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1. Definition of Singularity
A singularity (or singular point) of a complex function f (z) is a point z 0 in the complex plane where f (z) is not analytic, but it is analytic in some neighborhood around z 0 (except possibly at z 0 ).
Actual Meaning: It is a point where the function “misbehaves” and cannot be expressed as a convergent Taylor series.
How to Check:
- Identify points where the function is undefined (denominator = 0, log(0),
0, etc.)
- Check if the function can be redefined to become analytic (removable singularity)
- Expand in Laurent series; presence of negative powers indicates non-removable singularities.
Examples:
- f (z) = (^) z−^12 , singularity at z = 2
- f (z) = sinz z, singularity at z = 0 (removable)
- f (z) = e^1 /z^ , singularity at z = 0 (essential)
Trap: Do not assume every singularity is a pole; check carefully using limits or Laurent series.
Negation: If f (z) is analytic at z 0 , it is not a singularity.
2. Types of Singularities
2.1 Removable Singularity
Definition: A singularity z 0 is removable if limz→z 0 f (z) exists and is finite.
Laurent Series: No negative powers.
Geometrical Meaning: Function has a “hole” at z 0 which can be patched.
Examples:
- f (z) = sinz z, z = 0
- f (z) = 1 − ze z, z = 0
- f (z) = z^2 sinz z, z = 0
2.2 Pole
Definition: z 0 is a pole of order m if (z − z 0 )mf (z) is analytic and non-zero at z 0.
Formula: f (z) = (^) (z−g(zz 0 ))m , g(z 0 ) ̸= 0
Geometrical Meaning: Function blows up to infinity at z 0.
Examples:
- f (z) = (^) (z−^1 1) 2 , pole of order 2 at z = 1
- f (z) = (^) (zz−+23) 3 , pole of order 3 at z = 3
- f (z) = cot z = cos sin^ zz , simple pole at z = nπ
2.3 Essential Singularity
Definition: z 0 is essential if it is neither removable nor a pole.
Laurent Series: Infinite number of negative powers.
Geometrical Meaning: Function behaves wildly; dense in complex plane (Great Picard Theorem).
Examples:
- f (z) = e^1 /z^ , z = 0
- f (z) = sin(1/z), z = 0
- f (z) = tan(1/z), z = 0
3. Connection with Laurent Series
- Laurent series around singularity z 0 :
f (z) =
X^ −^1
n=−∞
an(z − z 0 )n^ +
X^ ∞
n=
an(z − z 0 )n
- Negative powers indicate non-removable singularity
- Finite negative powers → pole
- Infinite negative powers → essential
- No negative powers → removable
- Domain/definition check: Find where f fails to be defined (denominator zero, branch cut endpoints, logarithm points, etc.). Candidate singularities are only from that set.
- Local analyticity: Check whether there exists r > 0 such that f is analytic on 0 < |z − z 0 | < r. If no such r exists (singularities accumulate at z 0 ), then z 0 is a non-isolated singularity.
- Limit test (quick): Compute ℓ = limz→z 0 f (z) if it exists.
- If ℓ exists finite and the punctured disk is analytic, z 0 is removable.
- If ℓ = ∞, suspect a pole. Determine order by checking smallest m with limz→z 0 (z − z 0 )mf (z) finite nonzero.
- If the limit does not exist or is path-dependent, proceed to Laurent analysis (possible essential).
- Laurent expansion (definitive): Expand f in a Laurent series in 0 < |z − z 0 | < R. Inspect the principal part (a− 1 (z − z 0 )−^1 + · · ·): - No negative-power terms ⇒ removable. - Finite number of negative-power terms ⇒ pole (order = highest negative ex- ponent magnitude). - Infinite number of negative-power terms ⇒ essential.
- Reciprocal-zero dual test: If easier, check zeros of g(z) = 1/f (z). Zeros ↔ poles (order preserved).
