Whole function

An entire function is a function regular in the entire complex plane . A typical example of an entire function is a polynomial or exponent , as well as the sums, products, and superpositions of these functions. The Taylor series of an entire function converges in the entire plane of a complex variable. Logarithm, square root are not entire functions.

Note that an entire function can have a singularity (including even an essential singularity ) at infinity. As follows from the Liouville theorem , a function that does not have singular points on the entire extended complex plane must be constant (this property can be used to elegantly prove the main theorem of algebra ).

An entire function having a pole at infinity must be a polynomial. Thus, all entire functions that are not polynomials (in particular, identically constant) have an essentially singular point at infinity. Such functions are called transcendental entire functions.

Picard's small theorem significantly strengthens Liouville's theorem: an entire function that is not identically constant and takes all complex values, except, possibly, one. An example is an exponential function that takes all complex numbers except zero as values.

J. Littlewood, in one of his books, indicates the Weierstrass sigma function as a “typical” example of an entire function.

Content

1 The case of several complex variables
2 Decomposition into an infinite product
3 Space of entire functions
4 Order of an entire function
5 Type of whole function
6 Entire function of exponential type
- 6.1 Borel Associated Function

The case of several complex variables

The whole function can be seen in $\mathbb {C} ^{n}$ ${\ displaystyle \ mathbb {C} ^ {n}}$ . let be $k$ ${\ displaystyle k}$ - multi-index, $z\in \mathbb {C} ^{n}$ ${\ displaystyle z \ in \ mathbb {C} ^ {n}}$

The concept of series convergence

$\sum _{|k|=0}^{\infty }a_{k}z^{k}(*)$ ${\ displaystyle \ sum _ {| k | = 0} ^ {\ infty} a_ {k} z ^ {k} (*)}$

depends on the method of numbering the members, therefore, speaking of the convergence of this series, we mean absolute convergence : $\sum _{|k|=0}^{\infty }|a_{k}||z^{k}|<\infty$ ${\ displaystyle \ sum _ {| k | = 0} ^ {\ infty} | a_ {k} || z ^ {k} | <\ infty}$

Thus, if the series (*) converges in $\mathbb {C} ^{n}$ ${\ displaystyle \ mathbb {C} ^ {n}}$ , then the function represented by this series is called the whole.

Decomposition into an Endless Product

Just as meromorphic functions can be considered as generalizations of rational fractions, entire functions can be considered as generalizations of polynomials. In particular, if for meromorphic functions it is possible to generalize the decomposition into simple fractions ( the Mittag-Leffler theorem on the decomposition of a meromorphic function ), then for entire functions there is a generalization of factorization - the Weierstrass theorem on entire functions .

The space of entire functions

All entire functions form a linear space . The space of entire functions is denoted as $E$ ${\ displaystyle E}$ (from the word entire ) and $E_{n}$ ${\ displaystyle E_ {n}}$ for case $C^{n}$ ${\ displaystyle C ^ {n}}$ .

(In newer literature, the space of entire functions is denoted by $H$ ${\ displaystyle H}$ )

The order of an entire function

Let be $M(r)=\max _{|z|=r}\left|f(z)\right|$ ${\ displaystyle M (r) = \ max _ {| z | = r} \ left | f (z) \ right |}$

Whole function $f(x)$ ${\ displaystyle f (x)}$ called an entire function of finite order if exists $\mu >0$ ${\ displaystyle \ mu> 0}$ such that the asymptotic inequality holds $M(r)<\exp(r^{\mu })$ ${\ displaystyle M (r) <\ exp (r ^ {\ mu})}$ (*)

The order of the whole function $f(z)$ ${\ displaystyle f (z)}$ Is a number $\rho \geq 0:$ ${\ displaystyle \ rho \ geq 0:}$ $\rho =\inf \left\{\mu \right\}$ ${\ displaystyle \ rho = \ inf \ left \ {\ mu \ right \}}$

For an entire function with finite order $p$ ${\ displaystyle p}$ and come from $q$ ${\ displaystyle q}$ The following relation holds: $p\leq q\leq p+1$ ${\ displaystyle p \ leq q \ leq p + 1}$ . In fact, the finiteness of one of the characteristics implies the finiteness of the second.

Type of whole function

Whole function $f(z)$ ${\ displaystyle f (z)}$ has a finite type in order $\rho$ ${\ displaystyle \ rho}$ , if $\exists a>0$ ${\ displaystyle \ exists a> 0}$ , what

M(r)<_{ac.}e^{ar^{\rho }}

{\ displaystyle M (r) <_ {ac.} e ^ {ar ^ {\ rho}}}

Type of whole function $f(z)$ ${\ displaystyle f (z)}$ under the order $\rho$ ${\ displaystyle \ rho}$ is a number $\sigma \geq 0$ ${\ displaystyle \ sigma \ geq 0}$ :

\sigma =\inf \left\{a>0:M(r)<_{ac.}e^{ar^{\rho }}\right\}

{\ displaystyle \ sigma = \ inf \ left \ {a> 0: M (r) <_ {ac.} e ^ {ar ^ {\ rho}} \ right \}}

from the definition it follows that:

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

{\ displaystyle \ sigma = \ limsup _ {r \ rightarrow \ infty} {\ frac {\ ln (M (r))} {r ^ {\ rho}}}.}

If for a given $0<\rho <\infty$ ${\ displaystyle 0 <\ rho <\ infty}$ type of $f(z)$ ${\ displaystyle f (z)}$ endless then say that $f(z)$ ${\ displaystyle f (z)}$ maximum type.
If $0<\sigma <\infty$ ${\ displaystyle 0 <\ sigma <\ infty}$ then $f(z)$ ${\ displaystyle f (z)}$ - normal type.
If $\sigma =0$ ${\ displaystyle \ sigma = 0}$ then $f(z)$ ${\ displaystyle f (z)}$ - minimum type.

An entire function of exponential type

Whole order function $\rho =1$ ${\ displaystyle \ rho = 1}$ and of normal type is called an entire function of exponential type.

Space C.E.T.E. often referred to as $P$ ${\ displaystyle P}$ .

Borel Associated Function

Let C.E.F.E.T. is represented as:

f(z)=\sum _{k=0}^{\infty }{\frac {a_{k}}{k!}}z^{k}

{\ displaystyle f (z) = \ sum _ {k = 0} ^ {\ infty} {\ frac {a_ {k}} {k!}} z ^ {k}}

To each C.E. the function is mapped:

\gamma (t)=\sum _{k=0}^{\infty }{\frac {a_{k}}{t^{k+1}}}

{\ displaystyle \ gamma (t) = \ sum _ {k = 0} ^ {\ infty} {\ frac {a_ {k}} {t ^ {k + 1}}}}

the function $\gamma (z)$ ${\ displaystyle \ gamma (z)}$ called Borel associate. This series converges at $|t|>\sigma$ ${\ displaystyle | t |> \ sigma}$ , and at the boundary there is at least one feature of the function $\gamma (t)$ ${\ displaystyle \ gamma (t)}$