Complex Analysis

Question 0

If z = a + bi, what is the Re(z) and Im(z)?

Answer 0

Re(z) = a and Im(z) = b

Question 1

What are the properties of ℝ?

Answer 1

1) Field: multiplicative, associative, and commutative rules apply. There is a multiplicative inverse for every x ≠ 0
2) Ordered:
a. If x,y ∈ ℝ, then xy ∈ ℝ+ and x+y ∈ ℝ+.
b. If x ∈ ℝ, then either x ∈ ℝ+, -x ∈ ℝ+, OR x=0.
3) Complete: every sequence in ℝ that is bounded above has a least upper bound.

Question 2

Every z ∈ ℂ, z ≠ 0, can be written as:

Answer 2

z = r(cos(θ) + isin(θ)), r = |z| = √(x²+y²)
z = re^(iθ)

Question 3

Complex conjugate of z = x + iy and its properties

Answer 3

Complex conjugate z\bar = x-iy.
(z_1+z_2)\bar = z_1\bar + z_2\bar
(z_1z_2)\bar = z_1\bar*z_2\bar
re(z) = ½(z+z\bar)
im(z) = ½(z-z\bar)
|z|² = |zz\bar|
z is REAL ⇔ z=z\bar

Question 4

Difference between arg(z) and Arg(z)?

Answer 4

arg(z) is many (+2πk) and Arg(z) is unique

Question 5

What is De Moivre’s Theorem?

Answer 5

(cosθ + isinθ)ⁿ = cos nθ + i sin nθ

Question 6

What is the ε-neighborhood for a complex number z_0?

Answer 6

N_ε(z_0) = {z ∈ ℂ | |z-z_0| < ε}, ∀ ε>0

Question 7

What is N_ε(z_0) geometrically?

Answer 7

It is the disc centre z_0 consisting of all points less than ε from z_0.

Question 8

What does it mean to be “closed”?

Answer 8

Proposition: S is closed ⇔ S contains all of its limit points

Question 9

If ƒ: S → ℂ is an arbitrary complex function and z_0 is a limit point of S, what does it mean if lim{z→z_0}ƒ(z) = L?

Answer 9

Definition of limit: ∀ ε>0, ∃ δ>0 such that for all z ∈ S, if |z-z_0| < δ, then |ƒ(z)-L|<ε.

a) z_0 does not need to be in S, so ƒ(z_0) may not be defined. even if z_0 ∈ S, we may have ƒ(z_0) ≠ L.
b) it is essential that z_0 is a limit point of S.

Question 10

When is ƒ:S→ℂ continuous at z_0 ∈ S?

Answer 10

ƒ is continuous if for ∀ ε>0, there exists δ>0 such that for all z ∈ S, |z-z_0| < δ ⇒ |ƒ(z) - ƒ(z_0)|<ε.

Question 11

If z_0 is a limit point, this is equivalent to saying what?

Answer 11

It is equivalent to saying that lim{z→z_0}ƒ(z) exists AND lim{z→z_0}ƒ(z)=ƒ(z_0).

Question 12

What is a PATH in the complex plane?

Answer 12

A path is a continuous function γ: [a,b] → ℂ. The initial point is γ(a) and final point is γ(b).

Question 13

If {a_n} is a sequence of real numbers, what does it mean to say {a_n} is a Cauchy sequence?

Answer 13

{a_n} is Cauchy if for every ε>0, there is an N such that whenever n,m>N, we have |a_n - a_m| < ε.

Question 14

What are the key properties of |z| for z ∈ ℂ?

Answer 14

1) |z| ≥ |Re(z)| and |z| ≥ |Im(z)|
2) for z, w ∈ ℂ, |zw| = |z||w|
3) for z, w ∈ ℂ, |z+w| ≤ |z| + |w| (triangle inequality)

Question 15

Proposition: If z = r_1(cos(θ_1) + i sin(θ_1)) and w = r_2(cos(θ_2) + i sin(θ_2)), what is zw?

Answer 15

zw = (r_1)(r_2)(cos(θ_1 + θ_2) + i sin(θ_1 + θ_2))

Question 16

What is e^(iθ)?

Answer 16

e^(iθ) = cos(θ) + i sin(θ)

Question 17

What is e^z?

Answer 17

e^z = (e^x)(cos(y)+i sin(y))

Question 18

Let S be a subset of the complex numbers. What does it mean to say that S is open?

Answer 18

S is open if every point of S is an interior point

Question 19

What does it mean to say z ∈ S is an exterior point?

Answer 19

z is an exterior point if N_ε(z) is DISJOINT from S for some ε > 0.

Question 20

What does it mean to say that z ∈ S is a boundary point of S?

Answer 20

z is a boundary point if N_ε(z) intersects both S and its complement for every ε > 0.

Question 21

What is ∂S?

Answer 21

∂S is the set of boundary points of S

Question 22

If S ⊂ ℂ, what is the CLOSURE OF S, denoted S_bar?

Answer 22

S_bar = S U ∂S

Question 23

What is a domain in the complex plane?

Answer 23

A domain is a nonempty, open, connected subset of ℂ

Question 24

Rewrite a complex valued function ƒ on S.

Answer 24

ƒ(z) = u(x,y) + iv(x,y)

Question 25

If S is a set, when is z the limit point of S?

Answer 25

z is a limit point of S if every neighborhood of z contains infinitely many points of S.

Question 26

If z_0 ∈ S is a limit point of S, when is ƒ continuous at z_0?

Answer 26

ƒ is continuous at z_0 if for every ε>0, there is δ > 0 so that whenever |z-z_0| <ε.

Question 27

What does it mean if ƒ is continuous ON S?

Answer 27

ƒ is continuous on S if it is continuous at each point of S.

Question 28

Define the derivative of ƒ(z)

Answer 28

ƒ’(z) = lim{h→0} [ƒ(z+h)-ƒ(z)]/h

Question 29

What is the Cauchy-Riemann equations?

Answer 29

If ƒ is differentiable at z = x_0 + y_0 i , then at that point all of ∂u/∂x, ∂u/∂y, ∂v/∂x, ∂v/∂y exist and ∂u/∂x = ∂v/∂y & ∂v/∂x = -∂u/∂y