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Material Type: Exam; Professor: Markman; Class: Complex Variables; Subject: Mathematics; University: University of Massachusetts - Amherst; Term: Fall 2002;
Typology: Exams
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Solve the first 5 problems and only one out of problems 6 and 7. (If you solve both 6 and 7, then I will not grade 7).
C
ez¯^ dz,
where C is the boundary of the rectangle with vertices at the points 0, 2, 2 + i, and i, oriented counterclockwise. Caution: the exponent of the integrand is the complex conjugate z¯ of z.
C
ez z^2 + 1
dz.
C
z + 1 z − 1
dz
∣ ≤^3 π,
where C is the semi-circle, given by the parametrization z(t) = 2eit, 0 ≤ t ≤ π.
z(x) = (x + i sin(x)), 0 ≤ x ≤ 2 π.
Let C 2 be the piece of the graph of y = − sin(x), given by the parametrization
z(x) = (x − i sin(x)), 0 ≤ x ≤ 2 π.
Compute the difference (^) ∫
C 1
dz z − π 2
C 2
dz z − π 2
P ′(z) P (z)
f ′(z) f (z)
g′(z) g(z)
b) (8 points) Let P (z) be a polynomial of degree n. Let CR the circle of radius R > 0, centered at the origin, transversed counterclockwise. Prove, the equality ∫
CR
P ′(z) P (z)
dz = 2 nπi,
provided R is sufficiently large. Hint: Do the case n = 1 first.
