What Should (x-a) Look Like in This Series Expansion?

[Qubix]

Homework Statement

Ok so I have to expand in a power series of $$({\alpha} Z)^{2}$$, the equation

$$E_{nj}=mc^{2}\left\{ \left[1+\left(\frac{Z{\alpha}}{n-(j+1/2)+\sqrt{(j+1/2)^{2}-\alpha^{2}Z^{2}}}\right)^{2}\right]^{-\frac{1}{2}}-1\right\}$$

Homework Equations

I know that a series expansion of a function f(x) around a point a is of the form

$$f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n.$$
3. Question

In my above formula, if a is represented by $$({\alpha} Z)^{2}$$ , who is x ? What does E depend on? In other words , what should my (x-a) look like?

Thanks!

 

[praharmitra]

Actually in your formula a = 0 and x = $$(\alpha Z)^2$$. Though I am wondering where your Z is in the formula for E. I guess you set it to 1. Can you do the expansion now?

 

[Qubix]

hy, sorry about that, I modified now. So I should take x = $$(\alpha Z)^2$$ and a = 0.
My last question is, the variable of my equation, from what you're saying is x = $$(\alpha Z)^2$$ , so in my series expansion I must take the derivative with respect to this x ?

 

[praharmitra]

yes, absolutely.

 

[Qubix]

worked, thanks :)