Advanced Math (13 problems)

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Problem #1: Complex Integration

Evaluate the complex integral of the function f(z) = 1/(z^3+1)on the curve C, which corresponds to the arc of radius R that goes from theta = 0 to theta = pi/3.

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Problem #2: Complex Integration

Evaluate the integral

where C is the parametrized unit circle and

a) .

b) .

c)

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Problem #3: Complex Integration

Evaluate the integral

for

(a)

(b)

(c)

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Problem #4: Functional Analysis

Let f be a nonnegative bounded function on [a, b] with. Let

for n = 1, 2, … and and set. Prove thatand thatfor each n. Thusconverges uniformly to f on [a, b].

(Note that in this Exercise 1 there is. Given a subset A of some “universal” set S, we define : S ® R, the characteristic function of A, by (x) = 1 if x is in A, and (x) = 0 if x is not in A.)

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Problem #5: Measure Theory

Let. Show thatif and only if, for every n.

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Problem #6: Measure Theory

Given a subset E of R, prove that there is a -set containing E, such that

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Problem #7: Measure Theory

Ifis a countable union of pairwise disjoint intervals, prove that

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Problem #8: Measure Theory

Given any subset E of R, and any h in R, show that, where

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Problem #9: Measure Theory

If and, a.e., does it follow that? What is “a.e.” is weakened to “except at a countably many points”? Or “except at finitely many points”?

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Problem #10: Measure Theory

Suppose that E is measurable with. Show that

(a) There is a measurable setsuch that

(b) There is a closed set F consisting entirely of irrationals such thatand

(c) There is a compact set F with empty interior such thatand

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Problem #11: Measure Theory

Given show that, where the infimum is taken over all coverings of E by sequences of intervals, where eachhas a diameter less than.

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Problem #12: Measure Theory

If , show that, for any set A.

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Problem #13: Measure Theory

For which subsetsisRiemann integrable?

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