= Nevanlinna first main theorem
{c}
{title2=$T(r,f)=m(r,1/(f-a))+N(r,1/(f-a))+O(1)$}
= Nevanlinna first fundamental theorem
{c}
{synonym}
For a fixed finite target $a$, the displayed identity compares growth with proximity to and occurrences of $a$. If $g(z)=cz^\nu(1+O(z))$ near zero, where $\nu$ is its signed zero order, the exact reciprocal identity is $T(r,g)-T(r,1/g)=\log|c|$. It follows by factoring out zeros and poles on a disc, applying the <mean value property for harmonic functions> to the remaining nonvanishing factor, and using $\log^+u-\log^+(1/u)=\log u$. Translation changes $T$ by a bounded amount, proving the general assertion.
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