But in Jackson’s Classical Electrodynamics, III edition he notes the following:
This is an interesting form of the Laplacian that perhaps not everyone has encountered. This can obtained from the known form by making the substitution and simplifying. The steps to which have been outlined below:
Say what? This one blew my mind when I first encountered it. But it turns out Euler was the one who came up with it and it’s proof is just beautiful!
Say you have a quadratic equation whose roots are , then you can write as follows:
As for as this proof is concerned we are only worried about the coefficient of , which you can prove that for a n-degree polynomial is:
where are the n-roots of the polynomial.
Now begins the proof
It was known to Euler that
But this could also be written in terms of the roots of the equation as:
Now what are the roots of ?. Well, when i.e *
The roots of the equation are
Comparing the coefficient of y on both sides of the equation we get that:
* n=0 is not a root since
at y = 0
** Now if all that made sense but you are still thinking : Why on earth did Euler use this particular form of the polynomial for this problem, read the first three pages of this article. (It has to do with convergence)
One is commonly asked to prove in college as part of a linear algebra problem set that matrix multiplication is not commutative. i.e If A and B are two matrices then :
But without getting into the Algebra part of it, why should this even be true ? Let’s use linear transformations to get a feel for it.
If A and B are two Linear Transformations namely Rotation and Shear. Then it means that.
Is that true? Well, lets perform these linear operations on a unit square and find out:
You can clearly see that the resultant shape is not the same upon the two transformations. This means that the order of matrix multiplication matters a lot ! ( or matrix multiplication is not commutative.)
In this series of posts about Legendre differential equation, I would like to de-construct the differential equation down to the very bones. The motivation for this series is to put all that I know about the LDE in one place and also maybe help someone as a result.
The Legendre differential equation is the following:
We will find solutions for this differential equation using the power series expansion i.e
We will plug in these expressions for the derivatives into the differential equation.
** Note: Begin
Let’s take .
As n -> . , -> .
As n -> , -> .
Again performing a change of variables from to n.
** Note: End
(iii) can now be written as follows.
x = 0 is a trivial solution and therefore we get the indicial equation:
We get the following recursion relation on the coefficients of the power series expansion.