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The entire functions on the complex plane form an integral domain (in fact a Prüfer domain). They also form a commutative unital associative algebra over the complex numbers. Liouville's theorem states that any bounded entire function must be constant. [c]
Define the Hadamard canonical factors ():= = / Entire functions of finite order have Hadamard's canonical representation: [1] = = (/) where are those roots of that are not zero (), is the order of the zero of at = (the case = being taken to mean ()), a polynomial (whose degree we shall call ), and is the smallest non-negative integer such that the series = | | + converges.
Matsaev's theorem is a theorem from complex analysis, which characterizes the order and type of an entire function. The theorem was proven in 1960 by Vladimir Igorevich Matsaev . [ 1 ]
Since the gamma function is meromorphic and nonzero everywhere in the complex plane, its reciprocal is an entire function. As an entire function, it is of order 1 (meaning that log log | 1 / Γ(z) | grows no faster than log | z |), but of infinite type (meaning that log | 1 / Γ(z) | grows faster than any multiple of | z ...
It is clear that any finite set {} of points in the complex plane has an associated polynomial = whose zeroes are precisely at the points of that set. The converse is a consequence of the fundamental theorem of algebra: any polynomial function () in the complex plane has a factorization = (), where a is a non-zero constant and {} is the set of zeroes of ().
Similarly, if an entire function has a pole of order at —that is, it grows in magnitude comparably to in some neighborhood of —then is a polynomial. This extended version of Liouville's theorem can be more precisely stated: if | f ( z ) | ≤ M | z | n {\displaystyle |f(z)|\leq M|z|^{n}} for | z | {\displaystyle |z|} sufficiently large ...
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NURBS order, a number one greater than the degree of the polynomial representation of a non-uniform rational B-spline; Order of convergence, a measurement of convergence; Order of derivation; Order of an entire function; Order of a power series, the lowest degree of its terms; Ordered list, a sequence or tuple; Orders of approximation in Big O ...