Questions: Classification of Isolated Singularities

Question 1

The function f(z) = (1 - cos z)/z² near z = 0 has what type of singularity?

• A. A simple pole, because the denominator has a zero of order 2 and cos(0) = 1 makes the numerator vanish once
• B. A pole of order 2, because the denominator is z²
• C. A removable singularity, because expanding 1 - cos z = z²/2 - z⁴/24 + ... makes the Laurent series have no negative powers
• D. An essential singularity, because cos z involves an infinite series

Answer: (C)

Expanding: 1 - cos z = z²/2 - z⁴/24 + ..., so (1 - cos z)/z² = 1/2 - z²/24 + ... This is a pure power series with no negative powers of z, so the singularity is removable — f extends analytically to z = 0 by setting f(0) = 1/2. The tempting wrong answer is A or B: students often see z² in the denominator and conclude 'pole of order 2,' but the numerator vanishes to the same order, canceling the negative powers. Classification depends on the Laurent expansion, not just the denominator.

Question 2

Near z = 0, the function e^(1/z) takes values arbitrarily close to every nonzero complex number in every punctured neighborhood of 0. What type of singularity does e^(1/z) have at z = 0, and which theorem guarantees this dense-range behavior?

• A. A pole of infinite order; the Residue Theorem guarantees the behavior
• B. An essential singularity; the Casorati-Weierstrass theorem guarantees this dense-range behavior
• C. A removable singularity; the behavior follows from the analytic continuation principle
• D. A simple pole; the behavior reflects the 1/z term dominating the Laurent expansion

Answer: (B)

e^(1/z) = 1 + 1/z + 1/(2z²) + ... has infinitely many negative-power terms, making z = 0 an essential singularity. The Casorati-Weierstrass theorem states that near an essential singularity, f comes arbitrarily close to every complex value — the image of any punctured neighborhood is dense in ℂ. The stronger Picard Great Theorem says f actually achieves every value with at most one exception. No other singularity type has this wild behavior: removable singularities have finite limits, and poles blow up to ∞ in a controlled way.

Question 3

If (z - z₀)² · f(z) has a finite nonzero limit as z → z₀, then f has a pole of order exactly 2 at z₀.

• T. True
• F. False

Answer: True

This is the standard detection criterion for poles. A pole of order m at z₀ means the Laurent expansion of f has leading term a₋ₘ/(z - z₀)^m with a₋ₘ ≠ 0. Multiplying by (z - z₀)^m cancels the leading singularity, giving a function with a nonzero, finite limit. So (z - z₀)^m · f(z) → a₋ₘ ≠ 0 as z → z₀ iff f has a pole of exactly order m. If the limit were zero, the pole would be of lower order; if the limit were infinite, the order would be higher.

Question 4

Near an essential singularity z₀, the modulus |f(z)| is expected to tend to infinity as z → z₀, distinguishing it from a removable singularity where |f(z)| stays bounded.

• T. True
• F. False

Answer: False

This is false — and it is a common misconception. Near an essential singularity, |f(z)| does NOT tend to infinity along every path. By Casorati-Weierstrass, f(z) comes arbitrarily close to every complex value, including 0, so |f(z)| oscillates wildly and can be made close to 0 or any other value. For poles, |f(z)| → ∞ in a controlled way. For removable singularities, f(z) approaches a finite limit. Essential singularities are precisely the case where none of these controlled behaviors apply.

Question 5

Why is the principal part of the Laurent expansion — rather than, say, the behavior of |f(z)| as z → z₀ — the correct basis for classifying isolated singularities?

Model answer: The principal part (the negative-power terms in the Laurent expansion) is algebraically precise and directly determines every aspect of the singularity: zero negative-power terms means analytic extension is possible (removable); finitely many means controlled blow-up (pole); infinitely many means chaotic behavior (essential). Behavioral descriptions like |f(z)| → ∞ are consequences of the Laurent structure, not the cause, and they are ambiguous: knowing |f(z)| → ∞ tells you f has a pole but not its order; knowing the principal part tells you everything needed to compute residues and apply the Residue Theorem.

The Laurent expansion is also the practical tool: to classify, expand and count negative-power terms. This is why Laurent series are prerequisite to singularity classification — the expansion IS the classification. Behavioral tests (limit tests for removable singularities, the pole-order test) are shortcuts derived from the Laurent structure, and they work because the Laurent expansion is the ground truth.