Entire function

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In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic everywhere on the whole complex plane. Typical examples of entire functions are the polynomials, the exponential function, and sums, products and compositions of these. Every entire function can be represented as a power series which converges everywhere, uniformly on compacta. Neither the natural logarithm nor the square root functions can be continued to an entire function.

Liouville's theorem establishes an important property of entire functions — an entire function which is bounded must be constant. As a consequence, a (complex-valued) function which is entire on the whole Riemann sphere (complex plane and the point at infinity) is constant Thus an entire function must have a singularity at the complex point at infinity, either a pole or an essential singularity (see Liouville's theorem below). In the latter case, it is called a transcendental entire function, otherwise it is a polynomial.

Liouville's theorem can be used too for an elegant proof of the fundamental theorem of algebra. Picard's little theorem is a considerable strengthening of Liouville's theorem: a non-constant entire function takes on every complex number as value, except possibly one. The latter exception is illustrated by the exponential function, which never takes on the value 0.

J. E. Littlewood chose the Weierstrass sigma function as a 'typical' entire function in one of his books.

[edit] The order of an entire function

The order of an entire function $f (z)$ is defined using the limit superior as:

$\rho=\limsup_{r\rightarrow\infty}\frac{\ln(\ln(M(r)))}{\ln(r)},$

where $r$ is the distance from $0$ and $M (r)$ is the maximum absolute value of $f (z)$ when $\left|z\right| = r.$ If $0<\rho<\infty,$ one can also define the type: