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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
Entire_function
Topics referred to by the same term
up entire in Wiktionary, the free dictionary. Entire may refer to: Entire function, a function that is holomorphic on the whole complex plane Entire (animal)
Entire
Complex-differentiable (mathematical) function
referred to as regular functions. A holomorphic function whose domain is the whole complex plane is called an entire function. The phrase "holomorphic
Holomorphic_function
Branch of mathematics studying functions of a complex variable
the range of an entire function f {\displaystyle f} , then f {\displaystyle f} is a constant function. Moreover, a holomorphic function on a connected
Complex_analysis
Theorem in complex analysis
Liouville's theorem states that every bounded entire function must be constant. That is, every holomorphic function f {\displaystyle f} for which there exists
Liouville's theorem (complex analysis)
Liouville's_theorem_(complex_analysis)
Extension of the factorial function
the gamma function has no zeros, its reciprocal 1 Γ {\displaystyle {\frac {1}{\Gamma }}} is an entire function. In fact, the gamma function corresponds
Gamma_function
Notion from the theory of entire functions
indicator function of an entire function indicates the rate of growth of the function in different directions. Let us consider an entire function f : C →
Indicator function (complex analysis)
Indicator_function_(complex_analysis)
Theorem in complex analysis
analysis, the Weierstrass factorization theorem asserts that every entire function can be represented as a (possibly infinite) product involving its zeroes
Weierstrass factorization theorem
Weierstrass_factorization_theorem
Special mathematical function defined as sin(x)/x
analytic everywhere and hence an entire function. The normalized sinc function is the Fourier transform of the rectangular function with no scaling. It is used
Sinc_function
Class of mathematical function
"part". A holomorphic function resembles an entire function ("whole") in a domain of the complex plane while a meromorphic function, resembles a rational
Meromorphic_function
Functions in mathematics
imaginary part of any entire function will produce a harmonic function with the same singularity, so in this case the harmonic function is not determined
Harmonic_function
Statement in complex analysis
complex analysis, the Hadamard factorization theorem asserts that every entire function with finite order can be represented as a product involving its zeroes
Hadamard factorization theorem
Hadamard_factorization_theorem
Theorem about the range of an analytic function
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
Picard_theorem
Concept in complex dynamics
mathematics, and particularly complex dynamics, the escaping set of an entire function f {\displaystyle f} consists of all points that tend to infinity under
Escaping_set
Sigmoid shape special function
antiderivative of an even function which is zero at the origin is an odd function, and vice versa). Since the error function is an entire function which maps real
Error_function
Family of solutions to related differential equations
article. The Bessel function of the first kind is an entire function if α is an integer, otherwise it is a multivalued function with singularity at zero
Bessel_function
Mathematical function
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
Reciprocal_gamma_function
Function in analytic number theory
summable for any complex number. This serves to define the eta function as an entire function. Equivalently, we may begin by defining η ( s ) = 1 Γ ( s )
Dirichlet_eta_function
Analytic function that does not satisfy a polynomial equation
via the gamma function, and f 16 ( x ) {\displaystyle f_{16}(x)} is its reciprocal, an entire function. Finally, in the last function f 17 ( x ) {\displaystyle
Transcendental_function
analysis, the Bessel–Clifford function, named after Friedrich Bessel and William Kingdon Clifford, is an entire function of two complex variables that
Bessel–Clifford_function
Mathematical function
In some sense, the Mittag-Leffler function is the simplest entire function of its order. The indicator function of E α ( z ) {\displaystyle E_{\alpha
Mittag-Leffler_function
Theorem
if there are no singularities (i.e., if f {\displaystyle f} is an entire function), then the radius of convergence is infinite. Strictly speaking, this
Analyticity of holomorphic functions
Analyticity_of_holomorphic_functions
Mathematical function
In mathematics, the Riesz function is an entire function defined by Marcel Riesz in connection with the Riemann hypothesis, by means of the power series
Riesz_function
Evaluation of a function on its argument
continuous function on complete partial orders. Function application is also a continuous function in homotopy theory, and, indeed underpins the entire theory:
Function_application
Mathematical functions
exponential function. An alternative way of expressing the lemniscate functions as a ratio of entire functions involves the theta functions (see Lemniscate
Lemniscate_elliptic_functions
Two theorems about families of holomorphic functions
properties that imply that an entire function is constant correspond to properties that ensure that a family of holomorphic functions is normal. For example
