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.
Properties
Every entire function can be represented as a single power series
Any power series satisfying this criterion will represent an entire function.
If (and only if) the coefficients of the power series are all real then the function evidently takes real values for real arguments, and the value of the function at the complex conjugate of will be the complex conjugate of the value at Such functions are sometimes called self-conjugate (the conjugate function, being given by ).[1]
If the real part of an entire function is known in a neighborhood of a point then both the real and imaginary parts are known for the whole complex plane, up to an imaginary constant. For instance, if the real part is known in a neighborhood of zero, then we can find the coefficients for from the following derivatives with respect to a real variable :
(Likewise, if the imaginary part is known in a
neighborhood then the function is determined up to a real constant.) In fact, if the real part is known just on an arc of a circle, then the function is determined up to an imaginary constant.[b]
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Note however that an entire function is not determined by its real part on all curves. In particular, if the real part is given on any curve in the complex plane where the real part of some other entire function is zero, then any multiple of that function can be added to the function we are trying to determine. For example, if the curve where the real part is known is the real line, then we can add times any self-conjugate function. If the curve forms a loop, then it is determined by the real part of the function on the loop since the only functions whose real part is zero on the curve are those that are everywhere equal to some imaginary number.
As a consequence of Liouville's theorem, any function that is entire on the whole Riemann sphere[d]
is constant. Thus any non-constant entire function must have a
, for any transcendental entire function and any complex there is a sequence such that
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. When an exception exists, it is called a lacunary value of the function. The possibility of a lacunary value is illustrated by the exponential function, which never takes on the value 0 . One can take a suitable branch of the logarithm of an entire function that never hits 0 , so that this will also be an entire function (according to the Weierstrass factorization theorem). The logarithm hits every complex number except possibly one number, which implies that the first function will hit any value other than 0 an infinite number of times. Similarly, a non-constant, entire function that does not hit a particular value will hit every other value an infinite number of times.
Liouville's theorem is a special case of the following statement:
Theorem — Assume are positive constants and is a non-negative integer. An entire function satisfying the inequality for all with is necessarily a polynomial, of degree at most [e]
Similarly, an entire function satisfying the inequality for all with is necessarily a polynomial, of degree at least .
Growth
Entire functions may grow as fast as any increasing function: for any increasing function
there exists an entire function such that
for all real . Such a function may be easily found of the form:
for a constant and a strictly increasing sequence of positive integers . Any such sequence defines an entire function , and if the powers are chosen appropriately we may satisfy the inequality for all real . (For instance, it certainly holds if one chooses and, for any integer one chooses an even exponent such that ).
Order and type
The order (at infinity) of an entire function is defined using the
limit superior
as:
where is the disk of radius and denotes the
supremum norm
of on . The order is a non-negative real number or infinity (except when for all . In other words, the order of is the
infimum
of all such that:
The example of shows that this does not mean if
is of order .
If one can also define the type:
If the order is 1 and the type is , the function is said to be "of exponential type". If it is of order less than 1 it is said to be of exponential type 0.
If
then the order and type can be found by the formulas
Let denote the -th derivative of , then we may restate these formulas in terms of the derivatives at any arbitrary point :
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. The non-negative integer is called the genus of the entire function .
If the order is not an integer, then is the integer part of . If the order is a positive integer, then there are two possibilities: or .
For example, , and are entire functions of genus .
of functions (or distributions) with bounded support are entire functions of order and finite type.
Other examples are solutions of linear differential equations with polynomial coefficients. If the coefficient at the highest derivative is constant, then all solutions of such equations are entire functions. For example, the exponential function, sine, cosine,
dynamics of entire functions
.
An entire function of the square root of a complex number is entire if the original function is
even
, for example .
If a sequence of polynomials all of whose roots are real converges in a neighborhood of the origin to a limit which is not identically equal to zero, then this limit is an entire function. Such entire functions form the Laguerre–Pólya class, which can also be characterized in terms of the Hadamard product, namely, belongs to this class if and only if in the Hadamard representation all are real, , and
, where and are real, and . For example, the sequence of polynomials
converges, as increases, to . The polynomials
have all real roots, and converge to . The polynomials
also converge to , showing the buildup of the Hadamard product for cosine.