In Euclidean geometry, all lengths, areas and volumes are relative. Once a unit of length is chosen, all other lengths are given in terms of this unit. Classical geometry could determine the lengths of straight lines, the areas of polygons and the volumes of simple solids. However, the lengths of curved lines, areas bounded by curves and volumes with curved surfaces were mostly beyond the scope of Euclid. Only a few volumes — for example, the sphere, cylinder and cone — could be measured using classical methods.

## Posts Tagged 'Analysis'

### Circles, polygons and the Kepler-Bouwkamp constant

Published April 8, 2021 Occasional Leave a CommentTags: Analysis

If circles are drawn in and around an equilateral triangle (a regular trigon), the ratio of the radii is . More generally, for an N-gon the ratio is easily shown to be . Johannes Kepler, in developing his amazing polyhedral model of the solar system, started by considering circular orbits separated by regular polygons (see earlier post on the *Mysterium Cosmographicum* here).

Kepler was unable to construct an accurate model using polygons, but he noted that, if successive polygons with an increasing number of sides were inscribed within circles, the ratio did not diminish indefinitely but appeared to tend towards some limiting value. Likewise, if the polygons are circumscribed, forming successively larger circles (see Figure below), the ratio tends towards the inverse of this limit. It is only relatively recently that the limit, now known as the Kepler-Bouwkamp constant, has been established.

Continue reading ‘Circles, polygons and the Kepler-Bouwkamp constant’### Differential Forms and Stokes’ Theorem

Published March 11, 2021 Occasional Leave a CommentTags: Analysis

The theory of exterior calculus of differential forms was developed by the influential French mathematician Élie Cartan, who did fundamental work in the theory of differential geometry. Cartan is regarded as one of the great mathematicians of the twentieth century. The exterior calculus generalizes multivariate calculus, and allows us to integrate functions over differentiable manifolds in dimensions.

The fundamental theorem of calculus on manifolds is called Stokes’ Theorem. It is a generalization of the theorem in three dimensions. In essence, it says that the change on the boundary of a region of a manifold is the sum of the changes within the region. We will discuss the basis for the theorem and then the ideas of exterior calculus that allow it to be generalized. Finally, we will use exterior calculus to write Maxwell’s equations in a remarkably compact form.

### Mamikon’s Theorem and the area under a cycloid arch

Published February 25, 2021 Occasional Leave a CommentTags: Analysis, Geometry

**The Cycloid**

The cycloid is the locus of a point fixed to the rim of a circular disk that is rolling along a straight line (see figure). The parametric equations for the cycloid are

where is the angle through which the disk has rotated. The centre of the disk is at .

* * * * *

**That’s Maths II: A Ton of Wonders**

by Peter Lynch now available.

Full details and links to suppliers at

http://logicpress.ie/2020-3/

>> Review in *The Irish Times <<*

* * * * *

Continue reading ‘Mamikon’s Theorem and the area under a cycloid arch’

### Euler’s Product: the Golden Key

Published January 28, 2021 Occasional Leave a CommentTags: Analysis, Euler

The Basel problem was solved by Leonhard Euler in 1734 [see previous post]. His line of reasoning was ingenious, with some daring leaps of logic. The Basel series is a particular case of the much more general **zeta function**, which is at the core of the Riemann hypothesis, the most important unsolved problem in mathematics.

Euler treated the Taylor series for as a polynomial of infinite degree. He showed that it could also be expressed as an infinite product, arriving at the result

This enabled him to deduce the remarkable result

which he described as an unexpected and elegant formula.

### The Basel Problem: Euler’s Bravura Performance

Published January 14, 2021 Occasional Leave a CommentTags: Analysis, Euler

The Basel problem was first posed by Pietro Mengoli, a mathematics professor at the University of Bologna, in 1650, the same year in which he showed that the alternating harmonic series sums to . The Basel problem asks for the sum of the reciprocals of the squares of the natural numbers,

It is not immediately clear that this series converges, but this can be proved without much difficulty, as was first shown by Jakob Bernoulli in 1689. The sum is approximately 1.645 which has no obvious interpretation.

* * * * *

**That’s Maths II: A Ton of Wonders**

by Peter Lynch has just appeared.

