Posts Tagged 'Analysis'

Grandi’s Series: A Second Look

Grandis-Series
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.

Grandis-Series

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

Fourier’s Wonderful Idea – II

Solving PDEs by a Roundabout Route

Fourier-3

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 irishtimes.com].

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.”

Introduction

Fourier

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}.

LHopital-Bernoulli

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.

SinEvolute

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

Continue reading ‘The Evolute: Envelope of Normals’

Torricelli’s Trumpet & the Painter’s Paradox

 

Torricelli-03

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’


Last 50 Posts

Categories