Posts Tagged 'Analysis'

The Online Encyclopedia of Integer Sequences

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.


The Home page of OEIS:

Continue reading ‘The Online Encyclopedia of Integer Sequences’

Exponential Growth must come to an End

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].


“Flattening the curve” [image from ECDC].

Continue reading ‘Exponential Growth must come to an End’

Adjoints of Vector Operators

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:

\displaystyle \begin{array}{rcl} \mathbf{curl}\ \mathbf{grad}\ \chi &\equiv& 0\,, \quad \mbox{for all scalar functions\ }\chi \\ \mathrm{div}\ \mathbf{curl}\ \boldsymbol{\psi} &\equiv& 0\,, \quad \mbox{for all vector functions\ }\boldsymbol{\psi} \end{array}

Question: Is there a connection between these identities?


Continue reading ‘Adjoints of Vector Operators’

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

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.


Fig. 1. An idealized scalar field representing the mean sea-level atmospheric pressure over the North Atlantic area.

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

Divergent Series Yield Valuable Results

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.

Continue reading ‘Divergent Series Yield Valuable Results’

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


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 {\mathop\mathrm{sn} u}, {\mathop\mathrm{cn} u}, {\mathop\mathrm{dn} u} using the integral

\displaystyle u = \int_0^{\phi} \frac{\mathrm{d}\phi}{\sqrt{1-k^2\sin^2\phi}} \,.

He called {\phi} the amplitude and wrote {\phi = \mathop\mathrm{am} u}. It can be difficult to understand what motivated his definitions. We will define the elliptic functions {\mathop\mathrm{sn} u}, {\mathop\mathrm{cn} u}, {\mathop\mathrm{dn} u} 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

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.

Continue reading ‘Some Fundamental Theorems of Maths’

The Wonders of Complex Analysis


Augustin-Louis Cauchy (1789–1857)

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.

Continue reading ‘The Wonders of Complex Analysis’

Zeroing in on Zeros

Given a function {f(x)} of a real variable, we often have to find the values of {x} 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 {f(x) = 0}.


A rational function with five real zeros and a pole at x = 1.

Continue reading ‘Zeroing in on Zeros’

Bernard Bolzano, a Voice Crying in the Wilderness


Bernard Bolzano (1781-1848)

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!

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.


[Image Wikimedia Commons]

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

Grandi’s Series: A Second Look

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

\displaystyle G = 1 - 1 + 1 - 1 + 1 - 1 + \dots

This is a divergent series: the sequence of partial sums is {\{ 1, 0, 1, 0, 1, 0, \dots \}}, which obviously does not converge, but alternates between {0} and {1}.

Continue reading ‘Grandi’s Series: A Second Look’

Grandi’s Series: Divergent but Summable

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 {1} and {0}, the sequence of states over the first minute is {\{ 1, 0, 1, 0, 1, 0, \dots \}}. But how do we ascertain the final state from this sequence? This question is sometimes known as Thomson’s Lamp Puzzle.


Continue reading ‘Grandi’s Series: Divergent but Summable’

Fourier’s Wonderful Idea – II

Solving PDEs by a Roundabout Route


Joseph Fourier (1768-1830)

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].

Continue reading ‘Fourier’s Wonderful Idea – II’

Fourier’s Wonderful Idea – I

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.”



Joseph Fourier (1768-1830)

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?

Continue reading ‘Fourier’s Wonderful Idea – I’

Subtract 0 and divide by 1

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

\displaystyle \lim_{x\rightarrow 0} \frac{\sin x}{x} = 1

This is an essential step in deriving an expression for the derivative of {\sin x}.


Continue reading ‘Subtract 0 and divide by 1’

The Evolute: Envelope of Normals

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


Sin t (blue) and its evolute (red).

Continue reading ‘The Evolute: Envelope of Normals’

Torricelli’s Trumpet & the Painter’s Paradox



Torricelli’s Trumpet


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 {y=1/x} for {x\ge1} is rotated in 3-space about the x-axis.

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

Enigmas of Infinity

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]. 


Continue reading ‘Enigmas of Infinity’

Taylor Expansions from India


NPG 1920; Brook Taylor probably by Louis Goupy

FIg. 1: Brook Taylor (1685-1731). Image from NPG.

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 {\sin x}, {\cos x} and {\arctan x} 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?

Which is greater, {x^y} or {y^x}? Of course, it depends on the values of x and y. We might consider a particular case: Is {e^\pi > \pi^e} or {\pi^e > e^\pi}?


