entire function

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. The trigonometric and hyperbolic functions are also entire, but they are mere variations of the exponential function. Every entire function can be represented as a power series which converges everywhere. Neither the natural logarithm nor the square root function is entire. The most important fact about entire functions is Liouville's theorem: an entire function which is bounded must be constant. This 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.

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