How Do You Spell ENTIRE FUNCTION?

1. An entire function, in mathematics, refers to a complex-valued function that is holomorphic in the entire complex plane. More specifically, an entire function is a function f(z) that is defined and analytic at every point in the complex plane. This means that for any complex number z, the function is well-defined and differentiable at that point and in some neighborhood.

   Furthermore, an entire function is characterized by its power series representation, which converges for all complex numbers. This indicates that an entire function has an infinite radius of convergence and can be represented as an infinite sum of powers of z. Consequently, it is infinitely differentiable throughout the complex plane.

   Entire functions possess several remarkable properties. Firstly, they have no singularities or poles within the complex plane. Secondly, due to the Cauchy-Riemann equations, the real and imaginary parts of an entire function are harmonic functions. Moreover, their growth behavior is also noteworthy, as the order of an entire function (a measure of its growth rate) is an important concept in complex analysis.

   Examples of entire functions include polynomials, trigonometric functions, and exponential functions. In contrast, functions with singularities or poles, such as rational functions or functions with branch points, are not entire. The study of entire functions is a fundamental aspect of complex analysis and finds applications in many areas such as number theory, physics, and engineering.

The word "entire function" originates from the Latin word "integer" which means "whole" or "entire". It refers to a complex function that is defined and analytic in the entire complex plane. The concept was introduced by Karl Weierstrass in the 19th century, and the term "entire" was used to emphasize that these functions are analytic on the entire complex plane, as opposed to being only defined on a certain region or domain.