Entire function

From Wikipedia, the free encyclopedia

In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic over the whole complex plane. Typical examples of entire functions are the polynomials, the exponential function, and sums, products and compositions of these. Every entire function can be represented as a power series which converges compactly. Neither the natural logarithm nor the square root functions can be continued to an entire function.

Liouville's theorem establishes an important property of entire functions—an entire function which is bounded must be constant. As a consequence, a (complex-valued) function which is entire on the whole Riemann sphere (complex plane and the point at infinity) is constant. Thus a (non-constant) entire function must have a singularity at the complex point at infinity, either a pole or an essential singularity (see Liouville's theorem below). In the latter case, it is called a transcendental entire function, otherwise it is a polynomial.

Liouville's theorem may also be used to elegantly prove the fundamental theorem of algebra. Picard's little theorem is a considerable strengthening of Liouville's theorem: a non-constant entire function takes on every complex number as value, except possibly one. The latter exception is illustrated by the exponential function, which never takes on the value 0.

J. E. Littlewood chose the Weierstrass sigma function as a 'typical' entire function in one of his books.

[edit] The order of an entire function, and its growth

The order of an entire function $f (z)$ is defined using the limit superior as:

$\rho=\limsup_{r\rightarrow\infty}\frac{\ln(\ln(M(r)))}{\ln(r)},$

where $r$ is the distance from $0$ and $M (r)$ is the maximum absolute value of $f (z)$ when $\left|z\right| = r.$ If $0<\rho<\infty,$ one can also define the type:

$\sigma=\limsup_{r\rightarrow\infty}\frac{\ln(M(r))}{r^\rho}.$

In other words, the order of $f (z)$ is the infimum of all M such that $f(|z|)=O(\exp\left(|z|^M)\right)$ as $z\to\infty$ . However, the order need not to be finite; moreover, entire functions may grow as fast as any increasing function. Precisely, for every increasing function $g:[0,\infty)\to\R$ there exists an entire function $f (z)$ such that $f (x) > g ( | x | )$ for all real $x$ . Such a function $g$ may be easily found of the form:

$g(z)=\sum_{k=1}^{\infty}\left(\frac{z}{k}\right)^{n_k}$ ,

for a conveniently choosen strictly increasing sequence of positive integers $n k$ (any such sequence defines an entire series g(z); and if it is conveniently choosen, the inequality $f (x) > g ( | x | )$ also holds, for all real $x$ ).

[edit] See also

[edit] References

Ralph P. Boas (1954). Entire Functions. Academic Press. OCLC 847696.

Entire function

From Wikipedia, the free encyclopedia

[edit] The order of an entire function, and its growth

[edit] See also

[edit] References

Views

Personal tools

Navigation

Search

Interaction

Toolbox

Languages