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Section 1.5 Exercise set 1
Exercises Exercises
1. Let \(S\) be the square with corners \((0,1), (1,1), (1, 2), (0,2)\text{.}\) Find the image of the square under the map \(f(z) = z^2\text{.}\) Where does the interior of the square map? Is the image a square?
2. Let \(S\) be the unit square in \(\C\text{.}\) Find the image of the square under the map \(f(z) = z^2\text{.}\) Make sure to indicate where the interior maps.
3. Find the image of the first quadrant under the map \(f(z) = z^3\text{.}\)
4. Let \(z = x + iy\text{.}\) Plot the set \(A = \{xy>1 \text{ and } x, y >0\}\text{.}\) Find the image of the set under the map \(f(z) = z^2\text{.}\)
5.
Near which points are the following functions conformal?
\(f(z) = z/(1 + 2z)\text{;}\)
\(\displaystyle f(z) = \cc{z}\)
\(f(z) = \tan z\text{.}\)
6. Let \(a, b \in \C\) be constant. Show that \(f: \C \to \C\) defined by \(f(z) = az + b\) can be written as a rotation followed by a magnification followed by a translation.
7. The Riemann mapping theorem concerns simply connected sets that are not all of \(\C\text{.}\) Is there a conformal map from \(\C\) that is one-to-one onto the unit disk \(\D\text{?}\) Is there a conformal map of \(\D\) one-to-one onto \(\C\text{?}\)
