 Solution

Logarithmically differentiate the convergent product of the modular discriminant. On compact subsets of $\mathbb{H}$, absolute convergence permits termwise differentiation and rearrangement of the resulting divisor sums:The division is valid because $\Delta$ has no interior zeros. Its weight-twelve inversion law is $\Delta (-1/\tau )={\tau }^{12}\Delta (\tau )$. Taking a logarithmic derivative, including the derivative $1/{\tau }^2$ of $-1/\tau$, yieldsThus the Eisenstein series of weight two has the anomalous transformation

Let $y=\mathrm{Im}\tau$. Under inversion, $y$ becomes $y/|\tau {|}^2$. Using ${\tau }^2-|\tau {|}^2=2iy\tau$, the anomalous term cancels:The Fourier expansion gives $E_2(\tau +1)=E_2(\tau )$, and translation leaves $y$ unchanged. Since translation and inversion generate the projective modular group and weight two is unchanged by $-I$, the almost holomorphic weight-two Eisenstein series satisfiesIt is modular in the transformation sense, but is not a holomorphic modular form. Substituting $\mathrm{Im}(\gamma \tau )=y/|c\tau +d{|}^2$ into this law also gives the useful full-group anomaly