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Year 2 > Analysis III > Flashcards

Analysis III Flashcards

1
Q

Define a Partition

A
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2
Q

Define Upper Riemann Sums and Lower Riemann Sums

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3
Q

Define the Upper and Lower Riemann Integrals

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4
Q

When is a function Riemann integrable, and what is the integral equal to

A

Only when U(f) = L(f), and then the integral is equal to this value

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5
Q

Define a refinement

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6
Q

Theorem: Let f:[a,b] to R be a bounded function and P,Q refinements of [a,b] where Q is a refinement of P, then L(f,P) <= L(f,Q) <= U(f,Q) <= U(f,P)

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7
Q

Theorem: Let f:[a,b] to R bea bounded function and P,Q two partitions of [a,b]. Then L(f,P) <= U(f,Q)

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8
Q

Theorem: Let f:[a,b] to R be a bounded function. Then f is integrable if and only if for every ε > 0 there exists a partition P of [a,b] such that U(f,P) - L(f,P) < ε

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9
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10
Q

Define continuity of f

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11
Q

Define uniform continuity of f

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12
Q

Let f:[a,b] to R be a continuous function. Then it’s uniformly continuous

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13
Q

Let f:[a,b] to R be a continuous function. Then it’s Riemann integrable

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14
Q

Theorem: Let f:[a,b] to R be a monotonic function. Then it’s Riemann integrable

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15
Q

Let f,g:[a,b] to R be integrable. Then f + g is integrable

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16
Q

Theorem: Let f,g:[a,b] to R be integrable, and c in R. Then cf is integrable

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17
Q
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18
Q

Let f:[a,b] to R be integrable. Define m = inf f and M = sup f, whats the relationship between m(b-a), M(b-a) and the integral

A

m(b-a) <= integral <= M(b-a)

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19
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20
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21
Q

Let f:[a.b] to R and c in (a,b). Then f is integrable on [a,b] if and only if it’s integrable on [a,c] and [c,b]

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22
Q

Let f:[a,b] to R be bounded, integrable and Ф: R to R a continuous function. Then Ф o f is integrable

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23
Q

Theorem: Let f,g:[a,b] to R be integrable functions. Then fg is integrable, and if 1/g is bounded then f/g is integrable

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24
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25
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26
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27
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28
Q

Let f:[a,b] to R be integrable for every [c,b] with a < c. Define the improper integral of f on [a,b]

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29
Q

Let f:[a,b] to R be integrable for every [a,c] with c < b. Define the improper integral of f on [a,b]

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30
Q

Let f:[a,b] to R be a function integrable on any closed interval not containing c in [a,b], ie integrable on all [a,c - epsilon] and [c+ + delta, b]. Define the improper integral

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31
Q

Let f:[a, inf) to R be integrable for every interval [a,y] for a < y < inf. Define the improper integral

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32
Q

Let g:(inf, b) to R be integrable for every interval [y,b] for -inf < y < b. Define the improper integral

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33
Q

Let f:R to R be a function integrable on every bounded interval [a,b]. Define the improper integal

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34
Q

Define pointwise convergence of sequences

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35
Q

Define uniform convergence of sequences

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36
Q

Define uniformly cauchy for sequences

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37
Q

Theorem: A sequence (fn) is uniformly cauchy if and only if its uniformly convergent

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38
Q

Let (fn) be a sequence of continuous functions in Omega that converge uniformly to f: Omega to R. Then f is continuous

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39
Q

What does this represent?

A

The space of bounded, continuous functions with the uniform norm

40
Q
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41
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42
Q

Define continuity on f: R2 to R

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43
Q

Define uniform continuity on f: R2 to R

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44
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45
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46
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47
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48
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49
Q

For a sequence (fk) of functions define partial sums and both pointwise and uniform convergence

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50
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51
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52
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53
Q

Let I be an interval in R. When is a function f: I to R

i) increasing
ii) decreasing

A

i) f(x) <= f(y) when x < y
ii) f(x) >= f(y) when x < y

54
Q

Given f:[a,b] to R define total variation and state the conditions for bounded variation

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55
Q

Define absolute continuity for f:[a,b] to R

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56
Q

Define uniform lipschitz for a sequence of functions

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57
Q

Theorem: Suppose that fn converges pointwise to f and fn is uniformly lipschitz. Then f is lipschitz

A
58
Q

Theorem: Let f:[a,b] to R be an absolutely continuous function. Then f is of bounded variation

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59
Q

Define convergence for (zn) in C

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60
Q

Define open and closed for a set Omega in C

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61
Q

Define sequential compactness in C

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62
Q

What is the matrix representation of the complex number a + bi

A

a - b

b a

63
Q

Define continuity in C

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64
Q

Define complex differentiability on a open set Omega in C

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65
Q

What are the Cauchy-Riemann equations?

A

ux = vy

uy = -vx

66
Q

When is f analytic (or holomorphic) in C?

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67
Q
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68
Q

Define convergence of infinite sums, with entries in C

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69
Q

Define absolute convergence of infinite sums, with entries in C

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70
Q

State the Ratio test for infinite sums

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71
Q

State the root test for infinite sums

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72
Q
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73
Q
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74
Q

What is the derivative of the infinite sum (anzn)n=0n=inf with radius of convergence R

A
75
Q

Let (anzn) from n=0 to infinity be a power series with radius of convergence R. What is f(n)(0)

A

ann!

76
Q
A
77
Q

Define the power series for the following functions

i) ez
ii) cos(z)
iii) cosh(z)
iv) sin(z)
v) sinh(z)

A
78
Q

State the exponential identities for cos(z), sin(z), cosh(z) and sinh(z). Prove the result for cos(z)

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79
Q
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80
Q

For a function f:[a,b] to C define the complex integral

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81
Q
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82
Q
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83
Q
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84
Q
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85
Q

Define connected and simply connected on a subset omega of C

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86
Q
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87
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88
Q
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89
Q
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90
Q
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91
Q
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92
Q

Theorem: Every non-constant polynomial p on C has a root. ie there exists an a in C such that p(a) = 0

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93
Q
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94
Q

If f:C to C is harmonic, what two equations are satisfied

A

uxx + uyy = 0

vxx + vyy = 0

95
Q

Theorem: Let f: C to C be analytic. Then if |f| is also analytic, f must be a constant

A

Year 2 (15 decks)

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