Full details and links to suppliers at

http://logicpress.ie/2020-3/

* * * * *

Continue reading ‘The Basel Problem: Euler’s Bravura Performance’

### Will RH be Proved by a Physicist?

Published December 10, 2020 Occasional Leave a CommentTags: Analysis, Number Theory

The Riemann Hypothesis (RH) states that all the non-trivial (non-real) zeros of the zeta function lie on a line, the *critical line*, . By a simple change of variable, we can have them lying on the real axis. But the eigenvalues of any hermitian matrix are real. This led to the *Hilbert-Polya Conjecture*:

The non-trivial zeros of are the

eigenvalues of a hermitian operator.

Is there a *Riemann operator*? What could this operater be? What dynamical system would it represent? Are prime numbers and quantum mechanics linked? Will RH be proved by a physicist?

This last question might make a purest-of-the-pure number theorist squirm. But it is salutary to recall that, of the nine papers that Riemann published during his lifetime, four were on physics!

### Laczkovich Squares the Circle

Published November 26, 2020 Occasional Leave a CommentTags: Analysis, Logic

The phrase `squaring the circle’ generally denotes an impossible task. The original problem was one of three unsolved challenges in Greek geometry, along with trisecting an angle and duplicating a cube. The problem was to construct a square with area equal to that of a given circle, using only straightedge and compass.

### Cornelius Lanczos – Inspired by Hamilton’s Quaternions

Published July 16, 2020 Irish Times Leave a CommentTags: Analysis, Applied Maths, Mechanics, Relativity

In May 1954, Cornelius Lanczos took up a position as senior professor in the School of Theoretical Physics at the Dublin Institute for Advanced Studies (DIAS). The institute had been established in 1940 by Eamon de Valera, with a School of Theoretical Physics and a School of Celtic Studies, reflecting de Valera’s keen interest in mathematics and in the Irish language. Later, a School of Cosmic Physics was added. DIAS remains a significant international centre of research today [TM191 or search for “thatsmaths” at irishtimes.com].

Continue reading ‘Cornelius Lanczos – Inspired by Hamilton’s Quaternions’

### The Online Encyclopedia of Integer Sequences

Published June 25, 2020 Occasional Leave a CommentTags: Analysis, Arithmetic, Number Theory

Suppose that, in the course of an investigation, you stumble upon a string of whole numbers. You are convinced that there must be a pattern, but you cannot find it. All you have to do is to type the string into a database called OEIS — or simply “Slone’s” — and, if the string is recognized, an entire infinite sequence is revealed. If the string belongs to several sequences, several choices are offered. OEIS is a great boon to both professional mathematicians and applied scientists in fields like physics, chemistry, astronomy and biology.

Continue reading ‘The Online Encyclopedia of Integer Sequences’

### Exponential Growth must come to an End

Published April 16, 2020 Irish Times Leave a CommentTags: Analysis, Applied Maths

In its initial stages, the Covid-19 pandemic grew at an exponential rate. What does this mean? The number of infected people in a country is growing exponentially if it increases by a fixed multiple R each day: if N people are infected today, then R times N are infected tomorrow. The size of the growth-rate R determines how rapidly the virus is spreading. An example should make this clear [TM185 or search for “thatsmaths” at irishtimes.com].

### Adjoints of Vector Operators

Published January 23, 2020 Occasional Leave a CommentTags: Algebra, Analysis

We take a fresh look at the vector differential operators grad, div and curl. There are many vector identities relating these. In particular, there are two combinations that always yield zero results:

**Question: Is there a connection between these identities?**

### Grad, Div and Curl on Weather Maps: a Gateway to Vector Analysis

Published January 9, 2020 Occasional Leave a CommentTags: Analysis, Numerical Weather Prediction

Vector analysis can be daunting for students. The theory can appear abstract, and operators like Grad, Div and Curl seem to be introduced without any obvious motivation. Concrete examples can make things easier to understand. Weather maps, easily obtained on the web, provide real-life applications of vector operators.

