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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 C-algebra. Further, compositions of entire functions are also entire.
The inverse of an entire function has need not be entire. Usually, inverse of a non-trivial function is not entire. (The inverse of a linear function is entire). In particular, inverses of trigonometric functions are not entire.
The entire functions have all general properties of other analytic functions, but the infinite range of analyticity enhances the set of the properties, making the entire functions especially beautiful and attractive for applications.
Order of an entire function
As all entire functions (except the constants) are unbounded, they grow as the argument become large, and can be characterised by their growth rate, which is called order.
Let be entire function. Positive number is called order of function , if for all positive numbers , larger than , there exist positive number such that for all complex such that , the relation holds .
Range of values
For example, the exponential never takes on the value 0.
Entire function , at any complex and at any contour C enclosing the point just once, can be expressed the Cauchy's theorem
- Conway, John B. (1978). Functions of One Complex Variable I, 2nd edition. Springer. ISBN 0-387-90328-3.
- Boas, Ralph P. (1954). Entire Functions. Academic Press. OCLC 847696.
- Krantz (1999). Handbook of Complex Variables. Boston, MA: Birkhäuser. ISBN 0-8176-4011-8.