Montel's_theorem
Type of function in mathematics
an analytic function is a function that is locally represented by a convergent power series. More precisely, a real or complex function is analytic at
Analytic_function
Topics referred to by the same term
number of operands a function takes Type, any proposition or set in the intuitionistic type theory Type, of an entire function Exponential type Type
Type
Conjecture on zeros of the zeta function
Unsolved problem in mathematics Do all non-trivial zeros of the Riemann zeta function have a real part equal to one half? More unsolved problems in mathematics
Riemann_hypothesis
Hyperbolic analogues of trigonometric functions
cosine functions to an imaginary angle. The hyperbolic sine and the hyperbolic cosine are entire functions. As a result, the other hyperbolic functions are
Hyperbolic_functions
Generalization of the Riemann zeta function for algebraic number fields
Dedekind zeta function of an algebraic number field K, usually denoted ζ K ( s ) {\displaystyle \zeta _{K}(s)} , is an analytic function that represents
Dedekind_zeta_function
Solution of a confluent hypergeometric equation
a function of a, b, or z with the other two held constant, this defines an entire function of a or z, except when b = 0, −1, −2, ... As a function of
Confluent hypergeometric function
Confluent_hypergeometric_function
Mathematical function, denoted exp(x) or e^x
In mathematics, the exponential function is the unique real function which maps zero to one and has a derivative everywhere equal to its value. It is denoted
Exponential_function
Special functions of several complex variables
fixed τ, this is a Fourier series for a 1-periodic entire function of z. Accordingly, the theta function is 1-periodic in z: ϑ ( z + 1 ; τ ) = ϑ ( z ; τ
Theta_function
Special function in the physical sciences
mathematics, the Airy function (or Airy function of the first kind) A i ( x ) {\displaystyle \mathbf {Ai({\boldsymbol {x}})} } is a special function named after
Airy_function
Special function defined by an integral
the sinc function, and also the zeroth spherical Bessel function. Since sinc {\displaystyle \operatorname {sinc} } is an even entire function (holomorphic
Trigonometric_integral
Concept of complex analysis
{e^{itz}}{z^{2}+1}}\,dz.} Since eitz is an entire function (having no singularities at any point in the complex plane), this function has singularities only where the
Residue_theorem
Analytic function in mathematics
The Riemann zeta function or Euler–Riemann zeta function, denoted by the lowercase Greek letter ζ (zeta), is a mathematical function of a complex variable
Riemann_zeta_function
Extension of superfactorials to the complex numbers
^{2}{\frac {t}{2}}}}+{\frac {z^{2}}{2}}e^{-t}-{\frac {z}{t}}\right]} As an entire function, G {\displaystyle G} is of order two, and of infinite type. This can
Barnes_G-function
Special function in mathematics
previous integral, this integral is valid for all s, and indeed is an entire function of s. The contour integral representation provides an analytic continuation
Hurwitz_zeta_function
Attribute of a mathematical function
the residue of a function at a point of its domain is a complex number proportional to the contour integral of a meromorphic function along a path enclosing
Residue_(complex_analysis)
Functions of an angle
mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of
Trigonometric_functions
Types of special mathematical functions
Γ(z + k) when k → ∞, and the fact that the reciprocal of Γ(z) is an entire function, the coefficients in the rightmost sum are well-defined, and locally
Incomplete_gamma_function
Special function defined by an integral
Fresnel integrals S(x) and C(x), and their auxiliary functions F(x) and G(x) are transcendental functions named after Augustin-Jean Fresnel that are used in
Fresnel_integral
Mathematical approximation of a function
plane is called an entire function. Polynomials, the exponential function, and the sine and cosine functions are entire functions. For any power series
Taylor_series
Finnish mathematician (1907–1996)
work on Denjoy's conjecture on the number of asymptotic values of an entire function, and in 1929 published the first proof of this conjecture, now known
Lars_Ahlfors
Mathematical formula in complex analysis
an important statement in the study of entire functions. Suppose that f {\displaystyle f} is an analytic function in a region in the complex plane C {\displaystyle
Jensen's_formula
Function studied by Ramanujan
\Delta )} to all complex numbers s {\displaystyle s} , making it an entire function. Ramanujan conjectured that all nontrivial zeros of L ( s , Δ ) {\displaystyle
Ramanujan_tau_function
Exponential function of an exponential function
various functions. The inverse of the double exponential function is the double logarithm log(log(x)). The complex double exponential function is entire, because
Double_exponential_function
Product of numbers from 1 to n
large (but not entirely correct) tables of factorials, up to 64!, based on the work of Clavius. The power series for the exponential function, with the reciprocals
Factorial
Extension of the factorial function
H(n)=\Gamma (n)=(n-1)!} Unlike the classical gamma function, Hadamard's gamma function H(x) is an entire function, i.e., it is defined and analytic at all complex
