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’
, 
, 
using the integral
the amplitude and wrote 
. It can be difficult to understand what motivated his definitions. We will define the elliptic functions 
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 
.
, which obviously does not converge, but alternates between 
and 
.
.
for 
is rotated in 3-space about the x-axis.
and 
were known to mathematicians in India about 400 years before Taylor’s time.
or 
? Of course, it depends on the values of x and y. We might consider a particular case: Is 
or 
?
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.
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 sequence 
converges for 
. This result was first proved by Euler. For an earlier post on the power tower, click 
, the usual equation for a circle in Cartesian coordinates. In the figure, we plot the familiar sinusoid, which has a period of 
.
is beautifully behaved, oscillating regularly along the entire real line 
(it is also well-behaved for complex 
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}](https://s0.wp.com/latex.php?latex=%7B%5Cmathbf%7B%5Cgamma%7D+%3A+%5Ba%2Cb%5D%5Clongrightarrow+%5Cmathbb%7BR%7D%5E2%7D&bg=ffffff&fg=000000&s=0 "{\mathbf{\gamma} : [a,b]\longrightarrow \mathbb{R}^2}")
. 
this must blow up. Surprisingly, it has finite values for a range of x>1. 
. We call this function the power tower function. 