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Complex Analysis - Michaelmas > Complex Plane and Functions > Flashcards

Complex Plane and Functions Flashcards

1
Q

Define a complex number.

A

A complex number is quantity of the form z = x + iy, where x, y are real numbers and i is the imaginary unit.

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2
Q

What do we denote the set of complex numbers by?

A

ℂ

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3
Q

What do we denote the real and imaginary part of a complex number by?

A

If z = x + iy:

  • Re(z) = x
  • Im(z) = y
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4
Q

When dividing two complex numbers what do you have to multiply it by?

A

The complex conjugate of the denominator, ẑ = x - iy

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5
Q

What is the complex conjuagte of z?

A

ẑ = x - iy

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6
Q

What algebraic properites of ℝ don’t hold in ℂ? And why?

A

Any inequalitie of the form <, > etc. because ℂ isn’t “ordered”

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7
Q

What diagram do we draw complex numbers on?

A

Argand diagram

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8
Q

What does complex conjugation look like in an Argand diagram?

A

Reflection in the real axis

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9
Q

What is the modulus of z?

A
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10
Q

What is the Lemma about four of the useful and basic properties of complex numbers?

A
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11
Q

In what other types of coordinates can you write compelx numbers?

A

Polar coordiantes

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12
Q

What is the argument of z?

A

When written in polar form, it is θ

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13
Q

What are the two ways to write a comlex number in polar form?

A
  1. z = r(cosθ + i sinθ)
  2. z = reiθ
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14
Q

What is arg(z) only defined up to?

A

Multiples of 2π

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15
Q

Define the principle value of arg(z).

A

The principle value of arg(z) is the value in the interval (-π, π]

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16
Q

How is the principle value of arg(z) denoted?

A

Arg(z)

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17
Q

What does Re(z) look like on an argand diagram?

A

Projection onto the real axis

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18
Q

What does Im(z) look like on an argand diagram?

A

Projection onto imaginary axis

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19
Q

Finish the Lemma: Geometrically, multiplication in ℂ is given by a … ?

A
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20
Q

Prove the following Lemma.

A
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21
Q

What is de Moirve’s theorem?

A

(cos(θ) + isin(θ))n = cos(nθ) + isin(nθ)

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22
Q

What are three additional properties of the modulus |.|?

A
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23
Q

What are three additional properties of the argument?

A
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24
Q

Define a complex exponential function.

A
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25
Q

What is the propositiion about five properties of the complex exponential function?

A
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26
Q

What does exp(2πi) equal?

A

exp(2πi) = 1

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27
Q

What does exp(πi) equal?

A

exp(πi) = -1

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28
Q

What is Euler’s famous formula?

A

exp(πi) = -1

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29
Q

What equation shows that the complex exponential function is 2πi periodic?

A

exp(z) = exp(z + 2kπi) ∀ k ∈ ℤ

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30
Q

What does the following corollary imply?

A

exp is determined entirely by the values it takes on any horizontal strip of width 2π in the complex plane

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31
Q

Define sin(z) in terms of complex exponentials.

A
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32
Q

Define cos(z) in terms of complex exponentials.

A
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33
Q

Define sinh(z) in terms of complex exponentials.

A
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34
Q

Define cosh(z) in terms of complex exponentials.

A
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35
Q

What is a trig formula that relates cosh and cos?

A

cosh(iz) = cos(z)

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36
Q

What is a trig formula that relates sinh and sin?

A

sinh(iz) = isin(z)

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37
Q

How are the solutions to ez = w written?

A

z = ln(w) + i(Arg(w) + 2kπi)

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38
Q

Prove the following Lemma.

A
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39
Q

Define a logarithm of w.

A

Any solution z of the equation ez = w (with w ∈ ℂ٭) is called a logarithm of w and it is denoted log(w)

40
Q

Define the kth branch of log(w).

A

For any fixed k ∈ ℤ, we call the logk : ℂ \ ℝ≤0 ➝ ℂ defined by:

logk(w) = ln|w| + iArg(w) + i2kπ,

the kth branch of log(w).

41
Q

Define the principle branch of log(w).

A

When k = 0, we call the function Log(w) = log0(w) the principle branch of log(w), that is, Log : ℂ \ ℝ≤0 ➝ ℂ and:

Log(w) = ln|w| + iArg(w)

42
Q

The principle branch, Log agrees with what on the real line?

A

The natural logarithm, ln

43
Q

What are three properites when using any given branch of logarithm?

A
  1. elogz = z for any z ∈ ℂ \ ℝ≤0, but
  2. in general, log(zw) ≠ log(z) + log(w)
  3. in general, log(ez) ≠ z.
44
Q

Define complex powers.

A

For any branch of log, we define zw (for z ∈ ℂ \ ℝ≤0) by

zw = exp(w logz)

45
Q

What formula can you use to find all roots of z?

A
46
Q

What dimension would graphs of complex-valued functions be?