Continue reading ‘Grad, Div and Curl on Weather Maps: a Gateway to Vector Analysis’

### Divergent Series Yield Valuable Results

Published December 26, 2019 Occasional Leave a CommentTags: Analysis

Mathematicians have traditionally dealt with convergent series and shunned divergent ones. But, long ago, astronomers found that divergent expansions yield valuable results. If these so-called asymptotic expansions are truncated, the error is bounded by the first term omitted. Thus, by stopping just before the smallest term, excellent approximations may be obtained.

### Elliptic Trigonometry: Fun with “sun”, “cun” and “dun”

Published November 14, 2019 Occasional Leave a CommentTags: Analysis, Trigonometry

** Introduction **

The circular functions arise from ratios of lengths in a circle. In a similar manner, the elliptic functions can be defined by means of ratios of lengths in an ellipse. Many of the key properties of the elliptic functions follow from simple geometric properties of the ellipse.

Originally, Carl Gustav Jacobi defined the elliptic functions , , using the integral

He called the *amplitude* and wrote . It can be difficult to understand what motivated his definitions. We will define the elliptic functions , , in a more intuitive way, as simple ratios associated with an ellipse.

Continue reading ‘Elliptic Trigonometry: Fun with “sun”, “cun” and “dun”’

### Some Fundamental Theorems of Maths

Published October 24, 2019 Occasional Leave a CommentTags: Algebra, Analysis, Geometry

*Every branch of mathematics has key results that are so
important that they are dubbed fundamental theorems.*

The customary view of mathematical research is that of establishing the truth of propositions or theorems by rigorous deduction from axioms and definitions. Mathematics is founded upon axioms, basic assumptions that are taken as true. Logical reasoning is then used to deduce the consequences of those axioms with each major result designated as a theorem.

As each new theorem is proved, it provides a basis for the establishment of further results. The most important and fruitful theorem in each area of maths is often named as the *fundamental theorem* of that area. Thus, we have the fundamental theorems of arithmetic, algebra and so on. For example, the fundamental theorem of calculus gives the relationship between differential calculus and integral calculus.

### The Wonders of Complex Analysis

Published October 10, 2019 Occasional Leave a CommentTags: Analysis

If you love mathematics and have never studied complex function theory, then you are missing something wonderful. It is one of the most beautiful branches of maths, with many amazing results. Don’t be put off by the name: complex does not mean complicated. With elementary calculus and a basic knowledge of imaginary numbers, a whole world of wonder is within your grasp.

In the early nineteenth century, Augustin-Louis Cauchy (1789–1857) constructed the foundations of what became a major new branch of mathematics, the theory of functions of a complex variable.

Given a function of a real variable, we often have to find the values of for which the function is zero. A simple iterative method was devised by Isaac Newton and refined by Joseph Raphson. It is known either as Newton’s method or as the Newton-Raphson method. It usually produces highly accurate approximations to the roots of the equation .

### Bernard Bolzano, a Voice Crying in the Wilderness

Published June 13, 2019 Occasional Leave a CommentTags: Analysis, History

Bernard Bolzano, born in Prague in 1781, was a Bohemian mathematician with Italian origins. Bolzano made several profound advances in mathematics that were not well publicized. As a result, his mathematical work was overlooked, often for many decades after his death. For example, his construction of a function that is continuous on an interval but nowhere differentiable, did not become known. Thus, the credit still goes to Karl Weierstrass, who found such a function about 30 years later. Boyer and Merzbach described Bolzano as “a voice crying in the wilderness,” since so many of his results had to be rediscovered by other workers.

Continue reading ‘Bernard Bolzano, a Voice Crying in the Wilderness’

### Really, 0.999999… is equal to 1. Surreally, this is not so!

Published January 10, 2019 Occasional Leave a CommentTags: Analysis, Number Theory

The value of the recurring decimal 0.999999 … is a popular topic of conversation amongst amateur mathematicians of various levels of knowledge and expertise. Some of the discussions on the web are of little value or interest, but the topic touches on several subtle and deep aspects of number theory.

Continue reading ‘Really, 0.999999… is equal to 1. Surreally, this is not so!’

### Grandi’s Series: A Second Look

Published July 26, 2018 Occasional Leave a CommentTags: Analysis, History

In an earlier post, we discussed Grandi’s series, originally studied by the Italian monk Dom Guido Grandi around 1703. It is the series

This is a divergent series: the sequence of partial sums is , which obviously does not converge, but alternates between and .

