Editing Nevanlinna theory (section)

== First fundamental theorem ==
Let ''a''&nbsp;∈&nbsp;'''C''', and define

:<math>
\quad N(r,a,f) = N\left(r,\dfrac{1}{f-a}\right),
\quad m(r,a,f) = m\left(r,\dfrac{1}{f-a}\right).\,</math>

For ''a''&nbsp;=&nbsp;∞, we set ''N''(''r'',∞,''f'')&nbsp;=&nbsp;''N''(''r'',''f''), ''m''(''r'',∞,''f'')&nbsp;=&nbsp;''m''(''r'',''f'').

The '''First Fundamental Theorem''' of Nevanlinna theory states that for every ''a'' in the [[Riemann sphere]],

:<math> T(r,f) = N(r,a,f)+m(r,a,f) + O(1),\,</math>

where the bounded term ''O''(1) may depend on ''f'' and ''a''.<ref>Ru (2001) p.5</ref>  For non-constant meromorphic functions in the plane, ''T''(''r'',&nbsp;''f'') tends to infinity as ''r'' tends to infinity,
so the First Fundamental Theorem says that the sum ''N''(''r'',''a'',''f'')&nbsp;+&nbsp;''m''(''r'',''a'',''f''), tends to infinity at the rate which is independent of ''a''. The first Fundamental theorem is a simple consequence
of [[Jensen's formula]].

The characteristic function has the following properties of the degree:

:<math>\begin{array}{lcl}
T(r,fg)&\leq&T(r,f)+T(r,g)+O(1),\\
T(r,f+g)&\leq& T(r,f)+T(r,g)+O(1),\\
T(r,1/f)&=&T(r,f)+O(1),\\
T(r,f^m)&=&mT(r,f)+O(1), \,
\end{array}</math>

where ''m'' is a natural number. The bounded term ''O''(1) is negligible when ''T''(''r'',''f'') tends to infinity. These algebraic properties are easily obtained from Nevanlinna's definition and Jensen's formula.