If f and g are n-times differentiable functions, then :
Now, we would like to find out a generalized expression for the n-th derivative of fg. In order to arrive at that formulation lets calculate a few derivatives to see whether we can find a pattern:
You must have noticed a pattern in the above expressions. The coefficients seem are the one in the binomial expansion of
Therefore we can write the expression for the n-derivative of fg as the following expression:
where (i) means to differentiate i-times.
This is also known as Leibniz Formula.
** This plays an important role when we start discussing about the Associated Legendre Differential Equation.
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.
Next post: What do these coefficients mean ?