- Behavior along approaches: For suspected essential singularities, examine limits along various paths (z −z 0 ) = reiθ^ or sequences zn → z 0 to detect path-dependence.
- Singularity at infinity: Set w = 1/z. Study F (w) = f (1/w) at w = 0. Apply above tests.
Remarks (deep):
- Laurent expansion is local — its annulus of convergence matters. For classification, we only need the principal part in a punctured neighbourhood of z 0.
- Non-isolated singularities (accumulation points of singularities) break Laurent anal- ysis: no expansion exists in a punctured disk around the limit point.
- Branch points (e.g. z 0 = 0 for √z) are not isolated singularities in the sense used above — treat separately with Riemann branches.
B. Laurent series connection — rigorous statements
and consequences
Key fact (one-liner): For an isolated singularity z 0 ,
f (z) =
X^ ∞
n=−∞
an(z − z 0 )n^ (convergent in some 0 < |z − z 0 | < R).
Classification is read directly from the coefficients an:
- If a−n = 0 for all n ≥ 1, singularity is removable.
- If a−n = 0 for all n > m and a−m ̸= 0, then z 0 is a pole of order m.
- If infinitely many a−n ̸= 0 for n ≥ 1, then z 0 is essential.
Residue: Res(f ; z 0 ) = a− 1. Knowledge of the principal part gives residues directly — crucial in contour integration.
Construction remark: If f has a pole of order m at z 0 , there exists analytic g with g(z 0 ) ̸= 0 such that f (z) = g(z)/(z − z 0 )m. Expand g in Taylor series and divide term- by-term to obtain Laurent form.
C. Properties, hidden concepts, and traps (detailed)
Properties (concise)
- Poles are isolated (for meromorphic functions) but essential singularities can be isolated as well.
- For meromorphic f on a domain D, all singularities in D are poles (no essential singularities): meromorphic = analytic except for isolated poles.
- limz→z 0 |f (z)| = ∞ is a necessary but not sufficient distinguishing idea: this indi- cates a pole if the blow-up is of finite algebraic order. For essential singularities, modulus does not simply tend to infinity.
Hidden concepts & traps (exam-dangerous)
- Finite truncation trap: Seeing a few negative powers in a partial series expansion does not guarantee a pole — the Laurent series might have infinitely many negative terms you did not compute.
- Path-dependence trap: A limit that depends on the path to z 0 immediately rules out removable/pole — it suggests essentiality or non-isolated behaviour. Always test at least two different approaches (radial, spiral, along lines).
F. Long, difficult worked examples (3 per major type
+ 3 for non-isolated/infinity)
F.1 Removable singularities — 3 hard examples
Example F.1.1.
f (z) = e
z (^) − 1 − z − z^2 2 z^3
at z = 0.
Goal: Show 0 is removable and find the analytic extension.
Solution. Use the Taylor series ez^ =
P∞
n=
zn n!
. Then
ez^ − 1 − z − z 22 =
X^ ∞
n=
zn n!.
Divide by z^3 :
f (z) =
X^ ∞
n=
zn−^3 n!
X^ ∞
m=
zm (m + 3)!
This is a Taylor series (no negative powers) valid for all z. Hence z = 0 is removable and the analytic extension is
f˜ (z) =
X^ ∞
m=
zm (m + 3)!,^ f˜ (0) =^1 3!.
Key points: The use of full Taylor expansion avoids mistaken truncation; limit exists
finite: lim z→ 0 f (z) =^1 6
Example F.1.2.
g(z) = sin^ z^ −^ z^ +^
z^3 6 z^5
at z = 0.
Solution. Use sin z =
P∞
n=0(−1)n^
z^2 n+ (2n + 1)!. Subtract^ z^ −^
z^3 6 to get series starting at^ z
Precisely,
sin z − z + z 63 =
X^ ∞
n=
(−1)n^ z
2 n+ (2n + 1)!.
