This is a part of a series of posts which will deal with different types of counterexamples in topology and algebra.
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.
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
There is an interesting correspondence among the quadratic form of a symmetric linear transformation on a real Euclidean space,the extreme values of the sphere and the eigenvectors of
Let be the quadratic form associated with a symmetric linar transformation which maps into itself,then the set of elements in V satisfying is called the unit sphere of
There isn’t too much going in this question.Nonetheless,I just liked it for some bloody reason.
- Let and be positive integers where .If is prime,prove that is even and n is of the form where
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
This is the post excerpt.
The irrationality of e can be proved using the infinite series expansion.
Assume that e is rational.i.e where and .