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The irrationality of e can be proved using the infinite series expansion.
Assume that e is rational.i.e where and .
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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.
Continue reading “Using Legendre Polynomials in irrationality proofs and establishing irrationality measures”
There are numerous inequalities related to the classical gamma function() ,incomplete gamma function() 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 where is:
A stronger inequality can be given by
where is the Euler-Mascheroni constant.
with equality on the lower bound iff
Continue reading “Inequalities related to the Gamma Function and related functions”
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:
where are constants and
Now,to prove that any line from the origin which intersects the curve does so by making a constant angle(say ) with the curve(direction of tangent line),we consider the derivatives of the parameter equations which correspond to
Continue reading “The logarithmic spiral”