I learned about the connection of the Eisenstein series with the Weierstrass ℘ function.
For z near 0, the equality below holds. (Complex Analysis, Stein & Shakarchi, p.274)
$$℘(z) = 1/z^2 + 3E_4z^2 + 5E_6z^4 + \cdots$$
However, in this theorem, I found that this equality can be also shown by the power series expansion of ℘ near 0 where ℘ has a pole of order 2.
At the same time, I realized that I couldn't use this series expansion of ℘ to actually prove the theorem above because ℘ is not holomorphic at 0, so I couldn't guarantee the existence of power series expansion of ℘ at 0.
But I want to know what this implies. I mean, is power series expansion of ℘ has actually something to do with the connection between Eisenstein series and Weierstrass ℘ function?
Thanks in advance, and excuse me for having trouble with typing mathematical notation. I would be thankful for giving me a bit of guide about how to type notation correctly. (I'm also trying to find out how to do it)