Author Archives: Dr Christian P. H. Salas

The Legendre transform is a mechanism for converting a relationship like into a relationship like In other words, by going from a function to its Legendre transform and vice versa, the roles of and in (1) and (2) can be reversed. In moving from (1) to (2), the required Legendre transform is because differentiating bothContinue reading “Legendre-transforming a Lagrangian mechanics problem into a Hamiltonian one”

A heat flow problem led to the following general series solution: An initial condition at required , so it was necessary to find the coefficients in the following sine series: The sine functions here involve coefficients of of the form rather than the usual . They have period and by looking at graphs it isContinue reading “A note on a Fourier sine series solution to a heat flow problem”

Bessel’s equation arises in countless physics applications and has the form where the constant is known as the order of the Bessel function which solves (1). The method of Frobenius can be used to find series solutions for of the form where is a number to be found by substituting (2) and its relevant derivativesContinue reading “Equivalent integral formulae for Bessel functions of the first kind, order zero”

I needed to integrate increasing odd and even integer powers of the sine function repeatedly in a power series, with limits from to . Looking at the graph of the sine function, it is obvious that since sine is itself an odd function its odd powers must integrate to zero over the interval from toContinue reading “Useful formula for integrating integer powers of a sine function over a period”

A basic result from applying Gauss’ Law of electromagnetism to a conductor is that any excess charge always ends up distributed entirely on the surface of the conductor. Furthermore, the excess charge will always distribute itself there so as to produce an equipotential surface. The resulting surface charge density distribution per unit area is usuallyContinue reading “Surface charge density of a conducting disk from that of a conducting ellipsoid”

I recently needed to integrate the fifth power of a sine function from zero to and did this via a hypergeometric function, which I found quite interesting. I will make a quick note of it here. I came across the following entry in a table of integrals: The function is a special function known asContinue reading “Integrating integer powers of a sine function via a hypergeometric function”

When confronted with a first-order differential equation, try to convert it to one of these five basic types: 1). Separable: This is when can be written . If there is an initial condition , we can rearrange to get an equation involving two integrals: This incorporates the initial condition. It can sometimes be solved toContinue reading “Quick memo on five basic types of first-order differential equation”

My first academic job was as a lecturer in economics at Royal Holloway, University of London, where I was responsible for the second-year mathematical methods and econometrics course EC2203 Quantitative Methods in Economics II. I have lost all the original manuscripts of the lecture notes but still have scanned copies of most of the printoutsContinue reading “Royal Holloway lectures (Quantitative Methods in Economics II)”

The present note uses the setup outlined in a previous note about a 2016 paper in Physical Review Letters on the large deviation analysis of rapid-onset rain showers. This paper will be referred to as MW2016 herein. In order to apply large deviation theory to a collector-drop framework for rapid-onset rain formation, equation [14] in MW2016 presentsContinue reading “Derivation and manipulation of a generalised Erlang distribution”

In this note, I develop and implement a spline function approximation scheme that can quickly generate accurate benchmark solutions for large deviation theory problems. To my knowledge, spline function approximation methods have not been applied before in the context of large deviation theory. This note shows that such methods can be efficient and effective evenContinue reading “A spline approximation scheme for large deviation theory problems”