Divide by z^5 :
g(z) =
X^ ∞
n=
(−1)n^ z
2 n− 4 (2n + 1)!,
which is analytic at 0. Hence removable; g(0) = coefficient of z^0 term which occurs at 2 n − 4 = 0 ⇒ n = 2: g(0) = (−1)^2 5!^1 = 1201.
Example F.1.3. (Hard rational-trig combination)
h(z) = tan^ z^ −^ z^ −^
z^3 3 z^5
at z = 0.
Solution sketch. Expand tan z = z + z 33 + 215 z^5 + O(z^7 ). Substituting gives numerator 2 z^5 15 +^ O(z
(^7) ). Divide by z (^5) to get h(z) = 2 15 +^ O(z
(^2) ). Thus removable with h(0) = 2/15.
□
F.2 Poles — 3 challenging examples (with order determination
and Laurent parts)
Example F.2.1. (Pole of higher order via factorization)
f (z) = (^) (zcos − πz) 3 at z 0 = π.
Solution. Expand cos z about π: write z = π + w. Then
cos(π + w) = − cos w = −
1 − w
2 2 +^
w^4 24 − · · ·
So
f (z) =
1 − w 22 + w 244 − · · ·
w^3 =^ −^
w^3 +
w −^
w 24 +^ · · ·^. Hence z = π is a pole of order 3. Principal part is −w−^3 + 12 w−^1. Residue = a− 1 = 12.
Example F.2.2. (Rational function with cancellation — find order)
g(z) = e
z (z − 1)^4 (z − 2)
Goal: classify at z = 1 and z = 2.
Solution. At z = 1: ez^ is analytic and e^1 ̸= 0. Thus g(z) = e
(^1) + O(z − 1) (z − 1)^4 ·^
1 (1−(z−1)) — the factor (z − 2) is nonzero at 1. So z = 1 is a pole of order 4. Principal part coefficient of (z − 1)−^4 equals e · (^) −^11 times (careful expansion for exact principal part), but main classification is order 4.
At z = 2: denominator has simple zero; numerator e^2 ̸= 0 so z = 2 is a simple pole (order 1).
Example F.2.3. (Pole via reciprocal-zero test) Consider
h(z) = 1 sin^3 z
at z = 0.
We know sin z = z − z 63 + O(z^5 ). Thus
sin^3 z = z^3
1 − z 62 + O(z^4 )
= z^3
1 + O(z^2 )
So h(z) = z−^3 (1 + O(z^2 ))−^1 = z−^3 + O(z−^1 ). Therefore z = 0 is a pole of order 3. The residue is the coefficient of z−^1 in the full expansion (requires more terms if needed).
contains other singularities. Trap illustrated: Attempting to expand about 0 fails; one must treat 0 as accumulation point.
Example F.4.2. (Essential vs accumulation subtlety)
g(z) =
X^ ∞
n=
z − 1 /n.
Analysis. The function is meromorphic on C \ { 1 /n : n ∈ N} with simple poles at 1 /n, and these poles accumulate at 0. Therefore 0 is not an isolated singularity. The series defining g may diverge near large sets — the point 0 is a natural boundary for meromorphic continuation.
Example F.4.3. (Singularity at infinity) Consider the entire function f (z) = ez^. Classify the singularity at z = ∞.
Solution. Let w = 1/z. Consider F (w) = f (1/w) = exp(1/w). At w = 0, this has
Laurent expansion
P∞
n=
n!
w−n^ (infinitely many negative powers) so w = 0 is essential;
hence z = ∞ is an essential singularity of ez^. (Contrast: a polynomial has a pole at infinity of order equal to its degree.)
G. Short practice checklist (for exam use)
When asked to classify a singularity at z 0 :
- Locate where the function is undefined.
- Check whether there is a deleted neighbourhood free of other singularities (isolated).
- Try direct limit; test whether finite, infinite, or DNE.
- If limit inconclusive, compute Laurent series (or use reciprocal zero test).
- For ∞ replace z 7 → 1 /w.
- If suspected essential, try two distinct paths to detect path-dependence.
