The Arithmetic Mean – Geometric Mean inequality is an endless source of problems. A particularly sly use of it is contained it the attached paper

15 February 2023

I have done a lengthy paper on the basics of Bessel functions which explains the historical genesis of them in the context of astronomy as well as giving many very detailed proofs covering equivalence of various forms of the functions, uniform convergence, analyticity and so on. It is an understatement to say that I have only touched the surface in what is a 66 page paper. Anyone who has ever read Watson’s tome on the subject will understand just how sprawling the subject is. To access the paper click here.

03 October 2022

Bernoulli’s inequality is a useful little tool for various aspects of analysis and the following short article shows how to prove it and use it in a uniform convergence proof. Download the article here

27 August 2022

Exponentials and logarithms are fundamental to calculus and analysis. The following article explores some fundamental properties: Exponentials and logarithms

14 July 2022

Mean square convergence of functions plays important roles in mathematical physics and many other areas. To learn more about mean square convergence and its relationship to pointwise convergence even where discontinuities are involved read this short paper.

10 June 2022

Bessel functions are usually introduced in undergraduate Fourier analysis or engineering courses in the context of hanging chains and vibrating circular membranes, for example. However, the integral form of Bessel’s function actually arose from Bessel’s analysis of the eccentric anomaly in elliptic planetary motion and yet a modified form of Bessel’s function, known as a spherical Bessel function figures in the solution of a certain radial equation derived from the Schrodinger wave equation. A detailed paper on Bessel functions is coming soon. To learn more see this article: Spherical Bessel functions in quantum mechanics.pdf

17 March 2022

I have updated the paper on the Laplacian to contain a proof of a property that I had glossed over to my shame in previous iterations. A further expansion of the material covering the use of rotation matrices to obtain gradients and the Laplacian is coming. The updated paper can be accessed here

06 February 2022

The Gaussian is ubiquitous throughout mathematics and science. The fact that it maximises entropy is one of the reasons for this ubiquity. It also has a role in the Heisenberg Uncertainty Principle. To understand this in more detail read the following article.

17 July 2021

I recently noticed an integration substitution trick in a Youtube video https://www.youtube.com/watch?v=BfZObnTIsYk&t=110s that apparently came from an Indian high school exam. What was interesting was reverse engineering the trick (which was actually unnecessary anyway) and then showing how powerful simple substitutions are for the integral form of Bessel functions. View the article here.

12 April 2021

The Koide formula in particle physics is a fascinating experimental proposition that may suggest deeper things. In this article I explore some of the maths. Download here.

19 March 2021

Serious students of analysis will need to make various estimates of trigonometrical quantities and the following paper may assist in either refreshing or expanding knowledge. Download the paper here.

01 September 2020

In 1905 Einstein produced a remarkable paper on Brownian motion. In this paper he derived from meagre physical and probabilistic assumptions a partial differential equation for the heat equation which had a well known solution at the time. I have gone through his succinct derivation in detail in this article.

26 July 2020

The scientists at Hitachi did a double slit experiment some years ago and it is worth viewing the video they made. This is the quintessential quantum experiment. Follow this link

23 July 2020

Updated paper on the Laplacian by adding some material on natural frames for cylindrical coordinates and a section on how differential forms can be used to work out elementary areas and volumes purely algebraically. The updated paper is here.

23 May 2020

Trigonometrical integration is absolutely fundamental in higher mathematics and physics yet it is often treated in a superficial way. Salomon Bochner, who was an expert in Fourier theory, did a series of lectures in the 1950s in which he developed a really basic, yet rigorous approach, to trigonometric integrals which are, of course, at the core of Fourier theory. This is an “old school” approach which is not in favour today. It is reminiscent of how Frigyes Riesz did functional analysis almost like writing an airplane novel. My functional analysis professor, the late Alan McIntosh (of Kato’s square root fame), was taught using Riesz’s works, and that is the sort of flavour I bring to this article. There is no blizzard of epsilons, rather it is all about making some basic observations about trigonometric behaviour which any serious student will appreciate. The view the article click here.

26 April 2020

As a complement to my earlier paper on the intuition behind the Fourier and Laplace transforms I have done a detailed paper explaining how Fourier integrals (transforms) arise in the context of solving the heat equation for both discrete and continuous eigenvalues. The paper emphasises the basic point that the Fourier integral or transform owes its existence to solving differential equations. To read the paper click here

09 April 2020

Engineering and maths students frequently seek an explanation for the “intuition” behind the Fourier and Laplace transforms. The following paper explains the roots of the Fourier and Laplace transforms and provides some insights into why they exist. Download the paper here

18 February 2020

It is a standard homework problem in undergraduate physics to show how the Stefan-Boltzmann law can be derived from Planck’s radiation law. Underpinning the derivation one has to evaluate a certain type of integral and in this article I go through all the analytical steps involved in the evaluation. You need to know some analysis, Fourier theory and the properties of the gamma function. I also provide some historical information how Planck derived his law. Download the article here.

06 February 2020:

That the Gaussian can be extracted from an integral of cosines was proved by French functional analyst and probability theorist, Paul Levy, back in 1922 in his book “Lessons on Functional Analysis” ( this is the translated title but as far as I can tell there is no translation of the original French work). To see how this was done and appreciate that Norbert Wiener used Levy’s theory to develop his approach to Brownian motion, read this article.

18 January 2020:

The Gram-Schmidt orthogonalization process is an important tool in linear algebra and much more. It is based on a recursive process which can be visualised in 2 and 3 dimensions and inductively extended to n dimensions. To learn more about why it works read this article.

20 November 2019:

The theory of matrix exponentiation is covered in linear ordinary differential equations courses usually. If you want some practice at how the theory works, there are several problems in the Cambridge Tripos Part 1A exam from 30 May 2019 which may be of interest. Full solutions can be found here.

08 November 2019:

In the May 2019 Part 1A Cambridge Mathematical Tripos examination a couple of problems caught my eye for being like a “cheeky” white wine - inviting you to quaff them quickly in a mathematical sense. See what you think. As usual there are plenty of problems that require proficiency in technique to get through them in the space of 3 hours.

To view the problems and solutions click here.

27 August 2019:

The inequalities of Holder and Minkowski are fundamental to analysis and J E Littlewood devoted a whole book ( “Lectures on the Theory of Functions” ) to squeezing the mathematical pips out of them. To see how Littlewood proved these inequalities download this short article here.

11 July 2019:

Singular Value Decomposition (SVD) is an important part of machine learning algorithms and this article goes through the mechanics of SVD in a tutorial format. Download the article here.

A Powerpoint presentation can be accessed here

21 April 2019:

How did Maxwell derive his famous velocity distribution law, one of the foundational elements of statistical mechanics? As you will see, he used geometry and functional equation concepts to derive the law in only a few lines. To see how the gaps are filled in read the detailed paper here.

11 February 2019:

The Math Stack Exchange seems to be a refuge for some arrogantly offensive types so students who want some illumination through that forum do so at their own risk. Don’t be surprised if someone humiliates you. And mathematicians wonder why many people don’t like mathematics! For my views on this forum, read this paper

29 January 2019:

Chebyshev’s sum inequality is an important inequality and can be proved in various ways. Emile Picard proved it in the 1880s via concepts of centre of gravity. It can be proved easily once one has proved a more fundamental inequality. To see how read here.