that converges everywhere in the complex plane, hence uniformly on compact sets.
Any power series satisfying this criterion will represent an entire function.
[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
(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]} Note however that an entire function is not determined by its real part on all curves.
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.
The Weierstrass factorization theorem asserts that any entire function can be represented by a product involving its zeroes (or "roots").
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] 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 singularity at the complex point at infinity, either a pole for a polynomial or an essential singularity for a transcendental entire function.
Specifically, by the Casorati–Weierstrass theorem, for any transcendental entire function
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.
One can take a suitable branch of the logarithm of an entire function that never hits
, so that this will also be an entire function (according to the Weierstrass factorization theorem).
The order is a non-negative real number or infinity (except when
Then we may restate these formulas in terms of the derivatives at any arbitrary point
The type may be infinite, as in the case of the reciprocal gamma function, or zero (see example below under § Order 1).
Here are some examples of functions of various orders: For arbitrary positive numbers
one can construct an example of an entire function of order
According to the fundamental theorem of Paley and Wiener, Fourier transforms of functions (or distributions) with bounded support are entire functions of order
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.
The class of entire functions is closed with respect to compositions.
This makes it possible to study dynamics of entire functions.
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,