- If poles accumulate near the point, label it non-isolated.
H. Final remarks and recommended exercises
- Prove: if f is bounded in a punctured neighborhood of z 0 , then z 0 is removable (Riemann removable singularity theorem).
- Practice expansions: expand rational-trigonometric combinations to at least 5 or- ders to be safe.
- Explore Picard’s theorem examples (e.g., e^1 /z^ ) to cement intuition about essential singularities.
10 Conceptual MSQ Questions on Singularity
- Let f (z) be analytic in 0 < |z| < 1 and suppose f (z) =
P∞
n=−∞ anzn^ is its Laurent expansion around z = 0. Which of the following statements are true? (a) If a− 1 = 0 and a−n = 0 for all n > 1, then z = 0 is removable. (b) If a− 2 ̸= 0 and a−n = 0 for n > 2, then z = 0 is a pole of order 2. (c) If infinitely many a−n ̸= 0, then z = 0 is an essential singularity. (d) If f (z) has zeros accumulating at z = 0, then z = 0 is non-isolated.
- Consider f (z) = sinz 3 z. Which of the following are correct? (a) z = 0 is a pole. (b) The order of the pole is 3. (c) The principal part of Laurent expansion has terms up to 1/z^3. (d) z = 0 can be made analytic by redefining f (0).
- Let f (z) = e^1 /z^. Which statements are correct? (a) z = 0 is an essential singularity. (b) Principal part of Laurent expansion has infinitely many negative powers. (c) Zeros of f (z) accumulate at z = 0. (d) In any neighborhood of z = 0, f (z) attains all complex values except possibly one (Picard theorem).
- Suppose f (z) has a singularity at z = z 0 such that limz→z 0 (z − z 0 )f (z) = 2. Then which statements are correct? (a) z 0 is a removable singularity. (b) z 0 is a pole of order 1. (c) Laurent series around z 0 has a− 1 = 2. (d) f (z) is bounded in a neighborhood of z 0.
- Let f (z) = (^) (zz^2 −−1)^13. Which of the following are true?
(a) z = 1 is a pole. (b) The order of the pole is 3. (c) Laurent expansion around z = 1 has negative powers up to 1/(z − 1)^3. (d) z = −1 is a removable singularity.
- Consider f (z) = (^) sin(^1 πz). Which statements are correct?
(a) Poles occur at z = n, n ∈ Z. (b) All poles are simple. (c) Accumulation of poles at infinity implies ∞ is non-isolated.
(c) Let f (z) = log(sin z). Identify true statements: i. z = nπ, n ∈ Z are branch points. ii. z = nπ are isolated singularities. iii. f (z) requires branch cuts along real axis to define a single-valued branch. iv. z = π/2 is a singularity. (d) Consider f (z) = (^) z log^1 z. Determine correct statements:
i. z = 0 is a non-isolated singularity. ii. z = 0 is an essential singularity. iii. z = 1 is a regular point. iv. Laurent expansion exists in 0 < |z| < 1. (e) Let f (z) = e^1 /(z^2 −1). Which statements are correct? i. z = 1 and z = −1 are essential singularities. ii. Laurent series around z = 0 has only finitely many negative powers. iii. z = 0 is an ordinary point. iv. The function is unbounded near z = 1. (f) Consider f (z) = log((z−z2)−1) 2. Identify correct statements: i. z = 1 is a branch point. ii. z = 2 is a pole of order 2. iii. z = 0 is a removable singularity. iv. Branch cut needed from z = 1 to infinity.
(g) Let f (z) = (^1) z + (^) log^1 z. Determine correct statements: i. z = 0 is a non-isolated singularity. ii. z = 0 is an essential singularity. iii. Laurent expansion exists in punctured neighborhood of z = 0. iv. f (z) is bounded near z = 0.
