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properties of entire functions
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- If $f\!:\mathbb{C}\to\mathbb{C}$ is an entire function and $z_0\in\mathbb{C}$ , then $f(z)$ has the Taylor series expansion $$f(z) = a_0\!+\!a_1(z\!-\!z_0)\!+\!a_2(z\!-\!z_0)^2\!+\cdots$$ which is valid in the whole complex plane.
- If, conversely, such a power series converges for every complex value $z$ , then the sum of the series is an entire function.
- The entire functions may be divided in two disjoint classes:
a) The entire rational functions, i.e. polynomial functions; in their series expansion there is an $n_0$ such that $a_n = 0\,\,\forall n\geqq n_0$ .
b) The entire transcendental functions; in their series expansion one has $a_n \neq 0$ for infinitely many values of $n$ . Examples are complex sine and cosine, complex exponential function, sine integral, error function.
- A consequence of Liouville's theorem: If $f$ is a non-constant entire function and if $R$ and $M$ are two arbitrarily great positive numbers, then there exist such points $z$ that $$|z| > R \,\,\,\mathrm{and}\,\,\, |f(z)| > M.$$ This means that the non-constant entire functions are unbounded.
- The sum, the product and the composition of two entire functions are entire functions.
- The ring of all entire functions is a Prüfer domain.
- 1
- O. HELMER: ``Divisibility properties of integral functions''. - Duke Math. J. 6 (1940), 345-356.
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"properties of entire functions" is owned by pahio.
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Cross-references: Prüfer domain, ring, composition, points, numbers, positive, Liouville's theorem, consequence, error function, sine integral, complex exponential function, complex sine and cosine, series, polynomial functions, disjoint, complex, converges, power series, conversely, complex plane, valid, Taylor series, entire function
There are 5 references to this entry.
This is version 16 of properties of entire functions, born on 2004-11-30, modified 2009-04-30.
Object id is 6545, canonical name is PropertiesOfEntireFunctions.
Accessed 6296 times total.
Classification:
| AMS MSC: | 30D20 (Functions of a complex variable :: Entire and meromorphic functions, and related topics :: Entire functions, general theory) |
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Pending Errata and Addenda
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