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 z  n
y
x
lim1 +  = exp( )(cos( ) y + j sin( ))
n→∞ n
For what values of y is this quantity pure imaginary?

Homework Problems

Pb. 6.23 Plot the curves determined by the following parametric represen-
tations:
a. z = 1 – jt 0 ≤ t ≤ 2
b. z = t + jt 2 –∞ < t < ∞
π 3 π
c. z = 2(cos(t) + j sin(t)) <<t
2 2
d. z = 3(t + j – j exp(–jt)) 0 < t < ∞
Pb. 6.24 Find the expression y = f(x) and plot the families of curves defined
by each of the corresponding equations:
 1  1
a. Re  z  = 2 b. Im  z  = 2

c. Re( )z 2 = 4 d. Im( )z 2 = 4
z − 3  z −  3 π
e. = 5 f. arg =
z + 3  z +  3 4
2
g. z −= 3 h. z = Im( z +) 4
1
Pb. 6.25 Find the image of the line Re(z) = 1 upon the transformation z′ = z 2
+ z. (First obtain the result analytically, and then verify it graphically.)
az + b
Pb. 6.26 Consider the following bilinear transformation: ′ =z
cz + d
Show how with proper choices of the constants a, b, c, d, we can generate all
transformations of planar geometry (i.e., scaling, rotation, translation, and
inversion).

Pb. 6.27 Plot the curves C′ generated by the points P′ that are the images of
points on the circle centered at (3, 4) and of radius 5 under the transformation
of the preceding problem, with the following parameters:
Case 1: a = exp(jπ/4), b = 0, c = 0, d = 1
Case 2: a = 1, b = 3, c = 0, d = 1
Case 3: a = 0, b = 1, c = 1, d = 0

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