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# properties of entire functions

1. 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.

2. If, conversely, such a power series converges for every complex value $z$, then the sum of the series is an entire function.

3. 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.

4. 5. The sum, the product and the composition of two entire functions are entire functions.

6.

# References

- 1 O. Helmer: “Divisibility properties of integral functions”. – Duke Math. J. 6 (1940), 345–356.

## Mathematics Subject Classification

30D20*no label found*

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