# Homework Help: Simple Taylor or Multipole Expansion of Potential

1. Jun 17, 2008

### bavaji

Hullo,

Somehow, I couldn't get the TeX to come out right.

I have been trying to learn scheme theory (algebraic geometry) and completely forgotten how to do this simple calculus type stuff...

1. The problem statement, all variables and given/known data

Let V be a potential of the form

$$V = \left(\frac{1}{r} + \left(\frac{1}{\left|\vec{r} - \left(\vec{r_{1}} - vec{r_{2}}\right)\right|} - \left(\frac{1}{\left|\vec{r} + \vec{r_{2}}\right|} - \left(\frac{1}{\left|\vec{r} - \vec{r_{1}}\right|} \right)[\tex] where r = [tex]\left|\vec{r}\right|[\tex]. For large r >> 1 I am supposed to expand in powers of [tex]\frac{\vec{r_{i}}}{r}[\tex] to obtain the expression [tex] V \sim \frac{1}{r^3}\left(x_{1}x_{2} + y_{1}y_{2} - 2z_{1}z_{2} \right)[\tex] where higher order terms have been neglected. 2. Relevant equations 3. The attempt at a solution I tried using the expansion [tex]\left(\frac{1}{\left|\vec{r} + \vec{r'}\right|} = \frac{1}{r} + \frac{\vec{r}\vec{r'}}{r^3}[\tex] + terms of higher order, but somehow end up with V = 0 (up to order [tex]r^-3[\tex]. I have completely forgotten how to do this... Would be thankful for any help... Greetings from Germany, bavaji Last edited: Jun 17, 2008 2. Jun 18, 2008 ### dynamicsolo While you do use "backslashes" for the special TeX characters, you need to terminate the TeX command with regular "slash" [ / tex ]: [tex]V = \left(\frac{1}{r} + \left(\frac{1}{\left|\vec{r} - \left(\vec{r_{1}} - vec{r_{2}}\right)\right|} - \left(\frac{1}{\left|\vec{r} + \vec{r_{2}}\right|} - \left(\frac{1}{\left|\vec{r} - \vec{r_{1}}\right|} \right)$$

It looks like the expression may be missing something...

Last edited: Jun 18, 2008