Entire Function
المؤلف:
Knopp, K
المصدر:
Entire Transcendental Functions." Ch. 9 in Theory of Functions Parts I and II, Two Volumes Bound as One, Part I.New York: Dover
الجزء والصفحة:
...
18-11-2018
874
Entire Function
If a complex function is analytic at all finite points of the complex plane 
, then it is said to be entire, sometimes also called "integral" (Knopp 1996, p. 112).
Any polynomial 
is entire.
Examples of specific entire functions are given in the following table.
| function |
symbol |
| Airy functions |
 ,  |
| Airy function derivatives |
 ,  |
| Anger function |
 |
| Barnes G-function |
 |
| bei |
 |
| ber |
 |
| Bessel function of the first kind |
 |
| Bessel function of the second kind |
 |
| Beurling's function |
 |
| cosine |
 |
| coversine |
 |
| Dawsons integral |
 |
| erf |
 |
| erfc |
 |
| erfi |
 |
| exponential function |
 |
| Fresnel integrals |
 ,  |
| gamma function reciprocal |
 |
| generalized hypergeometric function |
 |
| haversine |
 |
| hyperbolic cosine |
 |
| hyperbolic sine |
 |
| Jacobi elliptic functions |
 ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  |
| Jacobi theta functions |
 |
| Jacobi theta function derivatives |
 |
| Mittag-Leffler function |
 |
| modified Struve function |
 |
| Neville theta functions |
 ,  ,  ,  |
| Shi |
 |
| sine |
 |
| sine integral |
 |
| spherical Bessel function of the first kind |
 |
| Struve function |
 |
| versine |
 |
| Weber functions |
 |
| Wright function |
 |
| xi-function |
 |
Liouville's boundedness theorem states that a bounded entire function must be a constant function.
REFERENCES:
Knopp, K. "Entire Transcendental Functions." Ch. 9 in Theory of Functions Parts I and II, Two Volumes Bound as One, Part I.New York: Dover, pp. 112-116, 1996.
Krantz, S. G. "Entire Functions and Liouville's Theorem." §3.1.3 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 31-32, 1999.
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