### Grandi’s Series: Divergent but Summable

Published July 12, 2018 Occasional 1 CommentTags: Analysis, History, Puzzles

** Is the Light On or Off? **

Suppose a light is switched on for a half-minute, off for a quarter minute, on for one eighth of a minute and so on until precisely one minute has elapsed. Is the light on or off at the end of this (infinite) process? Representing the two states “on” and “off” by and , the sequence of states over the first minute is . But how do we ascertain the final state from this sequence? This question is sometimes known as Thomson’s Lamp Puzzle.

### Fourier’s Wonderful Idea – II

Published April 5, 2018 Irish Times Leave a CommentTags: Analysis, History

**Solving PDEs by a Roundabout Route**

Joseph Fourier, born just 250 years ago, introduced a wonderful idea that revolutionized science and mathematics: any function or signal can be broken down into simple periodic sine-waves. Radio waves, micro-waves, infra-red radiation, visible light, ultraviolet light, X-rays and gamma rays are all forms of electromagnetic radiation, differing only in frequency [TM136 or search for “thatsmaths” at irishtimes.com].

### Fourier’s Wonderful Idea – I

Published March 29, 2018 Occasional Leave a CommentTags: Analysis, History

**Breaking Complex Objects into Simple Pieces**

“In a memorable session of the French Academy on the

21st of December 1807, the mathematician and engineer

Joseph Fourier announced a thesis which inaugurated a

new chapter in the history of mathematics. The claim of

Fourier appeared to the older members of the Academy,

including the great analyst Lagrange, entirely incredible.”

** Introduction **

The above words open the *Discourse on Fourier Series*, written by Cornelius Lanczos. What greatly surprised and shocked Lagrange and the other academicians was the claim of Fourier that an arbitrary function, defined by an arbitrarily capricious graph, can always be resolved into a sum of pure sine and cosine functions. There was good reason to question Fourier’s theorem. Since sine functions are continuous and infinitely differentiable, it was assumed that any superposition of such functions would have the same properties. How could this assumption be reconciled with Fourier’s claim?

### Subtract 0 and divide by 1

Published March 8, 2018 Occasional Leave a CommentTags: Analysis, History

We all know that division by zero is a prohibited operation, and that ratios that reduce to “zero divided by zero” are indeterminate. We probably also recall proving in elementary calculus class that

This is an essential step in deriving an expression for the derivative of .

### The Evolute: Envelope of Normals

Published February 22, 2018 Occasional Leave a CommentTags: Analysis, Geometry

Every curve in the plane has several other curves associated with it. One of the most interesting and important of these is the evolute.

### Torricelli’s Trumpet & the Painter’s Paradox

Published April 13, 2017 Occasional 1 CommentTags: Analysis, Geometry, Recreational Maths

Evangelista Torricelli, a student of Galileo, is remembered as the inventor of the barometer. He was also a talented mathematician and he discovered the remarkable properties of a simple geometric surface, now often called *Torricelli’s Trumpet*. It is the surface generated when the curve for is rotated in 3-space about the x-axis.

Continue reading ‘Torricelli’s Trumpet & the Painter’s Paradox’

### Enigmas of Infinity

Published March 2, 2017 Irish Times Leave a CommentTags: Analysis, History, Logic

Children sometimes amuse themselves searching for the biggest number. After trying millions, billions and trillions, they realize that there is no end to the game: however big a number may be, we can always add 1 to produce a bigger number: the set of counting numbers is infinite. The concept of infinity has intrigued philosophers since antiquity, and it leads to many surprises and paradoxical results [TM110 or search for “thatsmaths” at irishtimes.com].

### Taylor Expansions from India

Published November 24, 2016 Occasional Leave a CommentTags: Analysis, History

The English mathematician Brook Taylor (1685-1731) introduced the calculus of finite differences in his *Methodus Incrementorum Directa et Inversa*, published in 1715. This work contained the famous formula known today as Taylor’s formula. In 1772, Lagrange described it as “the main foundation of differential calculus” (Wikipedia: Brook Taylor). Taylor also wrote a treatise on linear perspective (see Fig. 1).