Hadamard's_gamma_function
Type of Dirichlet series associated to number field extensions
is entire function.. Assumption that representation is non-trivial is necessary, because for trivial representation we get Dedekind zeta function with
Artin_L-function
Every polynomial has a real or complex root
of R(z). Since, by assumption, A has no eigenvalues, the function R(z) is an entire function and Cauchy theorem implies that ∫ c ( r ) R ( z ) d z = 0
Fundamental theorem of algebra
Fundamental_theorem_of_algebra
an entire function of x. E-functions were first studied by Siegel in 1929. He found a method to show that the values taken by certain E-functions were
E-function
Special function defined by an integral
is a special function on the complex plane. It is defined as one particular definite integral of the ratio between an exponential function and its argument
Exponential_integral
Mathematical function
0 {\displaystyle \Re z>0} . The function ψ ( z ) / Γ ( z ) {\displaystyle \psi (z)/\Gamma (z)} is an entire function, and it can be represented by the
Digamma_function
Mathematical concept
entire function f(z) (that is, every function that is holomorphic over the entire complex plane) can be factored into an infinite product of entire functions
Infinite_product
Complex analysis function
the field of complex analysis, a Nevanlinna function is a complex function which is an analytic function on the open upper half-plane H {\displaystyle
Nevanlinna_function
Open problem on 3x+1 and x/2 functions
{\displaystyle h(z)} is any entire function. Since this expression evaluates to zero for real integers, the extended function f ( z ) = z 2 cos 2 ( π
Collatz_conjecture
Part of a computer program where a given name binding is valid
Assignment to a variable within a function causes it to be declared local to the function, hence its scope is the entire function, and thus using it prior to
Scope_(computer_programming)
Type of mathematical function
the corresponding Dirichlet L-function has a simple pole at s = 1 {\displaystyle s=1} . Otherwise, the L-function is entire. Since a Dirichlet character
Dirichlet_L-function
Conformal mappings in complex analysis
In complex analysis, the Schwarz triangle function or Schwarz s-function is a function that conformally maps the upper half plane to a triangle in the
Schwarz_triangle_function
Field of medical research
Gain-of-function research (GoF research or GoFR) is medical research that genetically alters an organism in a way that may enhance the biological functions of
Gain-of-function_research
Polish company
Entire M sp. z o.o. is a Polish company headquartered in Wrocław (Poland), operating internationally in the design and manufacture of military and outdoor
Entire_M
Geometric representation of the complex numbers
surface on which f(z) = z1/2 can be defined as a holomorphic function whose image is the entire w-plane (except for the point w = 0). Here's how that works
Complex_plane
Theorem in complex analysis
Goursat), is an important statement about line integrals for holomorphic functions in the complex plane. Essentially, it says that if f ( z ) {\displaystyle
Cauchy's_integral_theorem
Mathematical theorem
{\displaystyle f(\zeta )=\int _{-A}^{A}F(x)e^{ix\zeta }\,dx} is an entire function of exponential type A {\displaystyle A} , meaning that there is a constant
Paley–Wiener_theorem
Uniqueness theorem in complex analysis
one states that f vanishes on the non-negative integers. f(z) is an entire function of exponential type, meaning that | f ( z ) | ≤ C e τ | z | , z ∈ C
Carlson's_theorem
Provides integral formulas for all derivatives of a holomorphic function
} From Cauchy's estimate, one can easily deduce that every bounded entire function must be constant (which is Liouville's theorem). The formula can also
Cauchy's_integral_formula
Mathematical theory
invented by Anders Wiman as a tool to study the behavior of arbitrary entire functions. After the work of Wiman, the theory was developed by other mathematicians
Wiman–Valiron_theory
Theorem about entire functions
from complex analysis, which characterizes the order and type of an entire function. The theorem was proven in 1960 by Vladimir Igorevich Matsaev. Let
Matsaev's_theorem
Mathematical function that preserves angles
In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths. More formally, let U {\displaystyle U} and V
Conformal_map
Generalized function whose value is zero everywhere except at zero
Dirac delta function (or δ {\displaystyle {\boldsymbol {\delta }}} distribution), also known as the unit impulse, is a generalized function on the real
Dirac_delta_function
Numbers obtained by adding the two previous ones
F_{1}=F^{\prime }(0)=1} , the exponential generating function of the Fibonacci numbers is given by the entire function F ( x ) = e φ x − e ψ x 5 {\displaystyle F(x)={\frac
Fibonacci_sequence
domain is the entire complex plane. Quaternionic function: a function whose domain is quaternionic. Hypercomplex function: a function whose domain is
List_of_types_of_functions
In mathematics, the Hermite or Pólya class is a set of entire functions satisfying the requirement that if E(z) is in the class, then: E(z) has no zero
Hermite_class
Second-order partial differential equation
function. The Laplace operator therefore maps a scalar function to another scalar function. If the right-hand side is specified as a given function,
