This part of the site is devoted to analysis - the sort of thing that gives the struggling calculus student sweaty palms when they hear references to epsilon - delta proofs, uniform continuity etc. Analysis is often regarded as really hard but in my experience this is usually because it can be taught extremely poorly.  There are in fact several very lucid communicators in the area. Elias Stein for instance won the Wolf Prize in part for the quality of his exposition of functional analysis concepts (his Princeton Series of Lectures in conjunction with Rami Shakarchi are wonderful books). David Bressoud has also produced some superb books aimed at making analysis much more understandable at a practical level and in an historical context.

The aim is to give detailed but comprehensible explanations of fundamental analytical techniques.  Fourier Theory is the jewel in the crown when it comes to the application of analytical techniques and over time I will populate the site with material based around it, covering both the pure mathematics behind it but also the applications which are incredibly wide.

1. If you want to understand how to develop the binomial series for negative integral exponents read this article: The binomial series for negative integral exponents.pdf

2. If you want to develop a deeper knowledge of some fundamental analytical techniques of Fourier theory why not try my paper titled:   The good, the bad and the ugly of kernels: Why the Dirichlet kernel is not a good kernel.  It derives its inspiration from Elias Stein and Rami Shakarchi's book "Fourier Analysis: An Introduction". Download my paper here 
 
3.  Dirichlet’s test for the convergence of series.  In this short article I fully develop the test which is useful for oscillatory series. There is a derivation of the summation by parts formula. Click here to download  the article.  
 
4.  Applying Riemannian integration theory to some practical examples.  This is the mathematical equivalent of building a pyramid with a trowel, but having done it once (an only once!) you will appreciate the power of the theory.  The article can be accessed here: Riemann integration- some practical examples.pdf  
 
5. If you want to brush up on some more advanced inequality techniques have a look at this paper which takes you through the solution process for some more difficult types of inequalities.  You will need a knowledge of the Arithmetic Mean –Geometric Mean inequality and the Cauchy-Schwarz in equality.  The Rearrangement Theorem and Chebyshev’s inequality are also covered.  Download here: Advanced inequality manipulations.pdf
 
6. When Schwarz proved his famous inequality he used an insight that enables a one line proof. To understand that proof  click here: A one line proof of the Cauchy-Schwarz inequality.pdf
 
7. Proving the uniform continuity of sin x/x with some connections to Fourier transform theory. If you really want to get your hands dirty with epsilon-delta proofs go no further: Uniform continuity of sinc x.pdf

8. A short paper on logical manipulations for beginning analysis students:   Basic logic for first year analysis students.pdf

9. I have substantially expanded the existing paper on the wave equation and energy conservation to cover the general case of d dimensions where d > 1.This involves a level of mathematical sophistication far greater than the one dimensional case dealt with initially.  A fundamental part of the derivation involves differentiation under the integral sign, which requires a detailed discussion of the d dimensional Leibniz rule.  The Reynolds Transport Equation is effectively this rule and there is a detailed discussion of Harley Flanders’ reconciliation of mathematicians’ and physicists’ proofs of the Leibniz rule.  You are warned – this is not for the faint hearted but I have put in all the detail so you should not get lost. Download the new paper here: The wave equation and energy conservation.pdf

10. While it is possible to apply Fourier theory without knowing precisely why all the integrals converge, for those who worry about such things or actually have to demonstrate some understanding in an analysis exam, my detailed paper on “Basic Fourier Integrals” may help. It expands material covered in Elias Stein and Rami Shakarchi's Princeton Lectures on Fourier Theory. The concept of Schwartz space is developed in detail against the background of older approaches. Applications of Fourier theory are explained and the use of Fourier transforms to solve the Black-Scholes equation from finance is done in great detail. Download the paper here: Basic Fourier integrals.pdf

11. While soft sand running with Bondi Beach’s surfing physicist Ruben Meerman  of the ABC’s “Catalyst” program we were discussing, among other things, why 1-1+1-…=1/2.  My paper on Cesaro summability explains what is going on with such series and how mathematicians, like Mussolini redefining lateness so the trains ran on time (undoubtedly a myth but it sounds good), redefine what sums of such series are so that you get nice behaviour. The basics of Cesaro summability
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12. Laplace’s method of estimating the leading order behaviour of certain integrals is a powerful technique which can be used to prove Stirling’s formula among many other things.  The proof of Laplace’s method is a problem in Polya’s and Szego’s famous book “Problems and Theorems in Analysis 1”.  Their proof is rigorous but skips many fine details (which is not surprising given the level at which the book is pitched) which I have filled in in the download: Laplace’s method for integral asymptotics.pdf
 
