Entire function

An entire function is a holomorphic function whose domain is the whole complex plane.

A nonzero entire function can be described by its discrete zero locations and multiplicities, a product of Weierstrass elementary factors and a zero-free exponential factor. Conversely, consistent prescribed multiplicities at distinct points escaping every compact subset can be realized by choosing factor orders large enough for infinite product convergence from logarithmic tails. Repeated locations with incompatible exact multiplicities are not admissible data, and a finite accumulation of distinct zeros is excluded by the identity theorem.

An entire function of finite order factors as its canonical zero product times the exponential of a polynomial. For order at most one and nonzero value at zero, it has the form $f(z)=f(0)e^{Bz}{\prod }_j(1-z/a_j)e^{z/a_j}$. Genus-one factors converge when ${\sum }_j|a_j{|}^{-2}<\infty$. A zero-free function of this order is therefore the exponential of an affine polynomial.

If two nonzero entire functions have identical zeros and orders, their quotient extends to a nowhere-zero entire function. Its logarithmic derivative has a primitive on the simply connected plane, producing a holomorphic logarithm $h$ and the displayed relation. Any two choices differ by one constant integer multiple of $2\pi i$. On a multiply connected domain the logarithm can fail to exist globally.

Suppose that for every $z\in \mathbb{C}$, at least one derivative $f^{(n)}(z)$ of an entire function vanishes. The closed sets $E_n=\{z:f^{(n)}(z)=0\}$ cover $\mathbb{C}$. By the Baire category theorem, some $E_N$ has nonempty interior. The identity theorem gives $f^{(N)}=0$ everywhere, so $f$ is a polynomial of degree below $N$.

Every nonconstant entire function takes every complex value with at most one exception.

Near an essential singularity, a holomorphic function takes every complex value, with at most one exception, infinitely often.