Contour plot of x^y – y^x, positive in the yellow regions, negative in the blue ones.

Continue reading ‘Which is larger, e^pi or pi^e?’

Slicing Doughnuts


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.

Continue reading ‘Slicing Doughnuts’

Random Harmonic Series

We consider the convergence of the random harmonic series

\displaystyle R = \sum_{n=1}^{\infty}\frac{\sigma_{n}}{n}

where {\sigma_n\in\{-1,+1\}} is chosen randomly with probability {1/2} of being either plus one or minus one. It follows from the Kolmogorov three-series theorem that the series is “almost surely” convergent.


Continue reading ‘Random Harmonic Series’

Sigmoid Functions: Gudermannian and Gompertz Curves

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

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.


Detail of the power tower fractal.

Continue reading ‘The Power Tower Fractal’

The Imaginary Power Tower: Part II

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

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

\displaystyle Z(z) = {z^{z^{z^{.^{.^{.}}}}}} \,.

The sequence of successive approximations to this function is

z_0 = 1 \qquad z_{1} = z \qquad z_{2} = z^{z} \qquad \dots \qquad z_{n+1} = z^{z_n} \qquad \dots

If the sequence {\{z_n(z)\}} converges it is easy to solve numerically for a given {z }.

Pursuit-triangleIn Part I we described an attempt to fit a logarithmic spiral to the sequence {\{z_n(i)\}}. 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 {n}.

Continue reading ‘The Imaginary Power Tower: Part II’

The Imaginary Power Tower: Part I

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

\displaystyle y(x) = {x^{x^{x^{.^{.^{.}}}}}}

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

\displaystyle y_0 = 1 \qquad y_1 = x \qquad \dots \qquad y_{n+1} = x^{y_n} \,.

As {n\rightarrow\infty}, the sequence {\{y_n\}} converges for {e^{-e}<x<e^{1/e}}. This result was first proved by Euler. For an earlier post on the power tower, click here.

Continue reading ‘The Imaginary Power Tower: Part I’

Vanishing Zigzags of Unbounded Length

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.


Black: Frog hop. Blue: Cricket hops. Magenta: Flea hops.

Continue reading ‘Vanishing Zigzags of Unbounded Length’

Twin Peaks Entropy

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.

Continue reading ‘Twin Peaks Entropy’

Squaring the Circular Functions

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

\displaystyle x = a\cos t \qquad\mbox{and}\qquad y = a\sin t

then {x^2 + y^2 = a^2}, the usual equation for a circle in Cartesian coordinates. In the figure, we plot the familiar sinusoid, which has a period of {2\pi}.

Wild-Functions-01 Continue reading ‘Squaring the Circular Functions’

A Few Wild Functions

Sine Function: {\mathbf{y=\sin x}}

The function {y=\sin x} is beautifully behaved, oscillating regularly along the entire real line {\mathbb{R}} (it is also well-behaved for complex {x} but we won’t consider that here).

The sine function, the essence of good behaviour.

The sine function, the essence of good behaviour.

Continue reading ‘A Few Wild Functions’

Which Way did the Bicycle Go?

“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.

As front wheel moves along the positive {x}-axis the back wheel, initially at {(0,a)}, follows a tractrix curve.

As front wheel moves along the positive x-axis the back wheel, initially at (0,a), follows a tractrix curve (see below).

Continue reading ‘Which Way did the Bicycle Go?’

Tap-tap-tap the Cosine Button

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 … .

CosCosCos Continue reading ‘Tap-tap-tap the Cosine Button’

Café Mathematics in Lvov

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

The Scottish Café, Lvov in earlier times (left), now Hotel Atlas in Lviv.(image Wikimedia Commons).

The Scottish Café, Lvov in earlier times (left), now Hotel Atlas in Lviv (right).

Continue reading ‘Café Mathematics in Lvov’

The Birth of Functional Analysis

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.

A coin and a postage stamp commemorating Stefan Banach.

A coin and a postage stamp commemorating Stefan Banach.

Continue reading ‘The Birth of Functional Analysis’

New Curves for Old: Inversion

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 {\mathbb{R}^2} so that it generates a plane curve as it moves. This, a curve results from a mapping {\mathbf{\gamma} : [a,b]\longrightarrow \mathbb{R}^2}. Continue reading ‘New Curves for Old: Inversion’

Curves with Singularities

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.

Slinky traces a smooth helical curve in three dimensions.

Slinky traces a smooth helical curve in three dimensions.

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?

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

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.

