I am trying to find the coefficients of the $\wp$-function.
Right now I have the Laurent series about the pole $ z = 0$:
$$\wp(z) = \frac{c_{-n}}{z^n} + \cdots + \frac{c_{-1}}{z} + c_0 + c_1 z + \cdots$$ where $c_{-n} \neq 0$
I know $n \geq 1$, and that $\wp$ satisfies
$$\left(\frac{d\wp(z)}{dz}\right)^2 = 4(\wp)(z))^3 -g_2 \wp(z) -g_3$$
I know the end form I am looking form is the well known equation:
$$\wp(z) = z^{-2} +b_0 +b_1 z^2 +b_2 z^4+\cdots $$
But I am not sure how to show that $n = 2$ and that $c_{-n} = 1$.
I know I can substitute the Laurent Series into the differential equation, but that isn't helping me, at least not with my algebra skills.