Entire function - Knowlpedia

Definition

An entire function is a function $f:\mathbb C\to\mathbb C$ that is holomorphic at every point of the complex plane.

Power-series form

By analyticity of holomorphic functions, an entire function has a Taylor expansion about every $a\in\mathbb C$ with infinite radius of convergence:

f(z)=\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(z-a)^n.

Conversely, any complex power series of infinite radius defines an entire function.

Growth and rigidity

Polynomials, $e^z$ , $\sin z$ , and $\cos z$ are entire. Rational functions with poles are not. The Liouville theorem says that a bounded entire function is constant. More generally, if $|f(z)|\le C(1+|z|^m)$ , then $f$ is a polynomial of degree at most $m$ .

Behavior at infinity

Regard $f$ near $\infty$ through $g(w)=f(1/w)$ . The point $\infty$ is removable exactly when $f$ is constant, a pole exactly when $f$ is a nonconstant polynomial, and an essential singularity exactly when $f$ is transcendental entire.

References

John B. Conway, Functions of One Complex Variable I, 2nd ed., Springer, 1978. Publisher record. Relevant: Chapters III and V.