Express the following complex numbers in the form \(x + i y\) with \(x,y\) real:
(a) \(\frac{(1+2i)(5+2i)}{(2-3i)(1-i)},\) (b)
\(\frac{1+4i}{1-i} +
\frac{1-4i}{1+i}.\)
Evaluate the product \((1+i)i(1+i)\), first in the “usual algebraic way", then by writing \(i\) and \(1+i\) in polar form.
Write the following complex numbers in polar form:
(a) \(2-i2\sqrt 3,\) (b) \(\frac{1}{2+i} - \frac{1}{2-i} ,\) (c) \(\frac{2-i2\sqrt 3}{1+i}.\)
Find all complex numbers \(z\) for
which:
(a) \(|\mathop{\text{Re}}(z)| = |z|\),
(b) \(\mathop{\text{Im}}(z) = |z|\),
(c) \(|z|^2 = z^2\).
If \(w = \frac{z-1}{z+1}\) show that \(\mathop{\text{Re}}(w) = \frac{|z|^2-1}{|z|^2+2Re(z) +1}\) and \(\mathop{\text{Im}}(w) = \frac{2Im(z)}{|z|^2+2Re(z) +1}.\)
Show that (a) \(\overline{z_1+z_2}=\overline{z_1}+\overline{z_2}\). (b) \(\overline{z_1z_2}=\overline{z_1}\ \overline{z_2}\).
[You could prove these directly by using the definition of conjugation, but there are nicer ways: For (a) note complex conjugation viewed as a reflection in \(\mathbb R^2\) is a linear map. For (b), you could use the fact that \(\overline{z_1 z_2}(z_1z_2) = |z_1z_2|^2\).]
For each pair of complex numbers \(z_1\) and \(z_2\) prove the parallelogram identity: \[|z_1 + z_2|^2 + |z_1-z_2|^2 = 2 \bigl( |z_1|^2 + |z_2|^2 \bigr).\] Interpret this equation geometrically.
What is the geometric meaning of the following functions \(f(z)\) as transformations \(z\mapsto f(z)\) from \(\mathbb C\) to \(\mathbb C\)?
(a)\(\> f(z) = 2z, \qquad \> {\rm
(b)}\> f(z) = -z, \qquad\> {\rm (c)}\> f(z) =
(1+i)z,\)
(d)\(\> f(z) = -\bar{z}, \qquad{\rm (e)}
\>f(z) = z/|z|, \quad\>\>\, {\rm (f)}\> f(z) = 1-i+z,
\qquad\>\>{\rm (g)}\> f(z) =1-i+(1+i)z\).
Draw the following sets of points in the complex plane. \[(a) \ \ z+\bar z=2\ , \quad\qquad (b) \ \ z-\bar z=3i\ , \qquad\quad (c) \ \ {\left|\bar z\right|}=1\ ,\qquad \quad (d) \ \ {\left|z-i\right|}=1\ .\]
Solve the following equations in complex numbers, and mark the
solutions on a picture of the complex plane:
(a) \(|z+2|=|z-2|\), (b) \(\bar z = 1/z,\) (c) \(z =\frac{
\mathop{\text{Re}}z+\mathop{\text{Im}}z}{2},\) (d) \(|(z-2 )(\bar z-2)| = 1.\)
What do the following equations represent geometrically? Give
sketches.
(i) \(|z+2|=6\) (ii) \(|z-3i|=|z+i|\) (iii) \(|iz-1|=|iz+1|\) (iv) \(|z+1-i|=|\bar z- 1-i|\).
Describe geometrically the subsets of \(\mathbb C\) specified by
(i) \(\mathop{\text{Im}}(z+i)>2\)
(ii) \(1<\mathop{\text{Re}}z\le 2\)
(iii) \(|z-1-i|>1\)
(iv) \(|z-1+i|\ge |z-1-i|\) (v) \(|z+2-i| < |iz-1+2i|\) (vi) \(1< |z-1|<2\).
Show that the equation \(|z-a|= \lambda |z-b|\), where \(a\) and \(b\) are complex numbers and \(\lambda>0\), describes a circle in the complex plane if \(\lambda \neq 1\). [In fact, every circle in the complex plane can be written in this form!] What geometric figure is represented when \(\lambda = 1\)?
(i) Apply induction to show De Moivre’s formula: \((\cos(x)+i\sin(x))^n = \cos(nx) + i
\sin(nx)\).
(ii) Use this to write \(\cos(3x)\) as
a polynomial in \(\cos(x)\); namely
show that \(\cos(3x) = 4\cos^3(x) -
3\cos(x)\).
Write \((1+i\sqrt 3)^{100}\) in \(x+iy\) form.
Show that the inverse of the stereographic projection \(P:\mathbb{S}^{2}\setminus\{N\}\to\mathbb C\) is given by \[P^{-1}\left( z\right)=\left( \frac{2\mathop{\text{Re}}\left( z\right)}{1+{\left|z\right|}^2},\frac{2\mathop{\text{Im}}\left( z\right)}{1+{\left|z\right|}^2},\frac{{\left|z\right|}^2-1}{1+{\left|z\right|}^2}\right).\]
Consider the inverse stereographic projection \(P^{-1}:\mathbb C\to \mathbb{S}^2\setminus
\left\lbrace N\right\rbrace\).
(a) Show that \(P^{-1}\) takes the
circle \(\left\lbrace z\in\mathbb C\; | \;
{\left|z\right|}=c\right\rbrace\), where \(c> 0\) is a given positive number, to a
circle on \(\mathbb{S}^2\setminus \left\lbrace
N\right\rbrace\) that is parallel to the \(xy-\)plane.
(b)* Explain geometrically why the image of the line \(a \mathop{\text{Re}}\left(
z\right)+b\mathop{\text{Im}}\left( z\right)=0\), where \(a,b\in\mathbb R\) are not both zero, by
\(P^{-1}\) lies on a great circle on
\(\mathbb{S}^2\setminus \left\lbrace
N\right\rbrace\) that passes via the south pole (in fact – it is
the entire circle).
Show that \(P^{-1}(z) = -P^{-1}(w)\) (i.e. the point \(P^{-1}(z)\) and \(-P^{-1}(w)\) are diametrically opposite on the Riemann sphere) if and only if \(w=-\frac{1}{{\bar z}}\).