= Weight-two transformation from an invariant Eisenstein limit
{title2=$E_2(\gamma z)=(cz+d)^2E_2(z)+\dfrac{12c(cz+d)}{2\pi i}$}
The finite Laurent coefficient $C(z)$ in the <Kronecker limit formula> is invariant under the <modular group>. Its Wirtinger derivative satisfies $\partial_z C=-\pi i(E_2(z)-3/(\pi y))/12$, by differentiating the convergent product in that formula. Differentiating $C(\gamma z)=C(z)$ gives weight-two covariance for $E_2^*$. Since $\Im(\gamma z)=y/|cz+d|^2$, substitution gives the displayed transformation law. This proof needs no prior modularity assertion for the eta product.
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