Function that is holomorphic on the whole complex plane
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any finite sums, products and compositions of these, such as the trigonometric functions sine and cosine and their hyperbolic counterparts sinh and cosh, as well as derivatives and integrals of entire functions such as the error function. If an entire function has a
root at , then , taking the limit value at , is an entire function. On the other hand, the natural logarithm, the reciprocal function, and the square root are all not entire functions, nor can they be continued analytically to an entire function.
A transcendental entire function is an entire function that is not a polynomial.
Just as meromorphic functions can be viewed as a generalization of rational fractions, entire functions can be viewed as a generalization of polynomials. In particular, if for meromorphic functions one can generalize the factorization into simple fractions (the Mittag-Leffler theorem on the decomposition of a meromorphic function), then for entire functions there is a generalization of the factorization — the Weierstrass theorem on entire functions.
In complex analysis, an entirefunction, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane...
up entire in Wiktionary, the free dictionary. Entire may refer to: Entirefunction, a function that is holomorphic on the whole complex plane Entire (animal)...
the function has simple poles.[clarification needed] The gamma function has no zeros, so the reciprocal gamma function 1/Γ(z) is an entirefunction. In...
analysis, the Weierstrass factorization theorem asserts that every entirefunction can be represented as a (possibly infinite) product involving its zeroes...
The exponential function is a mathematical function denoted by f ( x ) = exp ( x ) {\displaystyle f(x)=\exp(x)} or e x {\displaystyle e^{x}} (where...
referred to as regular functions. A holomorphic function whose domain is the whole complex plane is called an entirefunction. The phrase "holomorphic...
analytic everywhere and hence an entirefunction. The function has also been called the cardinal sine or sine cardinal function. The term sinc /ˈsɪŋk/ was introduced...
arguments as functions C → C {\displaystyle \mathbb {C} \to \mathbb {C} } , are holomorphic over the entire complex plane, making them entirefunctions, while...
gamma function, a shifted generalization of the factorial function to non-integer values. The Bessel function of the first kind is an entirefunction if...
which is zero at the origin is an odd function and vice versa). Since the error function is an entirefunction which takes real numbers to real numbers...
mathematics, and particularly complex dynamics, the escaping set of an entirefunction ƒ consists of all points that tend to infinity under the repeated application...
imaginary part of any entirefunction will produce a harmonic function with the same singularity, so in this case the harmonic function is not determined...
of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the...
analytic function. They are named after Émile Picard. Little Picard Theorem: If a function f : C → C {\textstyle f:\mathbb {C} \to \mathbb {C} } is entire and...
the gamma function is meromorphic and nonzero everywhere in the complex plane, its reciprocal is an entirefunction. As an entirefunction, it is of order...
programming, an anonymous function (function literal, lambda abstraction, lambda function, lambda expression or block) is a function definition that is not...
Γ(z + k) when k → ∞, and the fact that the reciprocal of Γ(z) is an entirefunction, the coefficients in the rightmost sum are well-defined, and locally...
serves to define the eta function as an entirefunction. (The above relation and the facts that the eta function is entire and η ( 1 ) ≠ 0 {\displaystyle...
cosine functions to an imaginary angle. The hyperbolic sine and the hyperbolic cosine are entirefunctions. As a result, the other hyperbolic functions are...
integral over the entire real line is equal to one. Since there is no function having this property, modelling the delta "function" rigorously involves...
0 {\displaystyle \Re z>0} . The function ψ ( z ) / Γ ( z ) {\displaystyle \psi (z)/\Gamma (z)} is an entirefunction, and it can be represented by the...
expansion at x of the entirefunction z → e−z2 (in the physicist's case). One can also derive the (physicist's) generating function by using Cauchy's integral...
fixed τ, this is a Fourier series for a 1-periodic entirefunction of z. Accordingly, the theta function is 1-periodic in z: ϑ ( z + 1 ; τ ) = ϑ ( z ; τ...
mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of...
complex analysis, the Hadamard factorization theorem asserts that every entirefunction with finite order can be represented as a product involving its zeroes...
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