Posts Tagged 'Analysis'

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.
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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.

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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.

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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.


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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.


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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.

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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}.

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