The Construction of Entire and Meromorphic Functions

4.4.3 The Construction of Entire and Meromorphic Functions

It is common knowledge in algebra that any polynomial can be factored into linear factors. When can this factorization be extended to transcendental entire functions?

We will start by introducing the concept of infinite products. Let

be an infinite product. If the limit

exists and is finite, then the infinite product is said to be convergent.

For , since and , we can integrate over to get that . Therefore,

Since the convergence of is the same as that of , it follows that the convergence of is equivalent to that of . If is convergent, then is absolutely convergent. As with the order of summing an absolutely convergent series is unimportant, we may also rearrange terms in an absolutely convergent infinite product.

Similar to series, absolute convergence is a stronger condition than convergence:

Lemma 4.4.1.

An absolutely convergent infinite product is convergent.

Proof.

Let be a complex sequence such that is convergent. Then is absolutely convergent. Let denote the partial products of and let denote the partial products of . By the Cauchy Criterion (Theorem 1.2.12), we have that , such that , . Let us now analyze the absolute difference between and :

which therefore satisfies Theorem 1.2.12.

We will now provide the following assertions on the locally uniform convergence of infinite products:

Lemma 4.4.2.

Let be open and connected. Suppose uniformly converges on compact subsets of such that each is holomorphic on . Then the infinite product

is uniformly convergent on compact subsets of .

Proof.

Let be an arbitrary compact subset of . Since converges uniformly on , it follows that , such that , for all . Additionally, we have

By Theorem 4.1.1, the uniform limit is holomorphic on . By continuity and Theorem 1.2.13, this limit is bounded on . It follows that each partial sum is uniformly bounded on . Since the exponential function is Lipschitz continuous on compact subsets of , there exists a finite constant such that

Remark.

Uniform convergence on compact subsets is also known as compact convergence. In the case of (or in any topological space such that every point has a compact neighborhood), compact convergence is equivalent to locally uniform convergence.

We also have:

Lemma 4.4.3.

Let be open and connected. Suppose is uniformly convergent on compact subsets of such that each is holomorphic on . Then the infinite product

is uniformly convergent on compact subsets of to a holomorphic function, which vanishes only at a point if and only if for some . The multiplicity at each such zero is the sum of the multiplicities of at for all satisfying .

Proof.

Let be an arbitrary compact set. By the uniform convergence of on , it follows that the uniform limit is continuous by the Uniform Limit Theorem (Theorem 2.3.5). By continuity on a compact set, it follows that the limit is bounded by some constant . Additionally, , such that , . It follows that the partial sums are uniformly bounded on by

Similarly, by earlier discussion of infinite products, we have

or in other words, the partial products are uniformly bounded on . Let be arbitrary. By definition, there exists such that , for all . The difference between the non-absolute partial products satisfies

where the second inequality can be easily verified by expanding the product and the triangle inequality.

Since , it follows that

is uniformly convergent on . Let be a point satisfying . Since there exists an such that

is non-vanishing at , and from the fact that

we can analyze the zeros of the finite product to obtain the conclusion.

We will now study the construction of an entire function via its zeros. We have the following cases:

If has no zeros in , then the function defined by is entire, so it is the derivative of an entire function . Therefore, the function defined by has the vanishing derivative

It follows that is constant, and therefore for some constant . Since is entire, it follows that is also entire. Absorb the constant into , and .
If is entire and has finitely many zeros in , namely with respective multiplicities (if is not a zero, treat ), then at each zero , it has the local Taylor expansion

where . Therefore, we can divide by to obtain a new entire function with no additional zeros and no zero at . Repeating this for every zero, we can define

which is entire and has no zeros, where

We write in the above form rather than that of as we aim to generalize the construction to infinite products to study convergence. By the non-vanishing case above, for some entire function . Therefore, we can write

where is a polynomial with zeros at with respective multiplicities . The entire functions and both have the same zeros with matching multiplicities.
If is entire and has infinitely many zeros such that is not identically zero, then it follows that has countably many zeros (since the zeros are isolated). Let the zeros be indexed by , namely . Without loss of generality, assume that , (repeated elements representing multiplicities), and . The case for a zero at will be treated differently.

There exists a positive integer sequence such that for every positive and finite , converges. For example, let , and for sufficiently large , and the series is convergent. Consider the infinite product

Let

Therefore, we can rewrite (4.4.2) as

The expression in (4.4.3) is known as the -th Weierstrass elementary factor.

