Liouville's theorem establishes an important property of entire functions—an entire function which is bounded must be constant. As a consequence, a (complex-valued) function which is entire on the whole Riemann sphere (complex plane and the point at infinity) is constant. Thus an entire function must have a singularity at the complex point at infinity, either a pole or an essential singularity (see Liouville's theorem below). In the latter case, it is called a transcendental entire function, otherwise it is a polynomial.
Liouville's theorem may also be used to elegantly prove 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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Interpolation decomposition of Paley-Wiener-Schwartz space with application to signal theory and zero distribution.(Report)
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