Solution for X(T) and Y(T) for 30 time units. X(0)=0.5, Y(0)=0.2 and k=0.5.

Solution for X (blue) and Y (red) for 30 time units. X(0)=0.5, Y(0)=0.2 and k=0.5.

Continue reading ‘Predator-Prey Models’

The Unity of Mathematics

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.

Continue reading ‘The Unity of Mathematics’

The Langlands Program

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.

Canadian mathematician Robert Langlands, who formulated a series of far-reaching conjectures.

Canadian mathematician Robert Langlands, who formulated a series of far-reaching conjectures [image from Wikimedia Commons].

Continue reading ‘The Langlands Program’

The Prime Number Theorem

God may not play dice with the Universe, but something strange is going on with the prime numbers  [Paul Erdös, paraphrasing Albert Einstein]

The prime numbers are the atoms of the natural number system. We recall that a prime number is a natural number greater than one that cannot be broken into smaller factors. Every natural number greater than one can be expressed in a unique way as a product of primes. Continue reading ‘The Prime Number Theorem’

A Mathematical Dynasty

The idea that genius runs in families is supported by many examples in the arts and sciences. One striking case is the family of Johann Sebastian Bach, the most brilliant star in a constellation of talented musicians and composers.

In a similar vein, several generations of the Bernoulli family excelled in science and medicine. More that ten members of this Swiss family, over four generations, had distinguished careers in mathematics.

Continue reading ‘A Mathematical Dynasty’

Sonya Kovalevskaya

A brilliant Russian mathematician, Sonya Kovalevskaya, is the topic of the That’s Maths column this week (click Irish Times: TM029 and search for “thatsmaths”).

In the nineteenth century it was extremely difficult for a woman to achieve distinction in the academic sphere, and virtually impossible in the field of mathematics. But a few brilliant women managed to overcome all the obstacles and prejudice and reach the very top. The most notable of these was the remarkable Russian, Sonya Kovalevskya.

Continue reading ‘Sonya Kovalevskaya’

Ternary Variations

Georg Cantor (1845-1918) was led, through his study of trigonometric series, to distinguish between denumerably infinite sets like the rationals and uncountable sets like the reals. He introduced a set that is an abstract form of what we now call Cantor’s Ternary Set. In fact, the ternary set had been studied some ten years earlier by the Dublin-born mathematician Henry Smith and, independently, by the Italian Vito Volterra. General sets of this form are now called Smith-Volterra-Cantor sets (SVC sets).
Continue reading ‘Ternary Variations’

The Lambert W-Function

Follow on twitter: @thatsmaths

In a recent post ( The Power Tower ) we described a function defined by iterated exponentiation:

\displaystyle y(x) = {x^{x^{x^{.^{.^{.}}}}}}

It would seem that when {x>1} this must blow up. Surprisingly, it has finite values for a range of x>1. Continue reading ‘The Lambert W-Function’

The Power Tower

Look at the function defined by an `infinite tower’ of exponents:

\displaystyle y(x) = {x^{x^{x^{.^{.^{.}}}}}}

It would seem that for x>1 this must blow up. But, amazingly, this is not so.

In fact, the function has finite values for positive x up to {x=\exp(1/e)\approx 1.445}. We call this function the power tower function. Continue reading ‘The Power Tower’

Sharing a Pint

Four friends, exhausted after a long hike, stagger into a pub to slake their thirst. But, pooling their funds, they have enough money for only one pint.

Annie drinks first, until the surface of the beer is half way down the side (Fig. 1(A)). Then Barry drinks until the surface touches the bottom corner (Fig. 1(B)). Cathy then takes a sup, leaving the level as in Fig. 1(C), with the surface through the centre of the bottom. Finally, Danny empties the glass.

Question: Do all four friends drink the same amount? If not, who gets most and who gets least? Continue reading ‘Sharing a Pint’

The Root of Infinity: It’s Surreal!

Can we make any sense of quantities like “the square root of infinity”? Using the framework of surreal numbers, we can.

  • In Part 1, we develop the background for constructing the surreals.
  • In Part 2, the surreals are assembled and their amazing properties described.

Continue reading ‘The Root of Infinity: It’s Surreal!’

The Popcorn Function

Continuity is not what it seems. In 1875, the German mathematician Carl Johannes Thomae defined a function P(x) with the following extraordinary property:

P(x) is  discontinuous if x is rational

P(x) is continuous if x is irrational.

A graph of this function on the interval (0,1) is shown below. Continue reading ‘The Popcorn Function’

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