Laplace's_equation
Real function with secant line between points above the graph itself
derivative is nonnegative on its entire domain. Well-known examples of convex functions of a single variable include a linear function f ( x ) = c x {\displaystyle
Convex_function
Power series with negative powers
mathematics, the Laurent series of a complex function f ( z ) {\displaystyle f(z)} is a representation of that function as a power series which includes terms
Laurent_series
Topics referred to by the same term
transcendental numbers The Weierstrass factorization theorem asserts that entire functions can be represented by a product involving their zeroes The Sokhatsky–Weierstrass
Weierstrass_theorem
Concept in complex analysis
singularity of a complex-valued function of a complex variable. It is the simplest type of non-removable singularity of such a function (see essential singularity)
Zeros_and_poles
American country singer and actress (born 1955)
Reba Nell McEntire (/ˈriːbə ˈmækɪntaɪər/ REE-bə MAK-in-tire; born March 28, 1955), or simply Reba, is an American country singer and actress. Dubbed "The
Reba_McEntire
Fundamental trigonometric functions
sine and cosine. These are entire functions. It is also sometimes useful to express the complex sine and cosine functions in terms of the real and imaginary
Sine_and_cosine
Partial converse of Taylor's theorem
an entire function with simple zeros at 2 j . {\displaystyle 2^{j}.} The derivatives W '(2j) are bounded above and below. Similarly the function M (
Whitney_extension_theorem
Function defined by multiple sub-functions
mathematics, a piecewise function (also called a piecewise-defined function, a hybrid function, or a function defined by cases) is a function whose domain is partitioned
Piecewise_function
Linear combination of indicator functions of real intervals
mathematics, a function on the real numbers is called a step function if it can be written as a finite linear combination of indicator functions of intervals
Step_function
Formula in complex analysis
for example, we obtain Liouville's theorem, which says a bounded entire function is constant (indeed, let r → ∞ {\displaystyle r\to \infty } in the
Cauchy's_estimate
1859 mathematics paper by Bernhard Riemann
(ζ) for a function previously mentioned by Euler The analytic continuation of this zeta function ζ(s) to all complex s ≠ 1 The entire function ξ(s), related
On the Number of Primes Less Than a Given Magnitude
On_the_Number_of_Primes_Less_Than_a_Given_Magnitude
Bieberbach conjecture. A Hermite-Biehler function, also known as de Branges function is an entire function E from C {\displaystyle \mathbb {C} } to C
De_Branges_space
Concept in complex analysis
complex-valued function g is a function whose complex derivative is g. More precisely, given an open set U {\displaystyle U} in the complex plane and a function g
Antiderivative (complex analysis)
Antiderivative_(complex_analysis)
Indian botanist, civil servant and historian (1909–1986)
India was on the eve of independence. In 1947 he was in charge of the entire function where Jawaharlal Nehru delivered his famous Tryst with destiny speech
Mohinder_Singh_Randhawa
Mathematical function whose set of values is bounded
\rightarrow \mathbb {C} } must be unbounded since it is entire.[citation needed] The function f {\displaystyle f} which takes the value 0 for x {\displaystyle
Bounded_function
Family of power series in mathematics
coefficients indexed by n is a rational function of n. The series, if convergent, defines a generalized hypergeometric function, which may then be defined over
Generalized hypergeometric function
Generalized_hypergeometric_function
Description of continuous random distribution
the density function between the lowest and greatest values of the range. The PDF is nonnegative everywhere, and the area under the entire curve is equal
Probability_density_function
Generalisation of the generalised hypergeometric function pFq(z)
B_{q})\end{matrix}}\right.\right].} The entire function W λ , μ ( z ) {\displaystyle W_{\lambda ,\mu }(z)} is often called the Wright function. It is the special case
Fox–Wright_function
Mathematical function resembling a boxcar
boxcar function is any function which is zero over the entire real line except for a single interval where it is equal to a constant, A. The function is named
Boxcar_function
Meromorphic function on the complex plane
zeta-functions are L-functions. Most notably, the Prime zeta function is not an L-function, since they cannot be analytically extended to the entire complex
L-function
Quantum state of the entire universe
The universal wave function or the wave function of the universe is the wavefunction or quantum state of the entire universe. It is regarded as the basic
Universal_wave_function
Mapping arbitrary data to fixed-size values
A hash function is any function that can be used to map data of arbitrary size to fixed-size values, though there are some hash functions that support
Hash_function
~~.} Like the sine, cosine, and exponential functions, Stumpff functions are well-behaved entire functions : Their series converge absolutely for any finite
Stumpff_function
ENTIRE FUNCTION
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ENTIRE FUNCTION
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ENTIRE FUNCTION