when all singularities are poles
Proof. Suppose that is such an analytic function. The number of the poles of must be finite, since otherwise the set of the poles would have in the closed complex plane an accumulation point which is neither a point of regularity (http://planetmath.org/Holomorphic) nor a pole. Let and possibly be the poles of the function .
For every , the function has at the pole with the order , the Laurent expansion of the form
This is in in the greatest open disc containing no other poles. We write (1) as
which we write, corresponding to (2), as
defines a rational function having the same poles as . Therefore the function defined by
is analytic (http://planetmath.org/Analytic) everywhere except possibly at the points and . If we write
shows that is analytic in the infinity (http://planetmath.org/RiemannSphere), too. Thus is analytic in the whole closed complex plane. By Liouville’s theorem, is a constant function. We conclude that is a rational function. Q.E.D.
- 1 R. Nevanlinna & V. Paatero: Funktioteoria. Kustannusosakeyhtiö Otava, Helsinki (1963).
|Title||when all singularities are poles|
|Date of creation||2014-11-21 21:30:22|
|Last modified on||2014-11-21 21:30:22|
|Last modified by||pahio (2872)|