0
$\begingroup$

Let $A = \{z: 0 < arg(z) < \pi/2\}/\{w: |w| \ge 1, arg(w) = \pi/4\}$. Can someone give an example of a conformal map from $A$ onto the upper half plane? I'm terrible at these problems, so if you could explain your process for finding the map also I would really appreciate it.

  • 0
    Is that $/$ or \.? – 2017-02-12
  • 0
    The slash denotes the complement. – 2017-02-12
  • 0
    Perhaps if we find a map from $A$ to $\mathbb{C}-\{x:\,x<\frac{\pi}{4}\}$ then solution will clear, no? – 2017-02-12
  • 0
    Ah, that made it very clear, actually. Thank you! – 2017-02-12

1 Answers 1

1

First do $z \mapsto 1/z$, which maps $A$ to the lower right quadrant without the line segment $\{w: |w|<1, \arg w = -\pi/4\}$.

Then do $z \mapsto z^4$, and you will get the whole plane, minus the horizontal ray from $-1$ to $\infty$ which goes through $0$.

Then do $z \mapsto z+1$ and you will get the whole plane, without the horizontal ray from $0$ to $\infty$ which goes through $1$.

Then do $z \mapsto \sqrt z$ and you get the upper half plane.

  • 0
    I think last map must be $i\sqrt{z}$ – 2017-02-12
  • 0
    @MyGlasses It is correct as written. As $\arg z$ varies from $0$ to $2 \pi$, we know that $\arg \sqrt z$ will vary from $0$ to $\pi$ which is the upper half plane. – 2017-02-13
  • 0
    What's your final map.? – 2017-02-13
  • 0
    @MyGlasses Just compose all of them, you get $\sqrt{ 1+ z^{-4}}$ – 2017-02-13
  • 0
    With mathematica $z=2+i$ goes to $0.994572 - 0.0193048i$ – 2017-02-13
  • 0
    Sorry, also $z=\dfrac{3}{4}+\dfrac{1}{2}i$ goes to $0.7-0.76i$ – 2017-02-13
  • 0
    @MyGlasses Yes, of course Mathematica gives different answers because it uses the convention that $(- \pi, \pi)$ is the principal branch of the argument function, whereas this problem requires us to use $(0, 2 \pi)$ as the principal branch. There is no such canonical such branch. See the Wikipedia article on branch cuts: https://en.wikipedia.org/wiki/Principal_branch – 2017-02-13
  • 0
    For instance Mathematica will say $\sqrt {-i} = 0.71-0.71i$, whereas I will say that $\sqrt{-i} = -0.71+0.71i$. Which of us is correct? It depends entirely on the branch you use. – 2017-02-13
  • 0
    Similarly Mathematica will say that $(-i)^{1/3} = 0.866 - 0.5i$ whereas I will tell you that $(-i)^{1/3} = i$. Again, which of us is correct? Both numbers give you $-i$ if you raise to the third power. – 2017-02-13
  • 0
    @MyGlasses Like I asked before, which branch of the argument do you want to use for $\sqrt z$? If you want to use the Mathematica one, then try the function $i \sqrt{ -1-z^{-4}}$ instead. Then it should work. – 2017-02-13