Entire function

In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic at all finite points over the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any finite combinations of sums, products and compositions of these, such as the trigonometric functions sine and cosine and their hyperbolic counterparts sinh and cosh, as well as derivatives and integrals of entire functions such as the error function. If an entire function f(z) has a root at w, then f(z)/(z−w) is an entire function. On the other hand, neither the natural logarithm nor the square root is an entire function, nor can they be continued analytically to an entire function.

A transcendental entire function is an entire function that is not a polynomial.

Every entire function f(z) can be represented as a power series

that converges everywhere in the complex plane, hence uniformly on compact sets. The radius of convergence is infinite, which implies that