It is noteworthy that the series for , and were known to mathematicians in India about 400 years before Taylor’s time.

Continue reading ‘Taylor Expansions from India’

### Which is larger, e^pi or pi^e?

Published November 10, 2016 Occasional Leave a CommentTags: Analysis

Which is greater, or ? Of course, it depends on the values of x and y. We might consider a particular case: Is or ?

It is well-known that an ellipse is the locus of all points such that the sum of their distances from two fixed points, the foci, is constant. Thus, a gardener may map out an elliptical flower-bed by driving two stakes into the ground, looping a rope around them and pulling it taut with a pointed stick, tracing out a curve on the ground.

### Random Harmonic Series

Published July 28, 2016 Occasional Leave a CommentTags: Analysis, Number Theory, Probability

We consider the convergence of the random harmonic series

where is chosen randomly with probability of being either plus one or minus one. It follows from the Kolmogorov three-series theorem that the series is “almost surely” convergent.

### Sigmoid Functions: Gudermannian and Gompertz Curves

Published April 28, 2016 Occasional Leave a CommentTags: Analysis, modelling

The Gudermannian is named after Christoph Gudermann (1798–1852). The Gompertz function is named after Benjamin Gompertz (1779–1865). These are two amongst several *sigmoid functions*. Sigmoid functions find applications in many areas, including population dynamics, artificial neural networks, cartography, control systems and probability theory. We will look at several examples in this class of functions.

Continue reading ‘Sigmoid Functions: Gudermannian and Gompertz Curves’

### The Power Tower Fractal

Published April 14, 2016 Occasional Leave a CommentTags: Analysis, Fractals

We can construct a beautiful fractal set by defining an operation of iterating exponentials and applying it to the numbers in the complex plane. The operation is tetration and the fractal is called the power tower fractal or sometimes the tetration fractal. A detail of the set is shown in the figure here.

### The Imaginary Power Tower: Part II

Published March 31, 2016 Occasional Leave a CommentTags: Analysis, Recreational Maths

This is a continuation of last week’s post: LINK

The complex power tower is defined by an `infinite tower’ of exponents:

The sequence of successive approximations to this function is

If the sequence converges it is easy to solve numerically for a given .

In Part I we described an attempt to fit a logarithmic spiral to the sequence . While the points of the sequence were close to such a curve they did not lie exactly upon it. Therefore, we now examine the asymptotic behaviour of the sequence for large .

### The Imaginary Power Tower: Part I

Published March 24, 2016 Occasional Leave a CommentTags: Analysis, Recreational Maths

The function defined by an `infinite tower’ of exponents,

is called the *Power Tower function*. We consider the sequence of successive approximations to this function:

As , the sequence converges for . This result was first proved by Euler. For an earlier post on the power tower, click here.

### Vanishing Zigzags of Unbounded Length

Published February 18, 2016 Occasional Leave a CommentTags: Analysis

We will construct a sequence of functions on the unit interval such that it converges uniformly to zero while the arc-lengths diverge to infinity.

### Twin Peaks Entropy

Published January 14, 2016 Occasional Leave a CommentTags: Analysis, Probability

Next week there will be a post on tuning pianos using a method based on entropy. In preparation for that, we consider here how the entropy of a probability distribution function with twin peaks changes with the separation between the peaks.

### Squaring the Circular Functions

Published December 31, 2015 Occasional Leave a CommentTags: Analysis

The circular functions occur throughout mathematics. Fourier showed that, under very general assumptions, an arbitrary function can be decomposed into components each of which is a circular function. The functions get their name from their use in defining a circle in parametric form: if

then , the usual equation for a circle in Cartesian coordinates. In the figure, we plot the familiar sinusoid, which has a period of .

**Sine Function:**

The function is beautifully behaved, oscillating regularly along the entire real line (it is also well-behaved for complex but we won’t consider that here).

### Which Way did the Bicycle Go?

Published October 8, 2015 Occasional Leave a CommentTags: Analysis, Geometry, Mechanics

*“A bicycle, certainly, but not the bicycle,” said Holmes.*

In Conan-Doyle’s short story *The Adventure of the Priory School* Sherlock Holmes solved a mystery by deducing the direction of travel of a bicycle. His logic has been minutely examined in many studies, and it seems that in this case his reasoning fell below its normal level of brilliance.

