Entire function
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In complex analysis, an entire function is a function that is holomorphic everywhere on the whole complex plane. Typical examples of entire functions are the polynomials, the exponential function, and sums, products and compositions of these. Every entire function can be represented as a power series which converges everywhere. Neither the natural logarithm nor the square root function is entire.
Note that an entire function may have a singularity or even an essential singularity at the complex point at infinity. In the latter case, it is called a transcendental entire function. As a consequence of Liouville's theorem, a function which is entire on the entire Riemann sphere (complex plane and the point at infinity) is constant.
Liouville's theorem establishes an important property of entire functions — an entire function which is bounded must be constant. This property can be used for an elegant proof of the fundamental theorem of algebra. Picard's little theorem is a considerable strengthening of Liouville's theorem: a non-constant entire function takes on every complex number as value, except possibly one. The latter exception is illustrated by the exponential function, which never takes on the value 0.
J. E. Littlewood chose the Weierstrass sigma function as a 'typical' entire function in one of his books.
[edit] See also
[edit] References
- Ralph P. Boas (1954). Entire Functions. Academic Press. OCLC 847696.
