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, an.

Another important application of power series is to extend functions to different "domains". We typically learn how to evaluate functions like cos(x) or 3x for x any real number in secondary school. But suppose x is a complex number or a matrix. What would cos(x) or 3x 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.