rnielsen25
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- Homework Statement
- Expand $$\frac{1}{\sqrt{ (\boldsymbol{r-r'})^2+a} }$$ in legendre polynomials.
- Relevant Equations
- $$ \sum_{n=0}^{\infty} P_{n}(x) t^{n}=\frac{1}{\sqrt{1-2 x t+t^{2}}} $$
Using the generating function for the legendre polynomial: $$ \sum_{n=0}^{\infty} P_{n}(x) t^{n}=\frac{1}{\sqrt{1-2 x t+t^{2}}} $$ It's possible to expand the coulomb potential in a basis of legendre polynomials (or even spherical harmonic ) like this: $$ \begin{aligned} &\frac{1}{\left.\mid \vec{r}-\vec{r}^{\prime}\right]}=\frac{1}{\sqrt{r^{2}+r^{\prime 2}-2 r r^{\prime}\left(\hat{r} \cdot \hat{r}^{\prime}\right)}}= \sum_{\ell=0}^{\infty} \frac{r_{<}^{\ell}}{r_{>}^{\ell+1}} P\left(\hat{r} \cdot \hat{r}^{\prime}\right) \\ &=\sum_{\ell=0}^{\infty} \frac{4 \pi}{2 \ell+1} \frac{r_{<}^{\ell}}{r_{>}^{\ell+1}} \sum_{m=-\ell}^{\ell} Y_{\ell m}^{\star}\left(\vartheta^{\prime}, \varphi^{\prime}\right) Y_{\ell m}(\vartheta, \varphi) \end{aligned} $$ Where ##r_{<}## and ##r_{>}## represent the smaller and larger of ##r## and ##r^{\prime}##.
But I need to expand
$$\frac{1}{\sqrt{ (\boldsymbol{r-r'})^2+a} }=\frac{1}{\sqrt{r^{2}+r^{\prime 2}-2 r r^{\prime}\left(\hat{r} \cdot \hat{r}^{\prime}\right)+a}}$$
in a similar way. However, I can't seem to pull out a factor of ##r## or ##r'## to get the generating function as you can above, because of the addition of ##a##.
So how do I expand this expression in legendre polynomials?
But I need to expand
$$\frac{1}{\sqrt{ (\boldsymbol{r-r'})^2+a} }=\frac{1}{\sqrt{r^{2}+r^{\prime 2}-2 r r^{\prime}\left(\hat{r} \cdot \hat{r}^{\prime}\right)+a}}$$
in a similar way. However, I can't seem to pull out a factor of ##r## or ##r'## to get the generating function as you can above, because of the addition of ##a##.
So how do I expand this expression in legendre polynomials?