A

4 dimensions

47
Q

Define an open ball of radius r centred at z0.

A
48
Q

Define a closed ball of radius r centred at z0.

A
49
Q

Define when a subset is open.

A
50
Q

Define when a subset is closed .

A
51
Q

What is the (open) unit disc? And how is it denoted?

A

B1(0), 𝔻

52
Q

Is ℂ open or closed?

A

Open

53
Q

Is ∅ open or closed?

A

Open

54
Q

Is the set ℂ \ ℝ≤0, on which log and complex powers are defined open or closed?

A

Open

55
Q

Do open or closed sets contains their edges/boundaries?

A

Closed

56
Q

Define what is meant by converges to the limit z0.

A
57
Q

Define what is meant by tends to w ∈ ℂ as z tends to z0.

A
58
Q

What is another way to write

A
59
Q

Complete the following Lemma

A
60
Q

Define what is meant by continuous at z0 ∈ U and therefore continuous on U.

A
61
Q

How can you rewrite this definition to be in terms of open balls?

A
62
Q

How did we make Log(z) a continuous function?

A

By fixing the domains

63
Q

Define (complex) differentiable at z0 ∈ U.

A
64
Q

Define derivative of f at z0.

A
65
Q

What is the difference between the direction of limits in complex limits compared to real?

A

In complex the limit exists from every direction where as real limits only exist to the left and right of the real number line.

66
Q

Finish this sentance: if a function f is complex differentiable at z then .. ?

A

it is continuous at z.

67
Q

When calculating whether a function is complex differentiable using the limit what is one thing you have to do differently to if it was a real functions ? And why?

A

Approach z from two different direction, real and imaginary. And then check that they are the same.

68
Q

Write ux(x, y) in terms of limits.

A
69
Q

Write uy(x, y) in terms of limits.

A
70
Q

Write vx(x, y) in terms of limits.

A
71
Q

Write vy(x, y) in terms of limits.

A
72
Q

What is the proposition about the Cauchy-Riemann equations?

A
73
Q

Using the Cauchy-Riemann equations what are the four ways you can write the complex derivative of f(z0)?

A
74
Q

Prove the following proposition.

A
75
Q

If f is complex differentiable this implies f is real differentiable and what else?

A

That the C-R equations will hold

76
Q

What is the thereom that uses the C-R equations to prove something is complex differentiable?

A
77
Q

Define holomorphic on U.

A

A function f:U ➝ ℂ defined on an open set U ∈ ℂ is called holomorphic on U if it is complex differentiable everywhere in U.

78
Q

Define holomorphic at z0 ∈ U.

A

We say that f is holomorphic at z0 ∈ U if it is holomorphic on some open ball Bε(z0)

79
Q

Define a path.

A

A path (from a ∈ℂ to b ∈ℂ) is a continuous function Ɣ[0,1] ➝ ℂ such that Ɣ(0) = a and Ɣ(1) = b.

80
Q

Define piecewise smooth.

A

A path is piecewise smooth if it is continuously differentiable at all but finietly many points.

81
Q

Define path connected.

A

We say a subset U of ℂ is path connected if for every pair of points a, b ∈ U, there is a piecewise smooth path from a to b such that Ɣ(t) ∈ U for every t ∈ [0,1].

82
Q

Define a region.

A

A region, R is an open, path-connected subset of ℂ.

83
Q

What is the zero derivative therom?

A
84
Q

Prove the zero-derivative thereom.

A
85
Q

What is the proposition about the Laplace equations?

A
86
Q

What are the two Laplace equations?

A
87
Q

Prove the following propostion about the Laplace equations.

A
88
Q

Define a harmonic function.

A

A real-valued function U:𝑅 ➝ ℝ (defined on region 𝑅 ⊆ ℂ) is harmonic on 𝑅 if U is tqice continuously differentiable and Uxx + Uyy = 0

89
Q

What is the proposition about the harmonic conjugate?

A
90
Q

How do you find harmonic conjugates?

A
  • Partially integrate with respect to one variable (y)
  • Make a guess of the solution which includes some function independent of the variable you just integrated by
  • Differentiate and use the other C-R equations to find the remaining function
91
Q

What is the Dirichlet Problem?

A
92
Q

With the Dirichlet problem what does Ṝ stand for (minus the dot underneath)?

A

Ṝ = 𝑅 ∪ ∂𝑅

93
Q

What does ∂𝑅 stand for in the Dirichlet boundary problem?

A

The boundary of some region 𝑅.

94
Q

Finish this proposition: Suppose f:𝑅 ➝ ℂ is holomorphic on a region 𝑅 ⊆ ℂ, and suppose μ is harmonic on f(𝑅). Then …

A

μ∼ = μ ◦ f is harmonic on 𝑅.

95
Q

Prove the following proposition.

A

Need to do - see sheet 3, Q6

Complex Analysis - Michaelmas (4 decks)

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