By , for a fixed , such that , . Consider the product . For and , we have . The Taylor expansion

has a convergence disk of . Then,

By the definition of , the series converges. Therefore, is uniformly and absolutely convergent on by the Weierstrass -Test (Theorem 2.3.2). We then get that

and it uniformly converges on to a nonzero holomorphic function on by Lemma 4.4.2, Theorem 4.1.1, and Theorem 3.3.5.

The zeros of

are and lie in . To prove the absolute convergence of on , we will show that is convergent. Trivially, when , we have

By (4.4.4) above, we get that when .

Therefore, we have

which has a convergent series by definition. Therefore, is absolutely convergent on . Letting , we obtain the following result:

Theorem 4.4.6 (Weierstrass Product Theorem).

Let be a sequence of nonzero complex numbers satisfying as and (equality of and treated as multiplicities) for all . Then there exists a sequence of nonnegative integers such that , converges. For such a prescribed sequence, the function

defines an entire function with zeros at each element of the sequence of multiplicities matching the number of times an element is repeated. Moreover, the product converges uniformly on any compact disk .

The following result is then apparent:

Theorem 4.4.7 (Weierstrass Factorization Theorem).

Suppose is an entire function. Let be the sequence of all nonzero zeros of satisfying as and (equality of and treated as multiplicities) for all . Let be the multiplicity of at (let if there is no zero at ). Then there exists a sequence of nonnegative integers such that , converges. Then, we can write

on , where is the -th Weierstrass elementary factor defined in (4.4.3) and is an entire function. The infinite product converges uniformly on and converges absolutely on . If we let , we can write

Proof.

By the Weierstrass Product Theorem, construct to be entire and have zeros at . Thus, and have the same zeros and corresponding multiplicities. Then the function

has removable singularities on all of and has an analytic continuation (Theorem 3.2.6) to an entire and non-vanishing function. Therefore, it can be written as

where is entire. Let , and from rearrangement, we obtain (4.4.6).

Corollary 4.4.1.

Let be meromorphic on . Then can be written as the quotient of two entire functions.

Proof.

Let be any entire function with zeros only at each pole of (with multiplicities matching the order of each pole). If there are infinitely many poles, we can explicitly construct such a by the Weierstrass Product Theorem (Theorem 4.4.6). If there are finitely many poles, construct using (4.4.1). It follows that can be analytically continued on its removable singularities to an entire function with the same zeros as . Hence,

which is an explicit construction.

Therefore, any meromorphic function on can be expressed as the quotient of two infinite products. Hence, any meromorphic function on can be explicitly written in terms of its zeros and poles.

We will now study the construction of meromorphic functions from their poles and the principal parts of their Laurent expansions at each pole.

Suppose and is a sequence of distinct values. Let be a collection of functions in the form

where are finite integer constants and are complex constants.

Suppose that is meromorphic on such that has finitely many poles. Therefore, has an isolated singularity at . We have two cases:

If is a removable singularity or a pole, by the given proof of Theorem 4.3.1, we may construct to have poles at each of such that the principal parts of at each of are . It can be explicitly written as

(we can absorb the constant into the polynomial as used in the proof).
In the case that is a transcendental meromorphic function with an isolated essential singularity at , notice that the function defined by

has removable singularities at each of . Indeed, since the singularities are isolated, for a fixed , such that for any , . It follows that is holomorphic on . Notice that is the holomorphic part of the Laurent expansion at and is also holomorphic on the disk. Suppose has as the principal part of its Laurent expansion at . Then is holomorphic on . Since was arbitrarily chosen, is entire and transcendental.

Therefore, can be constructed by

for a transcendental entire function .
The existence of a transcendental meromorphic function whose poles have an accumulation point at is the concern of the following theorem:

Theorem 4.4.8 (Mittag–Leffler).

Let be a sequence of distinct complex numbers such that , and . Let be a function sequence, each in the form of (4.4.7). Then the following hold.

First, a meromorphic function on can be constructed such that , has a pole at with principal part at .

Second, the function satisfying the criteria above can be explicitly represented as

for some sequence of polynomials and an arbitrary entire function .

Proof.

