More practice for Exam 1

Here’s a bit more practice for our first exam this Wednesday. This does not replace the review sheet and you should definitely study that as well. The exam will certainly have a few definition statements on it, for example.

  1. Let \(\sqrt[3]{x}\) denote the principal cube root of \(x\). Evaluate

    1. \(\sqrt[3]{8}\) and
    2. \(-\sqrt[3]{-8}\)

    Express these in both Cartesian and polar form.

  2. Use the Cauchy-Riemann equations to determine which of the following defines an analytic function.

    1. \(f(x,y) = (x + x^2 - y^2) + i(y + 2 x y)\)
    2. \(g(x,y) = (x + x^2 - y^2) + i(y - 2 x y)\)

  3. Write down an \(\varepsilon\text{-}\delta\) proof of the fact that \[ \lim_{z \to -i} \left((3+4i)z + 2\right) = 6 - 3 I. \]

  4. Let \(C\) be the circle containing \(2\), \(-2\), and \(2i\) and let \[T(z) = \frac{z+2}{z-2}.\] Describe the image of \(C\) under the action of \(T\). What is the image of the interior of the circle?

  5. Suppose I’d like a Möbius transformation that maps the points \[(i,1,2) \text{ to } (-1,-i,4)\] in that order. Write down a specific equation in terms of variables \(w\) and \(z\) that you could solve to find that Möbius transformation and a brief description of why it should work. You do not need to solve the equation.

  6. Let \[f(z) = \frac{3 i z^3}{z^2-i}.\] We consider \(f\) as a map of the Riemann sphere to itself that fixes \(\infty\).
    Compute \(f'(\infty)\) and interpret the result.

  7. Describe the image of the vertical line \(\{z\in\mathbb{C}:\text{Re}(z)=1\}\) under the action of the exponential function.