Simple Taylor or Multipole Expansion of Potential

[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...

Homework Statement

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]<br /> <br /> where r = \left|\vec{r}\right|[\tex].&lt;br /&gt; &lt;br /&gt; For large r &amp;gt;&amp;gt; 1 I am supposed to expand in powers of \frac{\vec{r_{i}}}{r}[\tex] to obtain the expression &amp;lt;br /&amp;gt; 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. &amp;amp;lt;br /&amp;amp;gt; &amp;amp;lt;br /&amp;amp;gt; &amp;amp;lt;h2&amp;amp;gt;Homework Equations&amp;amp;lt;/h2&amp;amp;gt;&amp;amp;lt;br /&amp;amp;gt; &amp;amp;lt;br /&amp;amp;gt; &amp;amp;lt;br /&amp;amp;gt; &amp;amp;lt;h2&amp;amp;gt;The Attempt at a Solution&amp;amp;lt;/h2&amp;amp;gt;&amp;amp;lt;br /&amp;amp;gt; &amp;amp;lt;br /&amp;amp;gt; I tried using the expansion \left(\frac{1}{\left|\vec{r} + \vec{r&amp;amp;amp;amp;#039;}\right|} = \frac{1}{r} + \frac{\vec{r}\vec{r&amp;amp;amp;amp;#039;}}{r^3}[\tex] + terms of higher order, but somehow end up with V = 0 (up to order r^-3[\tex]. I have completely forgotten how to do this...&amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;gt; &amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;gt; Would be thankful for any help...&amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;gt; &amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;gt; Greetings from Germany, &amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;gt; &amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;gt; bavaji

 

[dynamicsolo]

Hullo,

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

While you do use "backslashes" for the special TeX characters, you need to terminate the TeX command with regular "slash" [ / 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...