Tap any number into your calculator. Yes, any number at all, plus or minus, big or small. Now tap the cosine button. You will get a number in the range [ -1, +1 ]. Now tap “cos” again and again, and keep tapping it repeatedly (make sure that angles are set to radians and not degrees). The result is a sequence of numbers that converge towards the value** 0.739085 …** .

### Café Mathematics in Lvov

Published March 5, 2015 Irish Times Leave a CommentTags: Analysis, History

For 150 years the city of Lvov was part of the Austro-Hungarian Empire. After Polish independence following World War I, research blossomed and between 1920 and 1940 a sparkling constellation of mathematicians flourished in Lvov [see this week’s *That’s Maths* column in *The Irish Times* (TM063, or search for “thatsmaths” at irishtimes.com).

### The Birth of Functional Analysis

Published February 26, 2015 Occasional Leave a CommentTags: Analysis, History, Set Theory

Stefan Banach (1892–1945) was amongst the most influential mathematicians of the twentieth century and the greatest that Poland has produced. Born in Krakow, he studied in Lvov, graduating in 1914 just before the outbreak of World War I. He returned to Krakow where, by chance, he met another mathematician, Hugo Steinhaus who was already well-known. Together they founded what would, in 1920, become the Polish Mathematical Society.

### New Curves for Old: Inversion

Published December 11, 2014 Occasional Leave a CommentTags: Analysis, Geometry, Maps

**Special Curves**

A large number of curves, called *special curves*, have been studied by mathematicians. A curve is the path traced out by a point moving in space. To keep things simple, we assume that the point is confined to two-dimensional Euclidean space so that it generates a plane curve as it moves. This, a curve results from a mapping . Continue reading ‘New Curves for Old: Inversion’

### Curves with Singularities

Published September 25, 2014 Occasional Leave a CommentTags: Analysis, Geometry

Many of the curves that we study are smooth, with a well-defined tangent at every point. Points where the derivative is defined — where there is a definite slope — are called regular points. However, many curves also have exceptional points, called singularities. If the derivative is not defined at a point, or if it does not have a unique value there, the point is singular.

Generally, if we zoom in close to a point on a curve, the curve looks increasingly like a straight line. However, at a singularity, it may look like two lines crossing or like two lines whose slopes converge as the resolution increases. Continue reading ‘Curves with Singularities’

### Invention or Discovery?

Published July 24, 2014 Occasional Leave a CommentTags: Analysis, Logic, Number Theory, Social attitudes

Is mathematics invented or discovered? As many great mathematicians have considered this question without fully resolving it, there is little likelihood that I can provide a complete answer here. But let me pose a possible answer in the form of a conjecture:

** Conjecture:*** Definitions are invented. Theorems are discovered.*

The goal is to prove this conjecture, or to refute it. Below, some arguments in support of the conjecture are presented. Continue reading ‘Invention or Discovery?’

### Predator-Prey Models

Published May 8, 2014 Occasional Leave a CommentTags: Analysis, Applied Maths, biology

Next week’s post will be about a model of the future of civilization! It is based on the classical predator-prey model, which is reviewed here.

### The Unity of Mathematics

Published March 20, 2014 Irish Times Leave a CommentTags: Algebra, Analysis, Geometry, Number Theory

This week, *That’s Maths* in *The Irish Times* ( TM041 ) is about an ambitious program to unify mathematics.

Mathematics expands! Results once proven to be true remain forever true. They are not displaced by subsequent results, but absorbed in an ever-growing theoretical web. Thus, it is increasingly difficult for any individual mathematician to have a comprehensive understanding of even a single field of mathematics: the web of knowledge grows so fast that no-one can master it all.

### The Langlands Program

Published March 13, 2014 Occasional Leave a CommentTags: Algebra, Analysis, Arithmetic, Group Theory, Number Theory

An ambitious programme to unify disparate areas of mathematics was set out some fifty years ago by Robert Langlands of the Institute for Advanced Study in Princeton. The “Langlands Program” (LP) is a set of deep conjectures that attempt to build bridges between certain algebraic and analytical objects.