(h) Consider f (z) = sin
log^1 z
. Which statements are correct? i. z = 1 is an essential singularity. ii. z = 0 is a branch point. iii. Laurent series around z = 1 has infinitely many negative powers. iv. f (z) is analytic in a neighborhood of z = 1. (i) Let f (z) = log
z− 1 z+
. Which of the following are true? i. z = 1 and z = −1 are branch points. ii. A branch cut is needed to define a single-valued function. iii. z = 0 is an isolated singularity. iv. f (z) is analytic elsewhere. (j) Consider the mapping f (z) = (^1) z + (^) z−^11. Determine true statements: i. z = 0 is a simple pole. ii. z = 1 is a simple pole. iii. z = ∞ is a removable singularity. iv. Jacobian determinant of the mapping f : C → C vanishes at z = 0.
10 Deep Conceptual MSQs on Singularities with
Analytic and Entire Functions
(a) Let f (z) be analytic in C \ { 0 } and non-constant, with |f (z)| ≤ 1 /|z| for |z| < 1. Which statements are true? i. z = 0 is a removable singularity. ii. z = 0 is a pole. iii. f (z) cannot be entire. iv. Laurent expansion around z = 0 has only finitely many negative powers. (b) Let f (z) be entire and non-constant. If f has zeros accumulating at z 0 ∈ C, which statements are correct? i. z 0 is a removable singularity. ii. z 0 must be an isolated singularity. iii. Accumulation of zeros is impossible for non-constant entire functions. iv. Identity theorem implies f ≡ 0. (c) Let f (z) be analytic in a punctured neighborhood of z = 0, non-constant, and bounded there. Which statements are true? i. z = 0 is removable. ii. z = 0 is a pole of order 1. iii. z = 0 can be an essential singularity. iv. Redefining f (0) makes f analytic at 0. (d) Suppose f (z) is entire, non-constant, and g(z) = 1/f (z). Which statements are correct? i. All zeros of f correspond to poles of g. ii. If f has infinitely many zeros accumulating at z 0 , then g has a non-isolated singularity there. iii. g is meromorphic. iv. g cannot have essential singularities in C. (e) Let f (z) be analytic on C \ { 1 } and non-constant, with |f (z)| → ∞ as z → 1. Which statements are true? i. z = 1 is a pole. ii. z = 1 is a removable singularity. iii. Laurent expansion around z = 1 has finitely many negative powers. iv. f is entire on C \ { 1 }. (f) Let f (z) be analytic in 0 < |z| < 1 and non-constant, with f (z) → 0 as z → 0. Which statements are correct? i. z = 0 is a removable singularity. ii. Laurent expansion has no negative powers. iii. z = 0 could be a pole. iv. Redefining f (0) = 0 makes f analytic at 0.
(b) Consider
f (z) = sin(1/(z^ −^ 1))^ ·^ e
1 /(z+1) z^2
, g(z) = log(f (z)), h(z) = g(z^2 − 1).
Which statements are true? i. z = 1 and z = −1 are essential singularities of h(z). ii. z = 0 is a pole of h(z). iii. h(z) has infinitely many negative powers in Laurent expansion at z = 0. iv. Branch cuts are required to make h(z) single-valued. v. h(z) is unbounded near z = 1 and z = −1. (c) Let
f (z) = e
tan(1/z) (z − 1)^2 ,^ g(z) = log(f^ (z)),^ h(z) =^ g
z + 1
· (^) z (^2 1) − 4.
Determine which statements are correct: i. z = 0 is an essential singularity of h(z). ii. z = 1 is a pole of order 2 of h(z). iii. z = −1 is an essential singularity of h(z). iv. Laurent series of h(z) around z = 0 has infinitely many negative powers. v. h(z) is unbounded near z = ±2.
