Questions tagged [entire-functions]

Definition: In mathematical analysis and, in particular, in the theory of functions of complex variable, an entire function, also called an integral function, is a function that is holomorphic in the whole complex plane (except, possibly, at the point at infinity).

It can be expanded in a power series $$f(z)=\sum_{k=0}^{\infty}a_k~z^k~,\qquad a_k=\frac{f^{(k)}(0)}{k!}~,\qquad k\ge 0$$which converges in the whole complex plane,$$\lim_{k\to \infty}|a_k|^{\frac{1}{k}}=0\qquad\text{or,}\qquad\lim_{n\to\infty}\frac{\ln|a_k|}{k}=-\infty~.$$

Examples:

• Examples of entire functions are polynomial and exponential functions. All sums, and products of entire functions are entire, so that the entire functions form a $\mathbb C$-algebra. Further, compositions of entire functions are also entire.

• All the derivatives and some of the integrals of entire functions, for example the error function $erf$, sine integral $Si$ and the Bessel function $J_0$ are also entire functions.

• In general, neither series nor limit of a sequence of entire functions need be an entire function.

• The inverse of an entire function need not be entire. Usually, inverse of a nonlinear function is not entire. (The inverse of a linear function is entire). In particular, inverses of trigonometric functions are not entire.

References:

https://en.wikipedia.org/wiki/Entire_function https://m.tau.ac.il/~tsirel/dump/Static/knowino.org/wiki/Entire_function.html https://www.encyclopediaofmath.org/index.php/Entire_function