MORERA'S THEOREM

In complex analysis, a branch of mathematics, 'Morera's theorem' gives one characterization of when a function is holomorphic. It reads as follows: let ''f'' be a complex-valued function of a complex variable defined on some open set ''D''. If for every ''x'' in ''D'', there exist an open neighborhood ''U'' of ''x'' on which ''f'' possesses an anti-derivative, then ''f'' is holomorphic on ''D''.

Contents

Proof

Uses

Notes

External links

Proof

A short proof of the theorem can be given. By assumption, on a ''U'' ⊂ ''D'', we have ''F' = f'' on ''U'' for some ''F''. But since holomorphic functions are analytic, i.e. once complex-differentiable functions are infinitely differentiable, ''f'' is holomorphic on ''U''. By the identity theorem, ''f'' is holomorphic on the connected component containing ''U''. Applying this to each component of ''D'' proves the theorem.
We note here the theorem does not assume that ''f'' has an anti-derivative ''everywhere'' on ''D''. This latter condition is equivalent to that the integral of ''f'' along every closed curve within ''D'' is zero^[1]. In symbols,
:$int\_C\; f(zeta)\; ,d\; zeta\; =0$
for ''C'' any closed curve. This is a more stringent requirement then holomorphy. For example, the function
:
is holomorphic on an open circular ring centered at 0 but fails the condition.

Uses

Morera's theorem can be used to show the analyticity of functions defined by sums or integrals, such as the Riemann zeta function
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or the Gamma function
:
It also leads to a quick proof of the general result that if a sequence ''f''_''n''(''z'') of analytic functions on a given open set ''D'' of complex numbers, converges to a function ''f''(''z'') uniformly on every compact subset ''K'', then ''f'' is analytic. The condition can easily be reduced to ''K'' being a closed disk.

Notes

1. The equivalence can be seen as follows: If ''F' = f'' for some ''F'', then for every curve ''γ'' from ''a'' to ''b'' in ''D'',
:$int\_\{gamma\}\; f(zeta)\; ,\; d\; zeta\; =\; F(a)\; -\; F(b).,$
This is due to the formal similarity between real and complex differentiability. So if ''γ'' is a closed curve, the integral is zero. For the converse, without loss of generality, assume ''D'' is connected. Fix a ''z''₀ in ''D''. Define a function ''F'' by
:$F(z)\; =\; int\_\{gamma\}\; f(zeta)\; ,\; d\; zeta,,$
where ''γ'' is a curve from ''z''₀ to ''z''. Assuming ''F'' is well-defined, i.e. independent of the ''γ'' chosen, it is clear that ''F' = f''. But the well-definedness of ''F'' can be verified using the fact ∫_''C'' ''f'' = 0 for all closed curve ''C''. This proves the claim.

External links

★

★ Module for Morera’s Theorem by John H. Mathews