(d) Consider
f (z) = log(sin(1 z − 1 /z )), g(z) = ef^ (z), h(z) = g
z^2 − 1
Which of the following are true? i. z = 0 is an essential singularity of h(z). ii. z = 1 and z = −1 are branch points of h(z). iii. h(z) requires branch cuts for single-valuedness. iv. Laurent series of h(z) around z = 0 has infinitely many negative powers. v. h(z) is unbounded near z = 0, 1 , −1. (e) Let
f (z) = e
1 /(z−1) (^) · sin(1/(z + 1)) log(z + 2) ,^ g(z) =^ f
z
· ez^ , h(z) = (^) (zg −(z 3)) 2.
Which statements are correct? i. z = 0 is an essential singularity of h(z). ii. z = 1 and z = −1 are essential singularities of h(z). iii. z = −2 is a branch point of h(z). iv. z = 3 is a pole of order 2 of h(z). v. h(z) is unbounded near all singular points.
10 Extremely Difficult MSQs on Singularities with
Composed Functions
(a) Let f (z) = e^1 /(z−1)^ · sin(1/(z + 2)) · log(z + 3). Which of the following statements are true? i. limz→ 1 (z − 1)f (z) exists; z = 1 is a pole of order 1. ii. limz→− 2 (z + 2) · f (z) does not exist; z = −2 is essential. iii. limz→− 3 (z + 3) · log(f (z)) exists; z = −3 is removable. iv. f (1/z) has an essential singularity at z = 0. (b) Consider f (z) = log((z −z^ + 1)2) 2 · e^1 /(z−3).
Which statements are correct? i. limz→ 2 (z − 2)^2 f (z) exists; z = 2 is a pole of order 2. ii. limz→− 1 (z + 1) · log(f (z)) does not exist; z = −1 is essential. iii. f (1/z) · e^1 /z^ has an essential singularity at z = 0. iv. limz→ 3 (z − 3) · f (z) does not exist; z = 3 is essential. (c) Let f (z) = sin(1 z/ (+ 2z^ − 1))· e^1 /(z+3). Which of the following are true? i. limz→ 1 (z − 1)f (z) does not exist; z = 1 is essential. ii. limz→− 2 (z + 2) · f (z) exists; z = −2 is a simple pole. iii. f (1/(z − 3)) has an essential singularity at z = 3. iv. limz→− 3 (z + 3) · log(f (z)) exists; z = −3 is removable. (d) Consider f (z) = e^1 /(z−1)^ · log(z + 2) · cos(1/(z + 3)). Which statements are correct? i. limz→ 1 (z − 1)f (z) does not exist; z = 1 is essential. ii. limz→− 2 (z + 2)f (z) exists; z = −2 is a pole. iii. f (1/z) · ez^ has an essential singularity at z = 0. iv. limz→− 3 (z + 3) · log(f (z)) does not exist; z = −3 is essential. (e) Let f (z) = log(z^ + 1)^ ·^ e
1 /(z−2) (z − 3) · sin(1/(z + 4)). Which of the following are correct? i. limz→ 2 (z − 2)f (z) does not exist; z = 2 is essential. ii. limz→ 3 (z − 3)f (z) exists; z = 3 is a simple pole. iii. f (1/z) has an essential singularity at z = 0. iv. limz→− 1 (z + 1) · log(f (z)) exists; z = −1 is removable.
iii. limz→− 3 (z + 3) · log(f (z)) exists; z = −3 is removable. iv. f (1/z) has an essential singularity at z = 0.
(b) Consider
f (z) = log((z −z^ + 1)2) 2 · e^1 /(z−3).
Which statements are true? i. limz→ 2 (z − 2)^2 f (z) exists; z = 2 is a pole of order 2. ii. limz→− 1 (z + 1) · log(f (z)) does not exist; z = −1 is essential. iii. f (1/z) · e^1 /z^ has an essential singularity at z = 0. iv. limz→ 3 (z − 3) · f (z) does not exist; z = 3 is essential. (c) Let f (z) = sin(1 z/ (+ 2z^ − 1))· e^1 /(z+3). Which statements are true? i. limz→ 1 (z − 1)f (z) does not exist; z = 1 is essential. ii. limz→− 2 (z + 2)f (z) exists; z = −2 is a simple pole. iii. f (1/(z − 3)) has an essential singularity at z = 3. iv. limz→− 3 (z + 3) · log(f (z)) exists; z = −3 is removable.
