Idea behind the series expansions

[torehan]

Hi,
In general, for most of the physical phenomenon, somewhere in the analytical procedure while developing the theory of the phenomenon, series expansions used (generally Taylor expansion) which is a key point of the analytical steps.

My question is why we should use such expansions I mean what is the general purpose of the these expansions?

Torehan

 

[HallsofIvy]

Polynomials, and their extension, power series, are the easiest kind of functions!

For example, to solve the differential equation, y''+ xy= 0, I can think of y as a power series, $y= \sum a_nx^n$, and differentiate "term by term": $y'= \sum na_n x^{n-1}$ and $y''= \sum n(n-1)x^{n-2}$.

Putting those into the differential equation,
$$\sum_{n= 2}^\infty n(n-1)a_n x^{n-2}+ \sum_{n=0}^\infty a_nx^{n+1}= 0$$
If we change the index in the first sum to i= n-2 it becomes
$$\sum_{i=0}^\infty (i+2)(i+1)a_{i+2}x^i$$
Changing the index in the second sum to i= n+ 1 we get
$$\sum_{i=1}^\infty a_{i- 1} a_{i- 1}x^i$$
The equation becomes
$$2a_0+ \sum_{i=1}^\infty \left[(i+2)(i+1)a_{i+ 2}+ a_{i-1}\right]x^i= 0$$
A polynomial or power series is equal to 0 for all x if and only if every coefficient is equal to 0. Setting each coefficient of that power series equal to 0 allows us to derive equations for the original coefficients, a_n.

Another important application of power series is to extend functions to different "domains". We typically learn how to evaluate functions like cos(x) or 3^x for x any real number in secondary school. But suppose x is a complex number or a matrix. What would cos(x) or 3^x be then? Answer- right the functions as power series in x. We know how to add and multiply complex numbers and matrices so we can evaluate such power series.