 # Complex Plane and Functions Flashcards Preview

## Complex Analysis - Michaelmas > Complex Plane and Functions > Flashcards

Flashcards in Complex Plane and Functions Deck (95):
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## What is the modulus of z? 10

## What is the Lemma about four of the useful and basic properties of complex numbers? 11

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## Finish the Lemma: Geometrically, multiplication in ℂ is given by a ... ? 20

## Prove the following Lemma.  21

## What is de Moirve's theorem?

### (cos(θ) + isin(θ))n = cos(nθ) + isin(nθ) 22

## What are three additional properties of the modulus |.|? 23

## What are three additional properties of the argument? 24

## Define a complex exponential function. 25

## What is the propositiion about five properties of the complex exponential function? 26

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## What does the following corollary imply? 31

## Define sin(z) in terms of complex exponentials. 32

## Define cos(z) in terms of complex exponentials. 33

## Define sinh(z) in terms of complex exponentials. 34

## Define cosh(z) in terms of complex exponentials. 35

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## Prove the following Lemma.  39

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## What formula can you use to find all roots of z? 46

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## Define an open ball of radius r centred at z0. 48

## Define a closed ball of radius r centred at z0. 49

## Define when a subset is open. 50

## Define when a subset is closed . 51

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## Define what is meant by converges to the limit z0. 57

## Define what is meant by tends to w ∈ ℂ as z tends to z0. 58

## What is another way to write  59

## Complete the following Lemma  60

## Define what is meant by continuous at z0 ∈ U and therefore continuous on U. 61

## How can you rewrite this definition to be in terms of open balls?  62

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## Define (complex) differentiable at z0 ∈ U. 64

## Define derivative of f at z0. 65

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## Write ux(x, y) in terms of limits. 69

## Write uy(x, y) in terms of limits. 70

## Write vx(x, y) in terms of limits. 71

## Write vy(x, y) in terms of limits. 72

## What is the proposition about the Cauchy-Riemann equations? 73

## Using the Cauchy-Riemann equations what are the four ways you can write the complex derivative of f(z0)? 74

## Prove the following proposition.  75

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## What is the thereom that uses the C-R equations to prove something is complex differentiable? 77

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## What is the zero derivative therom? 84

## Prove the zero-derivative thereom.  85

## What is the proposition about the Laplace equations? 86

## What are the two Laplace equations? 87

## Prove the following propostion about the Laplace equations.  88

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## What is the proposition about the harmonic conjugate? 90

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## What is the Dirichlet Problem? 92

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## Prove the following proposition. 