(d) Consider f (z) = e^1 /(z−1)^ · log(z + 2) · cos(1/(z + 3)). Which statements are true? i. limz→ 1 (z − 1)f (z) does not exist; z = 1 is essential. ii. limz→− 2 (z + 2)f (z) exists; z = −2 is a pole. iii. f (1/z) · ez^ has an essential singularity at z = 0. iv. limz→− 3 (z + 3) · log(f (z)) does not exist; z = −3 is essential. (e) Let f (z) = log(z^ + 1)^ ·^ e
1 /(z−2) (z − 3) · sin(1/(z + 4)). Which statements are correct? i. limz→ 2 (z − 2)f (z) does not exist; z = 2 is essential. ii. limz→ 3 (z − 3)f (z) exists; z = 3 is a simple pole. iii. f (1/z) has an essential singularity at z = 0. iv. limz→− 1 (z + 1) · log(f (z)) exists; z = −1 is removable. (f) Let f (z) = e
1 /(z−1) (^) · sin(1/(z + 2)) z + 3. Which statements are correct? i. limz→ 1 (z − 1)f (z) does not exist; z = 1 is essential. ii. limz→− 2 (z + 2)f (z) does not exist; z = −2 is essential. iii. limz→− 3 (z + 3)f (z) exists; z = −3 is a simple pole. iv. f (1/z) has an essential singularity at z = 0.
(g) Consider f (z) = log(z + 1) · e^1 /(z−2)^ · cos(1/(z + 3)). Which statements are correct? i. limz→ 2 (z − 2)f (z) does not exist; z = 2 is essential. ii. limz→− 1 (z + 1) · f (z) exists; z = −1 is a pole. iii. f (1/z) has an essential singularity at z = 0. iv. limz→− 3 (z + 3) · log(f (z)) does not exist; z = −3 is essential.
(h) Let
f (z) = sin(1/(z^ −^ 1))^ ·^ e
1 /(z+2) z − 3
Which statements are correct? i. limz→ 1 (z − 1)f (z) does not exist; z = 1 is essential. ii. limz→− 2 (z + 2)f (z) does not exist; z = −2 is essential. iii. limz→ 3 (z − 3)f (z) exists; z = 3 is a simple pole. iv. f (1/z) has an essential singularity at z = 0. (i) Consider f (z) =
log(z + 2) · e^1 /(z−1) (z − 3) · sin(1/(z + 4)). Which statements are correct? i. limz→ 1 (z − 1)f (z) does not exist; z = 1 is essential. ii. limz→ 3 (z − 3)f (z) exists; z = 3 is a simple pole. iii. limz→− 2 (z + 2) · f (z) does not exist; z = −2 is essential. iv. f (1/z) has an essential singularity at z = 0. (j) Let f (z) =
e^1 /(z−2)^ · sin(1/(z − 1)) z + 3 ·^ log(z^ + 4). Which statements are correct? i. limz→ 2 (z − 2)f (z) does not exist; z = 2 is essential. ii. limz→ 1 (z − 1)f (z) does not exist; z = 1 is essential. iii. limz→− 3 (z + 3)f (z) exists; z = −3 is a simple pole. iv. limz→− 4 (z + 4) · log(f (z)) exists; z = −4 is removable.
(k) Consider
f (z) = e^1 /(z−1)^ · (^) sin(1log(/z(^ z+ 2) − 3)).
Which statements are correct? i. limz→ 1 (z − 1)f (z) does not exist; z = 1 is essential. ii. limz→− 2 (z + 2)f (z) exists; z = −2 is a simple pole. iii. limz→ 3 (z − 3)f (z) does not exist; z = 3 is essential. iv. f (1/z) has an essential singularity at z = 0.