The classical proof for this theorem allows for a more explicit construction, as in (4.4.8). As for the existence statement, we can prove the first assertion by use of the -problem.

Fix , and let be an open neighborhood of such that where , . Let be a neighborhood of that is relatively compact in . By Theorem 3.2.12, for each , there is a function satisfying

Let

which is an element of . For a fixed , it is true that on . Hence, although is not meromorphic, it does have the required principal part near each . Let

Since and is on and vanishes on , . By the discussion preceding Theorem 3.1.6, there exists a function such that on . Since is , it follows that is also . Define . Then

which implies that is holomorphic on . Since has the desired principal part at each and is (and hence removable at each singularity), it follows that is meromorphic on with principal parts at each , as desired.

Let be a positive sequence such that is convergent. Without loss of generality, let (if is not a pole, set ). Choose (this can actually be any arbitrary polynomial). Fix . Since is a polynomial in terms of and has its only pole at , is holomorphic on and can be written as

By Theorem 2.3.3, this series is uniformly convergent on . Hence, such that

Let

Fix and let depend on such that for all and for all . Therefore, , . Then , we have

By the convergence of , by the Weierstrass -Test (Theorem 2.3.2), the series

converges uniformly on . Since when , the pole of , namely , is not in when . By Theorem 4.1.1, (4.4.9) is holomorphic on . Let

The poles of in are all of the with corresponding principal parts , where and . Since was arbitrarily chosen, has poles at each with the corresponding principal part on . Let be analytically continued onto each of . Then is an entire function (since the Laurent expansions of at each of vanish). By rearrangement, we obtain our desired result.

The Mittag–Leffler Theorem (Theorem 4.4.8) can also be generalized as follows:

Theorem 4.4.9.

Let be an open set with a simple closed boundary and let be a sequence of distinct complex numbers whose accumulation points lie on . Let be a sequence of functions in the form of (4.4.7). Then there exists a meromorphic function with poles at each with principal parts at each .

Indeed, since , each is not an accumulation point of . In other words, for each , there exist neighborhoods of that are relatively compact in with disjoint closures. The proceeding proof is analogous to that of the existence part in Theorem 4.4.8.

Finally, we will examine the construction of entire functions interpolating prescribed values and derivatives at given points.

Let be a sequence of distinct complex numbers and let be a sequence of complex numbers. We can then construct a polynomial such that , . One such explicit formula is given by the Lagrange interpolation formula:

Then, following the assumption that is a sequence of distinct complex numbers, let be a sequence where . Then we can find a polynomial such that , , (for clarity’s sake, selects the pair and selects the order of the derivative, whose upper bound varies for each ). Oftentimes, the factorial coefficient is absorbed into .

As it turns out, an entire function can in fact be constructed for infinitely many interpolation points, or when .

Theorem 4.4.10.

Let be a discrete set and let be a sequence where . Then there exists an entire function such that , ,

In other words, an entire function can be constructed by the given first coefficients of the Taylor expansion at each .

Proof.

According to the Weierstrass Product Theorem (Theorem 4.4.6), we can construct an entire function with zeros at each of with corresponding multiplicities . By the discreteness of , there exists a corresponding sequence of radii such that each is disjoint.

Define a complex function sequence by

where . By Theorem 3.2.12, we can construct a sequence of functions such that , , on , and on .

Let , and construct

Under what conditions on will be entire? Since the supports of each are disjoint, the summation contains at most one nonzero term and is convergent and well-defined. To construct to be entire, we must have . In other words, we want

on all of . Let

Since on , on . Consequently, on .

From rearrangement, we have

which has removable singularities at each . Define at . Under this assertion, we have . Since the support of is the union of disjoint compact sets, by Theorem 3.1.6, there exists a function satisfying

Since vanishes on , it follows that vanishes on , and is holomorphic on .

Fix and let . For , from (4.4.11), we have

Since has a zero at with multiplicity , vanishes at with multiplicity at least . Therefore, we have

as desired.

Remark.

For a general power series, there is no assurance that it corresponds to the Taylor expansion of an entire function. However, for any polynomial of degree , there always exists an entire function whose Taylor expansion agrees with the polynomial up to the first terms, which is the fundamental difference between a polynomial and a transcendental entire function.

Example 4.4.1.

Prove the pole expansion formula

for .

Proof.

Let the simple poles of at each integer be enumerated by