Using Legendre Polynomials in irrationality proofs and establishing irrationality measures

The Legendre polynomials are a set of polynomial solutions to the Legendre’s Differential equation:


The edifice of most irrationality arguments use the fact the repeated integration by parts can be used on generating functions involving the polynomial.Another useful property is that these polynomials have integer coefficients.

Now,I’ll present a few important examples which display this property.

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Inequalities related to the Gamma Function and related functions

There are numerous inequalities related to the classical gamma function(\Gamma (x)) ,incomplete gamma function(\Gamma (a;x)) and the psi function(also called the di-gamma function which is simply the derivative of the logarithm of the gamma function).

One of the simplest bounds for the function \Gamma (1+a) where 0 \leq a \leq 1 is:

2^{a-1} \leq \Gamma (1+a) \leq 1 .

A stronger inequality can be given by

e^{(x-1)(1- \gamma)} \leq \Gamma(1+a) \leq 1 where \gamma is the Euler-Mascheroni constant.


with equality on the lower bound iff x=1

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Connection between a symmetric linear transformation and the unit sphere


There is an interesting correspondence among the quadratic form of a symmetric linear transformation T:V \mapsto Von a real Euclidean space,the extreme values of the sphere and the eigenvectors of T

Let Q(x)=(T(x),x) be the quadratic form associated with a symmetric linar transformation which maps V into itself,then the set of elements u in V satisfying \langle u,u  \rangle=1 is called the unit sphere of V

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The logarithmic spiral

The logarithmic spiral has some very interesting properties and Bernoulli was especially fascinated by it.I’ll prove it’s most important property(the angle between the curve and the radius at every angle is constant) and proceed with an example.

In polar co-ordinates,the equation of the spiral is given by:

r(\theta)=ae^{k\theta} where a,k are constants and a>0

Now,to prove that any line from the origin which intersects the curve does so by making a constant angle(say \phi) with the curve(direction of tangent line),we consider the derivatives of the parameter equations which correspond to r(\theta)

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