Entire function

entireHadamard productorderEntire Functionsentire functions of finite orderentire holomorphic functionentire transcendental functionfonctions entièresholomorphic on a whole complex planetranscendental entire function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic at all finite points over the whole complex plane.wikipedia
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Holomorphic function

holomorphicholomorphic mapholomorphic functions
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic at all finite points over the whole complex plane.
A holomorphic function whose domain is the whole complex plane is called an entire function.

Polynomial

polynomial functionpolynomialsmultivariate polynomial
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.
Every polynomial function is continuous, smooth, and entire.

Exponential function

exponentialexponentiallyexp
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.
The exponential function extends to an entire function on the complex plane.

Trigonometric functions

cosinetrigonometric functiontangent
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.
Therefore, the sine and the cosine can be extended to entire functions (also called "sine" and "cosine"), which are (by definition) complex-valued functions that are defined and holomorphic on the whole complex plane.

Hyperbolic function

hyperbolic tangenthyperbolichyperbolic cosine
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.
The hyperbolic sine and the hyperbolic cosine are entire functions.

Sine

sine functionsinnatural sines
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.
This is an entire function.

Error function

complementary error functionerfcomplementary Gaussian error function
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.
The error function is an entire function; it has no singularities (except that at infinity) and its Taylor expansion always converges.

Weierstrass factorization theorem

Weierstrass productWeierstrass theoremHadamard factorization theorem
The Weierstrass factorization theorem asserts that any entire function can be represented by a product involving its zeroes (or "roots").
In mathematics, and particularly in the field of complex analysis, the Weierstrass factorization theorem asserts that every entire function can be represented as a (possibly infinite) product involving its zeroes.

Liouville's theorem (complex analysis)

Liouville's theoremLiouville TheoremLiouville's theorem.
Liouville's theorem states that any bounded entire function must be constant.
In complex analysis, Liouville's theorem, named after Joseph Liouville, states that every bounded entire function must be constant.

Prüfer domain

Prüfer
The entire functions on the complex plane form an integral domain (in fact a Prüfer domain).
The ring of entire functions on the open complex plane C form a Prüfer domain.

Radius of convergence

region of convergencedisc of convergencecircle of convergence
The radius of convergence is infinite, which implies that
Note that r = 1/0 is interpreted as an infinite radius, meaning that ƒ is an entire function.

Transcendental function

transcendentaltranscendental functionstranscendental equations
A transcendental entire function is an entire function that is not a polynomial.
The converse is not true: there are entire transcendental functions f such that f(\alpha) is an algebraic number for any algebraic \alpha.

Picard theorem

Picard's theoremPicard's great theoremPicard's little theorem
Picard's little theorem is a much stronger result: any non-constant entire function takes on every complex number as value, possibly with a single exception.
Little Picard Theorem: If a function f : C → C is entire and non-constant, then the set of values that f(z) assumes is either the whole complex plane or the plane minus a single point.

Fundamental theorem of algebra

boundedFundamental- of Algebranot an algebraic concept
Liouville's theorem may be used to elegantly prove the fundamental theorem of algebra.
Since, by assumption, A has no eigenvalues, the function R(z) is an entire function and Cauchy theorem implies that

Casorati–Weierstrass theorem

Weierstrass–Casorati theoremWeierstrass-Casorati theorem
Specifically, by the Casorati–Weierstrass theorem, for any transcendental entire function f and any complex w there is a sequence such that
In the case that f is an entire function and a=\infty, the theorem says that the values f(z)

Bessel function

Bessel functionsmodified Bessel functionBessel function of the first kind
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.

Reciprocal gamma function

reciprocal of Γ(''z'')
The type may be infinite, as in the case of the reciprocal gamma function, or zero (see example below under #Order 1). Other examples include the Fresnel integrals, the Jacobi theta function, and the reciprocal Gamma function.
Since the gamma function is meromorphic and nonzero everywhere in the complex plane, its reciprocal is an entire function.

Theta function

Jacobi theta functiontheta functionsJacobi theta functions
Other examples include the Fresnel integrals, the Jacobi theta function, and the reciprocal Gamma function.
If τ is fixed, this becomes a Fourier series for a periodic entire function of z with period 1; in this case, the theta function satisfies the identity

Bounded function

boundedunboundedbounded sequence
Liouville's theorem states that any bounded entire function must be constant.

Laguerre–Pólya class

Such entire functions form the Laguerre–Pólya class, which can also be characterized in terms of the Hadamard product, namely, f belongs to this class if and only if in the Hadamard representation all z n are real, p ≤ 1, and P(z) = a + bz + cz 2, where b and c are real, and c ≤ 0.
The Laguerre–Pólya class is the class of entire functions consisting of those functions which are locally the limit of a series of polynomials whose roots are all real.

Exponential type

If the order is 1 and the type is σ, the function is said to be "of exponential type σ".
has given a generalization of exponential type for entire functions of several complex variables.

Mittag-Leffler function

The exponential function and the error function are special cases of the Mittag-Leffler function.
In the case α and β are real and positive, the series converges for all values of the argument z, so the Mittag-Leffler function is an entire function.

Airy function

Airy equationAiryAiry differential equation
* Airy function Ai(z)
As explained below, the Airy functions can be extended to the complex plane, giving entire functions.

Fresnel integral

Fresnel integralsC(1)clothoid
Other examples include the Fresnel integrals, the Jacobi theta function, and the reciprocal Gamma function.
* C and S are entire functions.