Additional Problems to Section 12.3

 

 

1. Describe with words or a sketch the portion of the complex plane corresponding to the following equation or inequalities.  State which problems have no solutions

 

(a)    Im(z) >= Re(z)

(b)    Im(z+2i)=Re(z-3)

 

2. Represent the following regions by means of equations or inequalities in the variable z.

 

(a)    All the points occupying an annular region centered at 3+i.  The inner radius is 2; the outer is 4.  Exclude points on the inner boundary but include those on the outer one,

(b)    All points, except the center, on and within a circle of radius 2, centered at 3+4i.

 

3. Prove the continuity of the following functions in the domain indicated.  Assume z=x+iy.

 

(a)    f(z) = z³+z+1  all z

(b)    f(z)=|z|+x all z

(c)    f(z)=1/(x²-y²+z)  all z except 0 and -1

 

4. The function f(z)=z(z²-16)/(z²-4z) is defined and continuous for all z except z=0 and z=4.  How should we define f(0) and f(4) so that f(z) is continuous throughout the z-plane?