= Entire function
{title2=$f:\mathbb C\to\mathbb C$ holomorphic everywhere}
{wiki}
= Entire
{synonym}
An <entire function> is a <holomorphic function> whose <domain of a function> is the whole <complex plane>. Its <derivative> exists at every $z\in\mathbb C$. Every <polynomial>, the <complex exponential function> $e^z$, the product $ze^z$, and the <complex sine function> $\sin z$ are <entire functions>. The <Taylor series> of an <entire function> about any finite centre converges throughout the <complex plane>.
The <complex exponential function> is a particularly revealing <entire function>. Writing $z=x+iy$, <Euler's formula> gives
$$
e^{x+iy}=e^x(\cos y+i\sin y),\qquad |e^{x+iy}|=e^x>0.
$$
Thus the <complex exponential function> never takes the value zero. Conversely, for any $w\ne0$, choose an <argument> $\theta$ of $w$. All its <preimages> under the <complex exponential function> are
$$
z_k=\ln|w|+i(\theta+2\pi k),\qquad k\in\mathbb Z.
$$
Here $\ln|w|$ is the real <natural logarithm>, and $k$ ranges over the <integers>. Substitution gives $e^{z_k}=|w|e^{i\theta}=w$; equality of <moduli> and <arguments> shows these are all the solutions. The <complex exponential function> therefore has <image> exactly $\mathbb C\setminus\{0\}$, with infinitely many <preimages> for each nonzero value.
\Image[/complex-exponential-covering.png]
{title=Under the complex exponential, vertical lines become circles, horizontal lines become rays, and inputs separated by 2 pi i have the same image}
In the pictured vertical strip $-1\le x\le1$, the <real part> sets the <modulus> between $e^{-1}$ and $e$, while the <imaginary part> sets the <argument> modulo $2\pi$. The two drawn bands each make one full turn around the same <annulus>; their corresponding points differ by $2\pi i$ and have identical <images>. Allowing all real $x$ gives every positive <modulus>, so the whole <complex plane> covers every nonzero output.
An <entire function> has exceptionally strong global constraints. The <Liouville theorem> says that a <bounded function> which is also an <entire function> must be a <constant function>. Accordingly, the nonconstant <entire functions> above are unbounded somewhere in the <complex plane>, even when their restrictions to the real axis are bounded: the <complex sine function> satisfies $\sin(iy)=i\sinh y$, growing through the <hyperbolic sine>. The <Little Picard theorem> adds a different constraint: an unbounded nonconstant <entire function> still cannot leave two distinct complex values out of its <image>. These results make <entire functions> useful when global bounds or restrictions on the <image> can establish rigidity without solving for the <function> explicitly.
In the opening of his 1880 memoir, Émile Picard credits Karl Weierstrass with the terminology of <entire functions> and describes them by everywhere-convergent <power series>. https://numdam.org/articles/10.24033/asens.187/[Picard, Mémoire sur les fonctions entières (1880), p. 145]{ref}
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