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 a₀ + a₁z + · · · + a_nzⁿ 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 z₀. 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.