## zMap: Sample Questions

Here are a few sample questions to illustrate how this program could be used to answer questions of different degrees of difficulty. For further information on how to use ZMap, start the program and consult the online help feature.

**Question 1:**- Use the basic exponential function
*exp(z)*as your function and reset all parameters. Then map horizontal (-10 to 10) and vertical (-10 to 10) lines. Their image will be circles and radii. Make sure to zoom out on the range to see that there are circles of large radii.- Are the horizontal lines mapped to radii or circles?
- Are the vertical lines mapped to radii or circles?
- Explain mathematically why this mapping property is correct.
- What if all lines were in the left half plane? The right half plane?

**Question 2:**- Consider the function
*f(z) = z*. Show that angles between radii starting at^{2}*(0,0)*are doubled. Does the function also double angles for radii*not*starting at*(0,0)*, for example radii centered at, say,*(0.5, 0.0)*? What is so special about the origin for this function? Where would you expect the function*f(z) = z*to double angles?^{2}- 2z **Question 3:**- Discuss the mapping properties of the function
*f(z) = 1/z*by checking the images of different circles centered at*z = 0*and radial lines (but make sure to stay away from the singularity at*z = 0*). Verify your suspicions mathematically, then describe the map in your own (possibly non-mathematical) words. **Question 4:**- What is the image of the right-half plane under the map
*f(z) = exp(-z)*?*Hint:*pick horizontal and vertical lines with real parts from 0 to 20 and imaginary parts from -20 to 20, using plenty of lines and a suitably large number of points to get smooth curves. **Question 5:**- Show that the function
*f(z) = exp(z)*maps the half-strip*Re(z) > 0*,*-Pi/2 < Im(z) < Pi/2*onto the portion of the right half-plane that lies outside the unit circle. Next find a similar half-strip that is mapped into the right half of the unit disk. Finally, which half-strip is mapped into the left half of the unit disk? **Question 6:**- What is the image of the unit disk under the map
*f(z) = (z+1)/(z-1)*? In particular, what is the image of the linear segment from -1 to 1? What is the image of the unit circle?*Hint:*Make sure to use a suitably large number of points to get smooth curves, and to use a fair amount of radii and arcs to answer the first question. Be careful to avoid the pole of the function at*z = 1*(but get close to it). **Question 7:**- Find a function that is not constant but maps the unit disk into the unit
disk. Verify whatever map you come up with using zMap.
*Hint:*Perhaps some of the previous examples will do something for you if you combine them? **Question 8:**- Find a function that maps the right half-plane to the unit disk.
*Hint:*In a previous example we found a map that maps the unit disk to the right half-plane. The map we are looking for now is the "opposite" of that map. **Question 9:**- Show that the map
*f(z) = z + 1/z*maps circles onto ellipses. -
**Question 10:**- Find the image of the disk
*|z-2| = 2*under the transformation*f(z) = z/(2z-8)*. What about the inside of that disk? What about the outside? **Question 11:**- Discuss the image of the circle
*|z-2| = 1*and its interior under the following transformations:*f(z) = z-2i**f(z) = 3iz**f(z) = (z-2)/(z-1)**f(z) = (z-4)/(z-3)**f(z) = 1/z*

**Question 12:**- Prove by picture that the following functions are periodic with the periods
given. Make sure to use both polar and rectangular curves in the domain.
- The complex sine function
*sin(z)*has period*2*Pi* - The complex cosine function
*cos(z)*has period*2*Pi* - The complex exponential function
*exp(z)*has period*2*Pi*i*

*Hint:*ZMap can open two (or more) windows at the same time: select*File | New*to start another copy of ZMap. Now you can try to find two sets of different curves, each in its own window, whose images are the same. - The complex sine function
**Question 13:**- Prove by picture that the
*log(z)*is not continuous across the negative real axis. What about*z*, is that one continuous across the negative real axis?^{(1/2)}*Hint:*try radii only (but avoid the singularity at*z = 0*) that are close to the negative real axis, but "above" and "below" it. **Question 14:**- Show that the Taylor series for the
*sin(z)*function around*z = 0*starts with a*z*while the corresponding series for the*cos(z)-1*starts with a*c z*,^{2}*c*some constant. For extra credit determine whether the constant*c*in the Taylor series expansion is positive or negative.*Hint:*ZMap can open two (or more) windows at the same time: select*File | New*to start another copy of ZMap. If the transformations of, say,*small*circles in one window under the map*cos(z)*look*similar*to the transformations of the same circles in a second window under the map*c z*for some specific constants^{2}*c*, the first-order approximation of*cos(z)-1*would be*c z*.^{2} **Question 15:**- Find the first term of the Taylor series of
*tan(z)*,*exp(z*, and^{2})-1*z*around^{3}/(1-z)*z = 0*.