(Assignment sheet 4 problem 17) Let \(f(z)\) be a holomorphic function. Prove the
following variants of the Zero derivative
theorem, which says that, if any one of the following
conditions hold on a domain \(D\) then
\(f(z)\) is constant on \(D\).
(i) \(f(z)\) is a real number for all
\(z \in D\).
(ii) the real part of \(f(z)\) is
constant on \(D\).
(iii) the modulus of \(f(z)\) is
constant on \(D\).
(Assignment sheet 5 problem 5) 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 \(z_0\).
(Assignment sheet 6 problem 5) Is there a Möbius transformation which maps the sides of the triangle with vertices at \(-1,i\) and \(1\) to the sides of an equilateral triangle (all sides of equal length)? Either give an example of such a Möbius transformation, or explain why it is not possible.
(Assignment sheet 6 problem 8) If \(\alpha\) and \(\beta\) are the two fixed points of a Möbius transformation \(f(z)\), show that for all \(z\neq \alpha,\beta\) and \(f(z)\neq\infty\), we have \[\frac{f(z)-\alpha}{f(z)-\beta} =K\frac{z-\alpha}{z-\beta},\] where \(K\) is a constant.
(Assignment sheet 6 problem 9) Find the Möbius transformation taking the ordered set of points \(\{-i,-1,i \}\) to the ordered set of points \(\{-i,0,i\}\). What is the image of the unit disc under this map? Which point is sent to \(\infty\)?