Analysis III

Question 1

Define a Partition

Answer 1

Question 2

Define Upper Riemann Sums and Lower Riemann Sums

Answer 2

Question 3

Define the Upper and Lower Riemann Integrals

Answer 3

Question 4

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

Answer 4

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

Question 5

Define a refinement

Answer 5

Question 6

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)

Answer 6

Question 7

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)

Answer 7

Question 8

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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Question 9

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Question 10

Define continuity of f

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Question 11

Define uniform continuity of f

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Question 12

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

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Question 13

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

Answer 13

Question 14

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

Answer 14

Question 15

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

Answer 15

Question 16

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

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Question 17

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Question 18

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

Answer 18

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

Question 19

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Question 20

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Question 21

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]

Answer 21

Question 22

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

Answer 22

Question 23

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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Question 24

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Question 25

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Question 26

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Question 27

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Question 28

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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Question 29

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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Question 30

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

Answer 30

Question 31

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

Answer 31

Question 32

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

Answer 32

Question 33

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

Answer 33

Question 34

Define pointwise convergence of sequences

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Question 35

Define uniform convergence of sequences

Answer 35

Question 36

Define uniformly cauchy for sequences

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Question 37

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

Answer 37

Question 38

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

Answer 38

Question 39

What does this represent?

Answer 39

The space of bounded, continuous functions with the uniform norm

Question 40

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Question 41

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Question 42

Define continuity on f: R² to R

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Question 43

Define uniform continuity on f: R² to R

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Question 44

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Question 45

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Question 46

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Question 47

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Question 48

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Question 49

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

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Question 50

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Question 51

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Question 52

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Question 53

Let I be an interval in R. When is a function f: I to R i) increasing ii) decreasing

Answer 53

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

Question 54

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

Answer 54

Question 55

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

Answer 55

Question 56

Define uniform lipschitz for a sequence of functions

Answer 56

Question 57

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

Answer 57

Question 58

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

Answer 58

Question 59

Define convergence for (zn) in C

Answer 59

Question 60

Define open and closed for a set Omega in C

Answer 60

Question 61

Define sequential compactness in C

Answer 61

Question 62

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

Answer 62

a - b b a

Question 63

Define continuity in C

Answer 63

Question 64

Define complex differentiability on a open set Omega in C

Answer 64

Question 65

What are the Cauchy-Riemann equations?

Answer 65

u_x = v_y u_y = -v_x

Question 66

When is f analytic (or holomorphic) in C?

Answer 66

Question 67

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Question 68

Define convergence of infinite sums, with entries in C

Answer 68

Question 69

Define absolute convergence of infinite sums, with entries in C

Answer 69

Question 70

State the Ratio test for infinite sums

Answer 70

Question 71

State the root test for infinite sums

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Question 72

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Question 73

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Question 74

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

Answer 74

Question 75

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

Answer 75

a_nn!

Question 76

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Question 77

# Define the power series for the following functions i) e^z ii) cos(z) iii) cosh(z) iv) sin(z) v) sinh(z)

Answer 77

Question 78

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

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Question 79

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Question 80

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

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Question 81

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Question 82

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Question 83

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Question 84

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Question 85

Define connected and simply connected on a subset omega of C

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Question 86

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Question 87

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Question 88

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Question 89

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Question 90

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Question 91

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Question 92

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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Question 93

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Question 94

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

Answer 94

u_xx + u_yy = 0 v_xx + v_yy = 0

Question 95

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

Answer 95