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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.
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 can never define an entire function, because the infinite product does not converge. Thus one cannot, in general, define an entire function from a sequence of prescribed zeroes or represent an entire function by its zeroes using the expressions yielded by the fundamental theorem of algebra. Instead, the theorem replaces these with other factors.
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 ...
For an arbitrary function f define (,) = (()) (). f(x, λ) may be regarded as a differentiable map into the space of functions of bounded variation ρ; or equivalently as a differentiable map () into the Banach space E of bounded linear functionals dρ on C[α,β] whenever [α, β] is a compact subinterval of [1, ∞).
Corollary 1 — With the same ... Liouville's theorem, improved — If is an entire function, such that there ... has no roots and is of finite order ...
The fourteenth function () denotes the analytic extension of the factorial function via the gamma function, and () is its reciprocal, an entire function. Finally, in the last function f 16 ( x ) {\displaystyle f_{16}(x)} , the exponent x {\displaystyle x} can be replaced by k x {\displaystyle kx} for any nonzero real k {\displaystyle k} , and ...