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![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
when all singularities are poles
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(Theorem)
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In the parent entry we see that a rational function has as its only singularities a finite set of poles. It is also valid the converse
Theorem. Any single-valued analytic function, which has in the whole closed complex plane no other singularities than poles, is a rational function.
Proof. Suppose that $z\mapsto w(z)$ is such an analytic function. The number of the poles of $w$ 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 nor a pole. Let $b_1,\,b_2,\,\ldots,\,b_k$ and possibly $\infty$ be the
poles of the function $w$ .
For every $i = 1,\,2,\,\ldots,\,k$ , the function has at the pole $b_i$ with the order $n_i$ , the Laurent expansion of the form
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(1) |
This is in force in the greatest open disc containing no other poles. We write (1) as
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(2) |
where the first addend is the principal part of (1), i.e. consists of the terms of (1) which become infinite in $z = b_i$ .
If we think a circle having center in the origin and containing all the finite poles $b_i$ (an annulus $\varrho < |z| < \infty$ ), then $w(z)$ has outside it the Laurent series expansion $$w(z) = d_mz^m+d_{m-1}z^{m-1}+\ldots+d_0+\frac{d_{-1}}{z}+\ldots,$$ which we write, corresponding to (2), as
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(3) |
where $G_m(z)$ is a polynomial of $z$ and $Q\left(\frac{1}{z}\right)$ a power series in $\frac{1}{z}$ . Then the equation $$R(z)\, := \sum_{i=1}^kF_{n_i}\!\left(\frac{1}{z-b_i}\right)+G_m(z)$$ defines a rational function having the same poles as $w$ . Therefore the function defined by $$f(z)\, :=\, w(z)-R(z)$$ is analytic everywhere except possibly at the points $z = b_i$ and $z = \infty$ . If we write $$f(z) = \left[w(z)-F_{n_i}\!\left(\frac{1}{z-b_i}\right)\right]-\sum_{j\neq i}F_{n_j}\!\left(\frac{1}{z-b_j}\right)-G_m(z),$$ we see that $f(z)$ is bounded in a neighbourhood of the point $b_i$ and is analytic also in this point ($i = 1,\,2,\,\ldots,\,k$ ). But then again, the presentation $$f(z) = \left[w(z)-G_m(z)\right]-\sum_{j=1}^kF_{n_j}\!\left(\frac{1}{z-b_j}\right)$$ shows that $f$ is analytic in the infinity, too. Thus $f$ is analytic in the whole closed complex
plane. By Liouville's theorem, $f$ is a constant function. We conclude that $R(z)+f(z) = w(z)$ is a rational function. Q.E.D.
The theorem implies, that if a meromorphic function is regular at infinity or has there a pole, then it is a rational function.
- 1
- R. NEVANLINNA & V. PAATERO: Funktioteoria. Kustannusosakeyhtiö Otava, Helsinki (1963).
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"when all singularities are poles" is owned by pahio.
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Cross-references: regular at infinity, meromorphic, implies, constant function, Liouville's theorem, neighbourhood, bounded, points, equation, power series, polynomial, annulus, origin, center, circle, infinite, terms, principal part, disc, open, Laurent expansion, order, function, NOR, accumulation point, finite, number, proof, closed complex plane, analytic function, single-valued, theorem, converse, valid, poles, finite set, rational function
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This is version 12 of when all singularities are poles, born on 2007-11-16, modified 2007-11-20.
Object id is 10043, canonical name is WhenAllSingularitiesArePoles.
Accessed 900 times total.
Classification:
| AMS MSC: | 30A99 (Functions of a complex variable :: General properties :: Miscellaneous) | | | 30C15 (Functions of a complex variable :: Geometric function theory :: Zeros of polynomials, rational functions, and other analytic functions ) | | | 30D10 (Functions of a complex variable :: Entire and meromorphic functions, and related topics :: Representations of entire functions by series and integrals) |
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Pending Errata and Addenda
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