ECE 6382 Homework No. 1
Due September 4, 2002
1. In class we proved that if df/dz exists in a region (independent of path of
approach), then Cauchy-Riemann conditions are satisfied. Now prove the
reverse: if C-R conditions are satisfied in a region, and u and v have
continuous derivatives, then df/dz is independent of path of approach.
2. In class we showed that for an analytic function f = u + iv, u satisfies the
Laplace equation. Now show that v also satisfies the Laplace equation.
3. Let f(t) = cos(t) + 2 sin (t) – 2 cos (t+/4) + 3 cos(t-/4). Can f(t) be
expressed as f(t) = a cos (t+)? If so, find a and . If not, explain.
4. The following equations define some curves on the complex plane. Describe
these curves.
(a) z+1=2
(b) z+1-i=1.
5. Let f(z) = 1/z. Find the region where f is analytic.
6. It is given that for an analytic function f(z), the imaginary part is
v = e-y sin x.
Find f(z).
7. Find the result of the following integral for n=1,2, and 3.
In =
Cn
dz
z
z
)1(
,
where C is the circle defined by z=2.
8. Evaluate the following integral
I =
C
dz
z1
1
2
,
where C is the circle defined by z=2.
