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The solutions to problem 1, 2, and 3 from the math 412 week #11 practice quiz. Problem 1 involves evaluating integrals along straight lines and semicircular arcs. Problem 2 deals with proving a property of complex-valued functions. Problem 3 requires showing that the integral of a function over the boundary of a triangle is less than or equal to 60.
Typology: Quizzes
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Problem 1.
(a): (6 points) Evaluate โซ
C 1
z
3 dz,
where C 1 is a straight line from the origin to 2 + i.
(b): (4 points) Evaluate โซ
C 2
z
3 dz,
where C 2 is the path from 2 + i to the origin consisting of straight segments from 2 + i to 3 โ 2 i,
from 3 โ 2 i to โ 5 i, from โ 5 i to โ2, and the lower semicircular arc of C 1 (โ1) from โ2 to the origin.
Problem 2. (4 points) Let f (t) = u(t) + iv(t) be a complex-valued function of t โ R. Let
c + id โ C. Prove formally that
b
a
(c + id)f (t)dt = (c + id)
b
a
f (t)dt.
Problem 3. (6 points) Let C denote the boundary of the triangle with vertices at z = 0, z = โ4,
and z = 3i, with positive orientation. Show that
C
(e
z โ z ) dz
[Hint: Write e
z โ z = e
z
