Math 303 - Complex Variables
Homework due March 9
Question 1. Question 5 of your textbook in section 4.1 asks you to identify the interior of the simple
closed contour Γ given in Figure 4.13 on pg. 160. Draw the contour Γ, shade in the interior domain,
and decide if the contour Γ is positively oriented.
Recall that we stated in class that the length of an arc parameterized by z(t) = x(t)+iy(t) for a≤t≤b
is
Zb
asdx
dt 2
+dy
dt 2
dt.
Question 2. Consider a circle of radius ρcentered about the point z0=x0+iy0.
(a) Provide a parameterization z(t) for this circle (written in terms of the parameters ρand z0).
(b) Write z(t) in terms of its real and imaginary parts: z(t) = x(t) + iy(t). Find the derivatives dx
dt
and dy
dt .
(c) Use (b) and the formula above to find the arc length (i.e. circumference) of this circle.
Question 3. Consider the straight line from the point z0=x0+iy0to t z1=x1+iy1.
(a) Provide a parameterization z(t) for this line (written in terms of the parameters z0and z1).
(b) Write z(t) in terms of its real and imaginary parts z(t) = x(t) + iy(t). Find the derivatives dx
dt
and dy
dt .
(c) Use (b) and the arc length formula above to show that the length of this line is |z0−z1|.
Question 4. Consider the smooth arc parameterized by z(t) = tsin t+ cos t+i(sin t−tcos t) for
0≤t≤2π. Find the arc length of this curve.
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