Complex Analysis II, Michaelmas 2024. Assignment 4: Complex Functions and Complex Differentiation

Show that for any complex \(z\),
(a) \(\cos^2 z+\sin^2z=1\),
(b) \(\sin(2z)=2\sin z\cos z\)

(a) By writing \(\cos z=\frac{e^{iz}}{2} (1+e^{-2iz})\) or otherwise, determine all complex \(z\) for which \(\cos z =0\).
(b) Solve the equation \(\cosh z=0\) in complex numbers.
(c) Solve the equation \(\sin z +\cos z=0\) in complex numbers.
(d) Solve the equation \(e^{\frac 1 z}= \frac{e^2}{\sqrt 2}(1+i)\).

Write each of the following in \(x+iy\) form:
(a) \(4e^{i\pi/3} + \sqrt 2,\) (b) \(\cos i,\) (c) \(\sin (\pi/2+ 2i),\) (d) \(\sinh (i\pi/2),\) (e) \(\sinh i +\cosh i.\)

The real axis and the imaginary axis divide \(\mathbb C\) into four quadrants as follows:

By considering the modulus of the function, determine the images of \(\Omega_1, \Omega_2, \Omega_3, \Omega_4\) under the exponential map \(z\mapsto e^z\).

Using the principal branch of \(\log z\), determine the \(x+iy\) form of
\({\rm(a)}\> \log{(2i)}, \quad {\rm(b)}\> \sqrt{2i}, \quad {\rm(c)}\> i^i. \quad\)

Give examples to illustrate that, in general, for complex numbers \(z\), \(w\),
(a) \(\log e^z\ne z\),
(b) \(\log (zw)\ne \log z+\log w\),
(c) \(\sqrt{zw}\ne \sqrt{z}\sqrt{w}\).
Here all of the functions are defined with the principal branch of \(\log z\).

(a) Determine \(1^{1/4}\) if \(z^w\) is defined using the principal branch of logarithm.
(b) What are the other possible values of \(1^{1/4}\) if the branch is not principal?

(a) Determine the value of \(\sqrt{(2i)}\) according to the following three branches of the \(\log\) function: (i) the principal branch; (ii) \(\pi<\arg z<3\pi\); (iii) \(4\pi<\arg z<6\pi\).
(b) For any non-zero \(z=r e^{i \phi}\) and any branch of \(\log z\) for which \(\sqrt{z}\) is defined show that either \(\sqrt{z}=\sqrt{r}e^{i \phi/2}\) or \(\sqrt{z}=-\sqrt{r}e^{i \phi/2}\).
(c) More generally, for non-zero \(z=r e^{i \phi}\) and an integer \(n\ge 1\) show that there are exactly \(n\) possible “\(n\)-th roots of \(z\)”, that is values of \(z^{1/n}\) for various choices of the branch of \(\log z\).

(a) From the definition of complex differentiability, show that \(f(z)=1/z\) is complex differentiable for all non-zero complex \(z\), and determine its derivative.
(b) Verify the Cauchy-Riemann equations for \(f(z)=1/z\).

(a) Prove that \(f(z)=|z|\) is not complex differentiable anywhere.
(b) Show that \(g(z)=z \overline z = |z|^2\) is differentiable at the origin and nowhere else. Find \(g'(0)\).

Find out where the following functions are differentiable and give formulae for their derivatives (from the lectures we already know that exp, trigonometric functions and polynomials are differentiable everywhere):

Define \(f:\mathbb C\to \mathbb C\) by \(f(0)=0\), and \[f(z)=\frac{(1+i)x^3-(1-i)y^3}{x^2+y^2} \quad\text{for $z=x+iy\neq 0$.}\] Show that \(f\) satisfies the Cauchy-Riemann equations at \(0\) but is not differentiable there. [Hint: consider what happens as \(z\to 0\) along the line \(y=x\) and the line \(y=0\).]

At which points are the following functions differentiable?
(i) \(f(z)=x^2+2ixy\);
(ii) \(f(z)=2xy+i(x+\frac{2}{3}y^3)\);
(iii) \(f(z)=x\cosh y+\sin(iy)\cos x\);
(iv) \(f(z)=e^{-1/{\left|z\right|}^2}\) (\(z\ne0\)), \(f(0)=0\).

Find all complex differentiable functions defined on the whole of \(\mathbb C\) of the form \(f(z)=u(x)+iv(y)\) where \(u\) and \(v\) are both real valued.

Show that the principle branch of the complex logarithm function is complex differentiable at all points of \(\mathbb C\setminus\mathbb R_{\leq 0}\), and has derivative \(1/z\). [Hint: notice that if \(z=x+iy \neq 0\) we can write \[\mathop{\text{Arg}}(z) = \begin{cases} \arctan{(y/x)} \ & \text{ if } \quad x>0, \\ \arctan{(y/x)} + \mathop{\mathrm{sgn}}(y)\pi \ & \text{ if } \quad x<0, y \neq 0, \\ \mathop{\mathrm{sgn}}(y) \pi/2 \ & \text{ if } \quad x=0, y \neq 0, \end{cases}\] where \(\mathop{\mathrm{sgn}}(y)\) is the standard sign function taking values \(\pm 1\) depending on whether \(y\) is strictly positive or strictly negative.]

Are the following functions \(f(z)\!=\!f(x\!+\!iy)\) complex differentiable? [Remember to justify your responses.] For those that are, determine the derivative \(f'(z)\).
(a) \(\displaystyle f(z)=\frac{x}{x^2+y^2} - \frac{y}{x^2+y^2}\,i,\qquad (z\ne0\))
(b) \(f(z)=\sin(y)+i\cos(x)\),
(c) \(f(z)=\overline{e^{\bar{z}}}\),
(d) \(f(z)=\tan(z) \quad [= {\sin z \over \cos z}].\)

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 (connected) open set \(X\) of complex numbers then \(f(z)\) is constant on \(X\).
(i) \(f(z)\) is a real number for all \(z \in X\).
(ii) the real part of \(f(z)\) is constant on \(X\).
(iii) the modulus of \(f(z)\) is constant on \(X\).

Remark: for instance, (i) shows that the functions \(|z|\), \(\mathop{\text{Re}}z\), \(\mathop{\text{Im}}z\) and \(\arg z\) are not holomorphic.