Consider the expression
$$E(x,y)=\frac{x\hat{i}+(y-a)\hat{j}}{(x^2+(y-a)^2)^{3/2}}\tag{1}$$
where $r^2=x^2+y^2$. I'd like to investigate what happens when $r>>a$.
Let's consider the factor involving just the denominator
$$(x^2+(y-a)^2)^{-3/2}$$
$$=(x^2+y^2-2ay+a^2)^{-3/2}$$
$$=\left [ (x^2+y^2)(1+\frac{a^2-2ay}{x^2+y^2})\right ]^{-3/2}$$
$$=r^{-3}\left ( 1+ \frac{a^2-2ay}{r^2} \right )^{-3/2}\tag{2}$$
Let $s(r)=\frac{a^2-2ay}{r^2}$ and let $f(x)=(1+x)^{-3/2}$.
Then $f(s(r))=\left ( 1+ \frac{a^2-2ay}{r^2} \right )^{-3/2}\tag{3}$
Since $s\to 0$ as $r\to \infty$, apparently we can use a Taylor approximation to $f(x)$ near $0$ to find out what happens to (3) as $r\to\infty$.
$$P_{1,f,0}=1-\frac{3}{2}s\tag{4}$$
This is a linear approximation to $f$ near $x=0$. When $r\to\infty$ and $s\to 0$, $f(s(r))\approx 1-\frac{3}{2}s$.
Which means that when $r\to\infty$ the expression in (2) is approximately
$$r^{-3}(1-\frac{3}{2}s)$$
$$=r^{-3}\left (1-\frac{3}{2}\frac{a^2-2ay}{r^2}\right )\tag{5}$$
which we can sub into (1) to reach the desired expression for $E$ when $r>>a$.
What is the theorem or theorems that justify this type of calculation when we have a function composition?
What do I have to study to learn more about this specific topic of using Taylor approximations like they do in physics?
I recall from studying Taylor polynomials and Taylor's Theorem a formula for computing the Taylor polynomial of a function composition. The expression in (3) is a function composition.
If we apply the formula
$$P_{n,a,f\circ s}=\left [ P_{n,s(m),f}\circ P_{n,m,s} \right ]_n$$
where the $[]$ operator means truncation of terms that are of degree $\leq n$, then we reach the second-to-last result seen in the following Maple session (which includes all the calculations mentioned above):
Equation (12) above is the first-order Taylor polynomial of the function composition in (3). But it seems like this result isn't as useful as the previous one obtained in (5).
The variable $m$ in (12) represents the value about which we computed the Taylor polynomial of the function composition.
Is there some useful way to relate this calculation with the one that led to (5)?