# Entire Function

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## entire function

[en¦tīr ¦fəŋk·shən]## Entire Function

(or integral function), a function of a complex variable that is analytic throughout the entire complex plane. Examples of entire functions are the algebraic polynomial *a*_{0} + *a*_{1}*z* + · · · + *a _{n}z^{n}* and the functions sin

*z*, cos

*z*, and

*e*.

^{z}The point at infinity is in general an isolated singularity of an entire function. In order for the point at infinity to be a removable singularity of the entire function *f*(*z*), it is necessary and sufficient that *f*(*z*) be a constant. The point at infinity is a pole of *f*(*z*) if, and only if, *f*(*z*) is an algebraic polynomial. If the point *z* = ∞ is an essential singularity of *f*(*z*), then *f*(*z*) is said to be a transcendental entire function. Examples are the functions sin *z*, cos *z*, and *e ^{z}*.

In order for *f*(*z*) to be an entire function, it is necessary and sufficient that the relation

hold for at least one point *z*_{0}. In this case the expansion of *f*(*z*) in the Taylor’s series

converges throughout the entire complex plane.

The classification of transcendental entire functions is based on the rate of increase *M*(*r*) of the function; *M*(*r*) is defined by the equation

The quantity

is called the order of the entire function *f*(*z*). The relation between the order of an entire function and the distribution of its zeros was established by H. Poincaré, J. Hadamard, and E. Borel.

### REFERENCE

Markushevich, A. I.*Tselye funktsii*. Moscow, 1965.