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

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  • $\begingroup$ This is a polynomial in $a$, so its Taylor series is fairly straightforward. $\endgroup$
    – eyeballfrog
    Commented Jul 29, 2019 at 21:29
  • $\begingroup$ As far as I know, they mean the same thing. $\endgroup$
    – The Count
    Commented Jul 30, 2019 at 1:22
  • $\begingroup$ @nmasanta: In continuation of our previous conversation, if you're editing, don't forget titles. $\endgroup$
    – Asaf Karagila ♦
    Commented Jul 30, 2019 at 7:05

2 Answers 2

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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 $0$".)

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  • $\begingroup$ thank you @ethan-bolker for the example! $\endgroup$
    – nachofest
    Commented Jul 29, 2019 at 21:35
  • $\begingroup$ by the way, in your answer is it r/a or you meant a/r ? thanks $\endgroup$
    – nachofest
    Commented Jul 29, 2019 at 22:08
  • $\begingroup$ @nachofest Yes,. Fixed thanks. $\endgroup$
    – Ethan Bolker
    Commented Jul 29, 2019 at 22:49
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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.

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  • $\begingroup$ thank you @bounceback for your answer, specially for specifying the idea about rewriting it, that was they key point. $\endgroup$
    – nachofest
    Commented Jul 29, 2019 at 21:35

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