WeierstrassDecomposition.htm

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Weierstraß decomposition of entire functions

Functions holomorphic in the entire open plane are called entire functions. If the function is not constant then Liouville’s theorem implies that the point at infinity is a singular point.

As a holomorphic function every entire function can be represented by a power series

(1)

having an infinite radius of convergence.

The polynomials which form a special and important class of entire functions, can be characterized as those entire function which have at most a pole as a singularity at infinity. Entire functions which are not polynomials are called entire transcendental functions.

If is an entire function, then it can have generally an infinite number of roots.If an entire function has infinitely many zeros and is not identically vanishing, then its roots can be arranged in a sequence tending to infinity.

Given a finite set of non-zero complex numbers , , ..., , then we can easily construct a polynomial having these numbers for zeros:

(2)

where is a constant (by the way, ).

Now, given an arbitrary infinite number of non-zero complex numbers , , ..., , ... tending to infinity we can use the idea of this construction if the series is convergent. Then the function

(3)

is entire and vanishing at these prescribed points. If the numbers , , ..., , ... tend to ∞ so slowly that the series is divergent, then Weierstraß invented the following trick: Provide each factor in (3) with an additional factor which makes the product convergent but does not introduce new roots.

From this reason he introduced what is now called the Weierstraß primary factor

(4)

where .The following result proved by Weierstraß plays a fundamental role in the theory of entire functions.

Theorem. If , , ..., , ... is an arbitrary sequence of non-zero complex numbers tending to ∞, and is any positive integer, then there exists an entire function having roots at points , , ..., , ..., a root of multiplicity at the point 0, and otherwise non-vanishing. Moreover, if is an arbitrary sequence of positive integers such that the series

(5)

is almost uniformly convergent in the whole open plane, then such a function can be defined by an absolutely convergent product

$z^k Underoverscript[∏, n = 1, arg3] (1 - z/z _ n) e^{z/z _ n + 1/2 (z/z _ n)^2 + ... + 1/λ _ n (z/z _ n)^λ _ n} .$

(6)

If is an arbitrary nowhere vanishing entire function, then the function is also entire and has the same roots as the entire function . On the other hand, every entire function , everywhere different from zero, can be expressed in the form , where is also an entire function. Therefore we get

Corollary. If is an entire function having a -tuple root at the point 0, and , , ..., , ...is the sequence of roots different from zero, of function , then

$F(z) = e^h(z) z^k Underoverscript[∏, n = 1, arg3] (1 - z/z _ n) e^{z/z _ n + 1/2 (z/z _ n)^2 + ... + 1/λ _ n (z/z _ n)^λ _ n} ,$

(7)

where is an entire function, and the positive integers , , ... have the property that the series in (5) is almost uniformly convergent in the open plane. The product (7) is absolutely and almost uniformly convergent in the open plane, and consequently its value does not depend on the order of factors.

Since the sequence can be chosen in various ways, the above representation is not unique. Of particular importance is the case when we can take for λ‘s the same number, for instance if the series

(8)

is convergent.

Example 1. The entire function has simple roots at the points For the non-zero terms of this sequence the series (8) is convergent for we get

sin z = e^h(z) z Underoverscript[∏, n = 1, arg3] (1 - z/nπ) e^z/nπ . Underoverscri ... + z/nπ) e^(-z/nπ) = e^h(z) z Underoverscript[∏, n = 1, arg3] (1 - z^2/(n^2 π^2)) .

It can be shown that identically.

For we get Wallis’s formula from this expansion

(9)

Example 2. Construct an entire function having simple zeros at the points , and not vanishing anywhere except at these points.
Since we can take we get

F(z) = e^h(z) z Underoverscript[∏, n = 1, arg3] (1 + z/n)^e^(-z/n) .

If we take , where denotes the Euler-Mascheroni constant then we get the reciprocal of the Gamma function. Taking into account that

$FormBox[RowBox[{γ, , =, , RowBox[{lim _ (n -> ∞), , RowBox[{(Underoverscript[∑, k = 1, arg3] 1/k - ln n), Cell[]}]}]}], TraditionalForm]$

we get

1/Γ(z) = lim _ (n -> ∞) e^( (Underoverscript[∑, k = 1, arg3] 1/k - ln n) z ... eroverscript[∏, k = 1, arg3] ((z + k)/k) . e^(-z Underoverscript[∑, k = 1, arg3] 1/k),

from which we immediately get Gauß formula

Example 3. We also have

cos z = Underoverscript[∏, n = 1, arg3] (1 - ((2 z)/((2 n + 1) π))^2) ,

sinh z = z Underoverscript[∏, n = 1, arg3] (1 + z^2/(n^2 π^2)) ,

cosh z = Underoverscript[∏, n = 1, arg3] (1 + ((2 z)/((2 n + 1) π))^2) ,

e^z - 1 = ze^(z/2) Underoverscript[∏, n = 1, arg3] (1 + z^2/(4 n^2 π^2)) ,

a^az - e^bz = (a - b) z e^(1/2 (a + b) z) Underoverscript[∏, n = 1, arg3] (1 + ((a - b)^2 z^2)/(4 n^2 π^2)) .

Cite this web-page as:

Štefan Porubský: Weierstraß decomposition of entire functions.

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