Weierstrass's elliptic function's expansion

Question

I'm studying about elliptic functions. In Bergman's book (The Kernel function and conformal mapping - page 10), the author gave an expansion of the Weierstrass's $\wp$ function : $$\wp(u) = - \frac{\eta_1}{w_1}+(\frac{\pi}{2w_1})^2\frac{1}{\sin^2(\pi u/2w_1)}-2(\frac{\pi}{w_1})^2\sum_{n=1}^{\infty}\frac{nq^{2n}}{1-q^{2n}}\cos(\frac{n\pi u}{w_1})$$ where $w_1$,$w_2$ are periods, $q = \exp(i\pi w_2/w_1)$ and $2\eta_1$ is the increment of Weierstrass's zeta function related to the period $w_1$ (this means $\zeta (z+2w_1) = \zeta(z)+2\eta_1$, for more information, can see here. I do not know how to figure out this formula from the original formula :$$\wp(u)=\frac{1}{u^2}+\sum_{(m,n)\neq (0,0)} ((\frac{1}{u+mw_1+nw_2})^2-(\frac{1}{mw_1+nw_2})^2)$$ Any hints would be appreciated.