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regular at infinity
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(Definition)
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When the function $w$ of one complex variable is regular in the annulus $$\varrho < |z| < \infty,$$ it has a Laurent expansion
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If especially the coefficients $c_1,\, c_2,\,\ldots$ vanish, then we have $$w(z) = c_0+\frac{c_{-1}}{z}+\frac{c_{-2}}{z^2}+\ldots$$ Using the inversion $z = \frac{1}{\zeta}$ , we see that the function $$w\!\left(\frac{1}{\zeta}\right) = c_0+c_{-1}\zeta+c_{-2}\zeta^2+\ldots$$ is regular in the disc $|\zeta| < \varrho$ . Accordingly we can define that the function $w$ is regular at infinity also.
For example, $\displaystyle w(z) := \frac{1}{z}$ is regular at the point $z = \infty$ .
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"regular at infinity" is owned by pahio.
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Cross-references: point, disc, inversion, vanish, coefficients, Laurent expansion, annulus, regular, variable, complex, function
There are 2 references to this entry.
This is version 3 of regular at infinity, born on 2007-11-16, modified 2008-02-23.
Object id is 10045, canonical name is RegularAtInfinity.
Accessed 1465 times total.
Classification:
| AMS MSC: | 30D20 (Functions of a complex variable :: Entire and meromorphic functions, and related topics :: Entire functions, general theory) | | | 32A10 (Several complex variables and analytic spaces :: Holomorphic functions of several complex variables :: Holomorphic functions) |
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Pending Errata and Addenda
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