Applying a Taylor series "with respect to $a/r$" and "around $0$"

Question

My question is what does it mean

applying a Taylor series with respect to something and around a point.

What is the difference?

Please explain it with the following example:

Apply a Taylor series expansion to $r'$ with respect to $a/r$ around $0$. Only until the squared terms (inclusive).

Where $$r'^2 = a^2 + r^2 +2ar\sin(\theta)\cos(\alpha-\phi)$$

Answer 1

For an easier example, let $f(x) = \sin(x)$. The Taylor series expansion of (or, I guess, in your above language 'to') $f$ with respect to $x$ around $0$ is the familar $$ f(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \dots$$

So you want to rewrite $r'$ in terms of $a/r$ and expand around 0.

Answer 2

The "with respect to" specifies the variable whose powers appear in the series. "around" is the point near which you want an approximation. So your answer will look like $$ c_o + c_1(a/r - 0) + c_2(a/r - 0)^2 + \text{ higher order terms}. $$

(Of course the zeroes go away. I just put them in to illustrate "around $0$".)