13. The Fejer kernel figures prominently in the theory of convergence of Fourier series and, unlike the Dirichlet kernel, it is well behaved.  This good behaviour is explained by the Fejer kernel’s use of Cesaro sums.  My detailed paper sets out all the relevant derivations (with multiple styles of proof for the most important results). If you are doing some serious Fourier analysis this will be of interest. To download click  here: The nitty gritty of Fejer's Theorem.pdf

14. Tutorial on the uniform continuity of the Fourier Transform 
 
If you want to understand why, using classical analysis techniques, the Fourier transform of a function is uniformly continuous read this paper and watch the accompanying videos.  The  two part tutorial can be viewed here:
 
Part 1   https://youtu.be/wLxrfhcLxRY

 
Part 2  https://youtu.be/6OHsfjBYItw

or downloaded in lower definition (.wmv files) here:
 
Part 1  

​Part 2

The accompanying paper can be downloaded here
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15.   Chebyshev’s inequality for L^p spaces can be proved in 2 lines (as Steven Krantz does in his book “A Panorama of Harmonic Analysis” ) but it is instructive to expand the steps.  Download the paper: Proof of Chebyshev's inequality in L^p spaces.pdf
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16.  An Australian high school maths teacher, Eddie Woo, who seems to have some cachet with students,  did a video on the limit of x^x as x approaches  0.  I long ago gave up on high school maths but the video was so superficial I just had to fill the gaping holes: Is there a rigorous high school limit proof.pdf

17.    Eli Stein and Rami Shakarchi pose a meaty problem on Hermite functions in their book “Fourier Analysis – An Introduction”. Although their focus is not that of a quantum physics textbook ( they are functional analysts ), the problem is of interest from a number of perspectives.  A full solution is given and contains proofs of several intermediate steps.
Hermite functions - a solution to a Stein and Shakarchi problem.pdf
 
18.  Forgotten how to do square roots in your head with calculus?  squareroot.mov

19.  Euler was adept at using L’Hopital’s rule multiple times to calculate various “indeterminate” limits eg of the form 0/0.  The following article gives an example of what Euler did and how you can get the same result without use of L’Hopital’s rule. ​
Multiple uses of L'Hopital's rule


20.  It  is a fundamental result of Fourier theory that the Fourier transform of a Gaussian is a Gaussian. But what is the Laplace transform of a Gaussian? So here is a demonstration of both cases: The Laplace transform of a Gaussian.pdf

21. 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.

22. 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.

23.  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.

24.  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.

25. 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

26. 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

27. 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.

28. 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.

29.  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.

30.  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.
 
31. Exponentials and logarithms are fundamental to calculus and analysis.  The following article explores some fundamental properties: Exponentials and logarithms

32.  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

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

34.  Dirichlet’s convergence test is very useful for proving convergence of oscillatory series.  It can be proved by first proving the summation by parts formula. To see the details click here

35. Laplace transforms can be used to evaluate integrals in ways which might offer some simplification. They can also be used to perform abstract approximations of integrals and Renyi’s parking problem is an example of that type of use. The following article deals with an integral from the American Mathematical Monthly which is solved using a Laplace Transform and for comparison by another method. Download the article here.

36. I have produced a lengthy paper titled “The basics of Tauberian theory”. This paper is really only of interest to hard core analysis students probably at the Cambridge Tripos level.

This paper deals with the history of Tauberian methods in analysis in the  context of Abelian and Cesaro methods of summation. Hardy’s book on “Divergent Series” is used as the foundation of this history. Tauber’s original proof of his first theorem is given based on a translation of his paper. This is followed by an undergraduate analysis course style of proof. Hardy’s “epsilon free” proof of Tauber’s first theorem is explained in detail and its relationship to Laplace transform theory is fleshed out to lay the foundation for the use of Laplace transforms and Tauberian techniques to obtainan an  asymptotic limit for the Renyi Parking Problem.  This process unites analysis and Laplace transform theory with probability theory. The original papers of Tauber, Renyi and Karamata are included in the Appendix. I give a heuristic explanation for Karamata's 1931 simplification of the earlier Hardy-Littlewood Tauberian theorem. I sent the Rigour Police on a long boozy holiday for the purposes of that explanation.

The paper can be accessed here

37. Carlson’s inequality is as follows:  if  the a_k  are positive real numbers not all zero then
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Carlson's inequality

​He proved it by rather intricate methods in 1935 and then shortly after Hardy came along with a truly gobsmacking  proof of a few lines that can only be described as magical.  I have gone through how Carlson proved the inequality (and there is a continuous version as well) and then compare  it with Hardy’s proof. Having done the hard work of getting into Carlson’s mindset I have to say that Hardy’s proof is just immediately convincing while Carlson’s approach is so intricate it is easy to get lost.
To understand Carlson’s approach you will need to know something about the Cauchy-Schwartz and Holder inequalities as well as harmonic/subharmonic theory, Fourier theory and Mittag-Leffler expansions. You have been warned! The paper is here