Taylor Series, Laurent Series and Residues - One Line Questions
1.
What is the residue of f(z) = cos(z) / z³ at z = 0? —
-1/2
2.
What is the residue of f(z) = 1 / (z(z-2)²) at z = 2? —
-1/4
3.
If a function f(z) is analytic in a simply connected domain D, and C is any simple closed contour in D, then by Cauchy's Integral Theorem: —
∫_C f(z) dz = 0
4.
For a pole z₀ of order m > 1, the residue can be calculated using the formula: Res(f, z₀) = (1 / (m-1)!) * lim_(z→z₀) d^(m-1)/dz^(m-1) [ (z - z₀)ᵐ f(z) ]. What is the value of m for a simple pole? —
1
5.
If a function f(z) is analytic inside and on a simple closed contour C, except for a finite number of isolated singularities inside C, the Residue Theorem is applicable. What is the value of the integral ∫_C f(z) dz in this case? —
2πi times the sum of residues inside C
6.
Consider the function f(z) = 1 / (z - a). What is the residue of f(z) at z = a? —
1
7.
The Taylor series expansion for e^z about z=0 is Σ[n=0 to ∞] zⁿ/n!. What is the radius of convergence for this series? —
∞
8.
What is the residue of f(z) = e^z / z at z = 0? —
1
9.
The Taylor series of f(z) = 1/(1-z) about z=0 is Σ zⁿ. What is the radius of convergence? —
1
10.
If f(z) has a pole of order m at z₀, then (z - z₀)ᵐ f(z) has a removable singularity at z₀. What is the limit of (z - z₀)ᵐ f(z) as z approaches z₀? —
A finite non-zero value
11.
What is the residue of f(z) = z e^(1/z) at z = 0? —
1/2
12.
If f(z) has a removable singularity at z₀, what is its residue at z₀? —
0
13.
What is the order of the pole at z=0 for the function f(z) = 1/z²? —
2
14.
What is the residue of f(z) = z² / (z-1)³ at z = 1? —
3
15.
What is the residue of f(z) = 1 / (z² + 1) at z = i? —
-1/(2i)
16.
What is the residue of f(z) = sin(z) / z³ at z = 0? —
-1/2
17.
What is the residue of f(z) = z / (z - 1)(z + 2) at z = 1? —
2/3
18.
What is the principal part of the Laurent series of f(z) = 1/sin(z) around z=0? —
1/z + z/6 + ...
19.
If f(z) has an essential singularity at z₀, the Laurent series expansion about z₀ has: —
An infinite number of negative power terms
20.
If a function f(z) has a pole of order m at z₀, its Laurent series expansion in the punctured neighborhood 0 < |z - z₀| < R will have: —
A finite number of terms with negative powers, the highest being m
21.
The coefficients c_n in the Laurent series expansion of f(z) about z₀ in an annulus 0 < |z - z₀| < R are given by the integral formula: c_n = (1/2πi) ∫_C f(ζ) / (ζ - z₀)ⁿ⁺¹ dζ. What is C in this formula? —
A simple closed contour in the annulus
22.
Consider the integral ∫_(-∞ to ∞) dx / (x² + 1). Which contour is typically used to evaluate this integral using residues? —
A large semi-circle in the upper half-plane
23.
The Taylor series expansion of a function f(z) about a point z₀ is given by Σ[n=0 to ∞] a_n (z - z₀)ⁿ. What is the formula for the coefficient a_n? —
a_n = f⁽ⁿ⁾(z₀) / n!
24.
If the integral ∫_C f(z) dz = 0 for every simple closed contour C in a domain D, then f(z) is: —
Analytic in D
25.
A Laurent series expansion of a function f(z) about a point z₀ is given by Σ[n=-∞ to ∞] c_n (z - z₀)ⁿ. The part of the series with negative powers of (z - z₀) is called the: —
Principal Part
26.
What is the fundamental theorem that allows a function to be represented by a power series in a neighborhood of a point? —
Taylor's Theorem
27.
The integral of a function around a closed contour is related to the residues inside the contour. This is the statement of which theorem? —
Residue Theorem
28.
What is the residue of f(z) = e^(iz) / (z² + a²) at z = ia, where a > 0? —
e^(-a) / (2a)
29.
Which of the following functions is analytic everywhere in the complex plane? —
f(z) = e^z
30.
What is the condition for a function f(z) to be analytic in a domain D? —
f(z) is differentiable at every point in D
31.
A singularity z₀ is called a removable singularity if the principal part of the Laurent series expansion of f(z) about z₀ is: —
Zero
32.
What is the significance of the principal part of a Laurent series in classifying singularities? —
It indicates the nature of the singularity at z₀
33.
If a function f(z) has a removable singularity at z₀, how does its Taylor series behave around z₀? —
It is the same as the Laurent series
34.
In the context of Taylor series, what does the term 'analytic' imply about a function? —
It is differentiable at every point in an open set
35.
The Cauchy principal value of an integral is often used when dealing with improper integrals. How does it relate to residues? —
It can be calculated using residues when the integrand has poles on the path of integration
36.
If lim_(z→z₀) (z - z₀)ᵐ f(z) = L, where L is a finite non-zero complex number, and lim_(z→z₀) (z - z₀)ᵐ⁺¹ f(z) = 0, then z₀ is a pole of order: —
m
37.
How can the residue at a simple pole z₀ of a function f(z) = P(z)/Q(z) be calculated if P(z₀) ≠ 0, Q(z₀) = 0, and Q'(z₀) ≠ 0? —
P(z₀) / Q'(z₀)
38.
For a function f(z) to be analytic at a point z₀ and in a disk |z - z₀| < R, its Taylor series expansion about z₀ converges to f(z) within that disk. What is the minimum value of R called? —
Radius of Convergence
39.
If the principal part of the Laurent series of f(z) about z₀ contains only a finite number of terms, what type of singularity is z₀? —
Pole
40.
If the principal part of the Laurent series of f(z) about z₀ contains infinitely many terms, what type of singularity is z₀? —
Essential Singularity
41.
Which of the following is NOT a type of isolated singularity for a complex function? —
Branch Point
42.
If f(z) has a pole of order 1 at z₀, the residue is given by lim_(z→z₀) (z - z₀) f(z). This is also known as the residue at a: —
Simple pole
43.
Which type of series expansion is used to represent a function in an annulus (a region between two concentric circles)? —
Laurent Series
44.
The Laurent series expansion of a function f(z) about an isolated singularity z₀ is unique in any annulus 0 < |z - z₀| < R where f(z) is analytic. This uniqueness is guaranteed by: —
The integral formula for Laurent coefficients
45.
The Taylor series expansion of a function f(z) about z₀ is unique if the function is analytic in a neighborhood of z₀. This uniqueness is a consequence of: —
Cauchy's Integral Formula
46.
What is the residue of a function f(z) at an isolated singularity z₀? —
The coefficient c₋₁ of the Laurent series
47.
The Residue Theorem states that ∫_C f(z) dz = 2πi * (Sum of residues of f(z) at poles inside C). What condition must the poles satisfy? —
They must be inside the contour C
48.
Consider the function f(z) = 1 / (z² + 4). What are the poles of this function? —
z = 2i, z = -2i
49.
The Laurent series expansion of f(z) = 1/(z-1)(z-2) in the annulus 1 < |z| < 2 contains terms of the form: —
Σ a_n zⁿ and Σ b_n z⁻ⁿ