Math 519 (Spring 2012) Assignment 2
Due Wednesday, February 15
1. S&S 2.2.
2. S&S 2.12a (not b).
3. S&S 2.13.
4. S&S Complete the proof of Theorem 4.4, stated on page 49 (the bit
about an ).
5. Let C be the boundary of the triangle with vertices
at the points 0, 3i,
R
and −4, with positive orientation. Show that | C (ez − z̄)dz| ≤ 60.
6. Evaluate these integrals, using any path between the limits of integration:
Ri
(a) i 2 eπz dz.
R3
(b) 1 (z − 2)3 dz.
(HInt: Think about the complex version of the fundamental theorem
of calculus.)
7. Let Cauchy help you evaluate these integrals, where C1 is the square
with sides along x = ±2 and y = ±2 (positively oriented) and C2 is
the positively-oriented circle |z − i| = 2:
R −z dz
(a) C1 ez− πi
2
R zdz
(b) C1 2z+1
R
(c) C2 z21+4
R
1
(d) C2 (z2 +4)
2
1