= Area formula for a univalent power series
{title2=$\operatorname{Area}(f(r\mathbb D))=\pi\sum_{n\geq1}n|c_n|^2r^{2n}$}
For an <injective> <holomorphic function> $f(z)=\sum_{n\geq0}c_nz^n$ on the <unit disc>, its real <Jacobian determinant> is $|f'(z)|^2$. <Change of variables> gives image <area> $\int_{|z|<r}|f'(z)|^2dA$. Angular <orthogonality> removes cross terms of the differentiated <power series>; integrating $2\pi n^2|c_n|^2\rho^{2n-1}$ radially gives the title formula. For $r<1$, uniform convergence on a slightly larger <closed disk> justifies termwise operations. Injectivity is essential for this to be image <area> rather than <multiplicity of a parametrized surface integral>.
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