Let \(f(z)=\left(-1-i\sqrt 3 \right)z+3-2i\). Describe \(f\) as a rotation followed by a dilation followed by a translation. Hence draw the image under \(f\) of the unit circle \(|z|=1\) and of the line \(x=y\).
Show that the function \(f(z)=az+b\) (with \(a\ne 0\)) may be described as a translation followed by a rotation followed by a dilation. Describe \(f(z)=\sqrt{2}\left(1-i\right)z+2-4i\) in this way.
In what subset of the complex plane is \(4z^3-3iz^2 + 4 - 3i\) conformal?
In what subset of the complex plane is \(2z^3-3\left(1+i\right)z^2+6iz\) conformal?
In what subset of the complex plane is \(\sinh z\) conformal? For every point \(z_0\) at which the function is not conformal, give an example of two paths (lines) through \(z_0\) such that the angle (or the orientation of the angle) between them is not preserved by \(f(z)\) at \(z_0\).
At which points in \(\mathbb C\) are
the following maps conformal?
(a) \(f(z)=z^3+2i\) (b) \(f(x+iy)=x-3yi\)
In both cases, for every point \(z_0\)
at which the function is not conformal, give an example of two paths
(lines) through \(z_0\) such that the
angle (or the orientation of the angle) between them is not preserved by
\(f(z)\) at \(z_0\).
Let \(f(z)=z^2+2z\). Show that \(f\) is conformal at \(z=i\) and describe the effect of \(f'(z)\) on the tangent vectors of curves passing through this point.
Let \(f(z)=2z^3+3z^2\). Show that \(f\) is conformal at \(z=i\) and describe the effect of \(f'(z)\) on the tangent vectors of curves passing through this point.
Is the following true or false? If \(f,g\) are conformal at a point \(z_0\) then \(f+g\) is conformal at \(z_0\). Give a proof or a counter-example.
Let \(f(z) = \overline{g(z)}\) with \(g(z)\) holomorphic (such functions \(f\) we call anti-holomorphic). Describe geometrically what happens to tangent vectors of paths passing through a point under the map \(f\). Conclude that \(f\) is angle-preserving, but reverses the orientation.
Let \(f: \mathcal D\to \mathcal D'\) be a biholomorphic map between two domains \(\mathcal D\) and \(\mathcal D'\). By considering the equation \(f (f^{-1}(w)) = w\) (for \(w \in \mathcal D'\)), show that \(f\) is conformal.
Let \(\Omega:= \{z \in \mathbb C: \mathop{\text{Re}}(z) > \sqrt{3}\, |\mathop{\text{Im}}(z)| \}\). Sketch the domain \(\Omega\) and find its image \(f(\Omega)\) under the map \(f(z)=z^6\). Hence show that \(f\) is a biholomorphic map from \(\Omega\) onto its image, and give the inverse function.