Math 421 – Practice Exam II
April, 2009
The notation HCf(z)dz indicates the simple closed contour Cis positively oriented.
1. True or False. Explain why.
(a) ∀z∈C,1z= 1.
(b) ∀z∈C, z1=z.
(c) f(z) = zcis a single-valued function if and only if c∈Z.
(d) An analytic function fin a domain Dmust have an antiderivative everywhere in D.
2. Find all value(s) the following expressions.
(a) log(3 + 3i).
(b) (ei)πi.
3. Find the principal value of (1 + i)i.
4. Solve the equation sin z=5
3for z.
5. Show that |I|z|=3
z
z4+ 9z2+ 18 dz| ≤ π.
6. Compute ZC
Rez dz, where Cis the line segment from 0 to 1 + 2i.
7. Evaluate ZC
cos z dz, where Cstarts at the origin, traverses the bottom half of a unit circle
centered at z0= 1 and then the line segment from to z= 2 to z=iπ.
8. Compute I|z|=2
sin z
zdz.
9. Compute I|z|=2
1
z(z+ 1)2(z+ 3) dz.
10. Compute ZC
z(z2+ 9)
(z2+ 1)(z6+z2+ 100) dz, where Cis the upper half circle |z|= 2 from z= 2
to z=−2.
11. Compute I|z|=1
ez2
zdz. Then use it to evaluate Zπ
0
ecos 2θcos(sin 2θ)dθ.
12. Show that the area of a region enclosed by a simple closed contour Cis equal to 1
2iIC
¯z dz.
13. Let fbe an entire function. Show that if there exist positive Aand Bsuch that |f(z)| ≤
A|z|1
2+Bfor all z, then fis constant.
1
