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Section 1.7 Exercises 2
Exercises Exercises
1.
Let \(f(z) = (z-1)/(z+1)\text{.}\) Find the image under \(f\) of the following sets:
the real line;
the imaginary axis
the unit circle;
the circle centered a 0 with radius 2;
2. Is the image of a triangle under a conformal map a triangle? Why or why not? How about on the sphere? (What is a “triangle” on a sphere?)
3.
Let \(f(z) = (z- i)/(z+i)\text{.}\) Find the image under \(f\) of the following sets:
the real line;
the imaginary axis
the unit circle;
the circle centered a 0 with radius 2;
the triangle with vertices \(0, i, 1+i \in \C\text{.}\)
4. Find a linear fractional transform that takes the unit disk to the right half plane with \(f(0) = 2\text{.}\)
5. Find a linear fractional transform that takes the unit disk to itself and maps \(1/2\) to \(1/3\text{.}\)
6. Determine the image of the unit disk under the map \(f(z) = \frac{z+1}{3z + 1}\text{.}\) Where is the image of the real axis? What about the imaginary axis?
7. Find a conformal map taking the quarter of unit disk in the first quadrant to the entire unit disk. Hint: \(f(z) = z^4\) isn't good enough. Why? Can you do it with a bijective conformal map?
8. Show that a composition of two linear fractional maps is a linear fractional map.
