4.3 Entireness and Meromorphy
We have previously defined the concept of an entire function in the chapter on complex differentiation. Let be entire with the unique Taylor expansion . Since is an isolated singularity, by the uniqueness of the Laurent expansion, the expansion at has the same form as the expansion at . We will now analyze the implications on the entire function given an isolated singularity.
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If the infinity point is a removable singularity, then exists and is finite.
Proposition 4.3.1.
If is entire and has a removable singularity at , then is constant.
Proof.
Let , and let , which has a removable singularity at . By Theorem 3.2.6, can be analytically continued to all of , especially at . Let . Then, , such that , . It follows that , , and is bounded. For the complement, , is continuous on a compact set, and by Theorem 1.2.13, is also bounded.
Then by Liouville’s Theorem (Theorem 3.2.3), is constant.
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If has a pole at of order , then is a polynomial of degree .
Proof.
By the classification of a pole at , can be written as
Since is entire, it is holomorphic at and has a convergent Taylor expansion. By the uniqueness of Laurent expansions (Theorem 4.1.2), the two expansions are equivalent and therefore all terms with negative exponents vanish, and
and since , the statement is confirmed.
- If has an essential singularity at , is known as a transcendental entire function.
Example 4.3.1.
The entire functions , , , , and are transcendental.
Definition 4.3.1 (Meromorphy).
Let be open, and let be a set of isolated points. Suppose is holomorphic and has a pole at each of . Then is meromorphic in .
Similar to holomorphy, meromorphy on a compact set can be defined as meromorphy on a neighborhood of the set. In general, we imply for the set to be open unless stated otherwise. If the set is not implicitly specified, we assume meromorphy on .
All holomorphic functions are meromorphic functions (with poles on ). Consequently, entire functions are meromorphic on . All rational functions (including polynomials) are also meromorphic on . In the study of meromorphic functions with an isolated singularity at , rational functions are of important interest.
Let be rational, written as , where and are polynomials. Let
where . Trivially, the poles of are the zeros of . Since
we have
Conversely, we have:
Theorem 4.3.1.
If is meromorphic on and has a pole or removable singularity at , then is a rational function.
Proof.
Since is meromorphic on , its singularities are isolated poles. The assumption that has either a pole or a removable singularity at implies that this singularity is also isolated. Thus, there exists some such that is holomorphic on the punctured neighborhood of .
Consider the Laurent expansion of at , obtained by substituting and expanding around :
where the series converges for sufficiently large . If is a removable singularity, the coefficients for all . If is a pole of order , then for all , and . In either case, the principal part at is
which is a polynomial (identically zero if degree is ).
Next, observe that has only finitely many poles in the closed disk . Suppose otherwise. Then the set of poles in would be infinite. By Bolzano–Weierstrass (Theorem 1.1.2), this set would have an accumulation point in . At such an accumulation point, would have a non-isolated singularity, a contradiction of the meromorphy of on .
Let denote these finitely many poles in . For each , the Laurent expansion of at has principal part
where is the order of the pole at . Define the auxiliary function
which is meromorphic on , with potential singularities only at and .
We now show that each of these singularities is removable. First, fix arbitrarily. Since the poles are isolated, there exists such that the punctured disk contains no other poles for .
- Since is the holomorphic part of the Laurent expansion at , it is holomorphic on (including at ).
- is holomorphic on , as each has its singularity elsewhere.
- is a polynomial, hence entire.
Thus,
is holomorphic on , including at . Therefore, we can define to make holomorphic at .
Since is the holomorphic part of the expansion at , consisting of terms with nonpositive powers of , exists and is finite. Additionally, each consists of negative powers of , so for each , and thus . Therefore, exists and is finite, so is a removable singularity of . Without the finite singularities at each , is entire. Since has a finite limit at , it is bounded on . By Liouville’s theorem, for some constant .
Hence,
The right-hand side is a sum of a constant, a polynomial, and finitely many principal parts (each a rational function with a single pole), so is rational.
If is not a pole or removable singularity of a meromorphic function , then it is either an essential singularity or an accumulation point of poles. In this case, is not rational and is known as a transcendental meromorphic function.