Topic 2 - Complex Numbers

Week 4 Material

Question 2.1 $\;$ If $z_1=2+3i$ and $z_2=-1+i$, find the real and imaginary parts of

$\;\;(a)\;\;z_1+z_2,\hspace{1cm}$ $(b)\;\;z_1-z_2,\hspace{1cm}$ $(c)\;\;z_1z_2,\hspace{1cm}$ $(d)\;\;z_1/z_2,$

$\;\;(e)\;\;\overline{z_1}z_2,\hspace{1.5cm}$ $(f)\;\;z_1\overline{z_2},\hspace{1.5cm}$ $(g)\;\;z_1\overline{z_1},\hspace{1.cm}$ $(h)\;\;z_2\overline{z_2}$.

Quick Check

$(a)\;\;$ $\Re(z_1+z_2)=1\quad\text{and}\quad \Im(z_1+z_2)=4.$

$(b)\;\;$ $\Re(z_1-z_2)=3\quad\text{and}\quad\Im(z_1+z_2)=2.$

$(c)\;\;$ $\Re(z_1z_2)=-5\quad\text{and}\quad\Im(z_1z_2)=-1.$

$(d)\;\;$ $\Re(z_1/z_2)=1/2\quad\text{and}\quad\Im(z_1/z_2)=-5/2.$

$(e)\;\;$ $\Re(\overline{z_1}z_2)=1\quad\text{and}\quad\Im(\overline{z_1}z_2)=5.$

$(f)\;\;$ $\Re(z_1\overline{z_2})=1\quad\text{and}\quad\Im(z_1\overline{z_2})=-5.$

$(g)\;\;$ $\Re(z_1\overline{z_1})=13\quad\text{and}\quad\Im(z_1\overline{z_1})=0.$

$(h)\;\;$ $\Re(z_2\overline{z_2})=2\quad\text{and}\quad\Im(z_2\overline{z_2})=0.$

Question 2.2 $\!{}^*\;$ Find the real and imaginary parts of

$\;\;(a)\;\;\dfrac{2+5i}{3+2i},\hspace{1cm}$ $(b)\;\;\dfrac{1}{i^7},\hspace{1cm}$ $(c)\;\;\left(-\dfrac{1}{2}+i\dfrac{\sqrt{3}}{2}\right)^2,\hspace{1cm}$ $(d)\;\;\left(-\dfrac{1}{2}+i\dfrac{\sqrt{3}}{2}\right)^{33}.$

Quick Check

$(a)\;$ The real part is $16/13$ and the imaginary part is $11/13$.

$(b)\;$ The real part is $0$ and the imaginary part is $1$.

$(c)\;$ The real part is $-1/2$ and the imaginary part is $-\sqrt{3}/2$.

$(d)\;$ The real part is $1$ and the imaginary part $0$.

Question 2.3 $\;$ Suppose $z=x+iy$ where $x$ and $y$ are real. Express each of the following in terms of $x$ and $y$.

$\;\;(a)\;\;\Re(z^3),\hspace{1cm}$ $(b)\;\;\Im(z^3),\hspace{1cm}$ $(c)\;\;\Re\left(1/\overline{z}\right),\hspace{1cm}$ $(d)\;\;\Im(z+z^{-1}).$

Quick Check

$(a)\;$ $\;\Re(z^3)=x^3-3xy^2$. $\hspace{1cm}$ $(b)\;$ $\;\Im(z^3)=3x^2y-y^3$.

$(c)\;$ $\;\Re\left(1/\overline{z}\right)=\dfrac{x}{x^2+y^2}$. $\hspace{1cm}$ $(d)\;$ $\;\Im(z+z^{-1})=y-\dfrac{y}{x^2+y^2}$.

Question 2.4 $\;$ Calculate the modulus and principal argument of each of the following complex numbers:

$\;\;(a)\;\;1+\sqrt{3}i,\hspace{2.2cm}$ $(b)\;\;3+2i,\hspace{3cm}$ $(c)\;\;-6i,$

$\;\;(d)\;\;\dfrac{2+i}{3-i}-\dfrac{4+i}{1+2i},\hspace{1cm}$ $(e)\;\;\dfrac{2+3i}{i}+\dfrac{5+10i}{2+i},\hspace{1cm}$ $(f)\;\;\dfrac{3+2i}{1+i}+\dfrac{5-2i}{-1+i}.$

Quick Check

$(a)\;$ $|1+\sqrt{3}i|=2$, $\Arg(1+\sqrt{3}i)=\dfrac{\pi}{3}$.

$(b)\;$ $|3+2i|=\sqrt{13}$, $\Arg(3+2i)=\arctan\left(\dfrac23\right)$.

$(c)\;$ $|-6i|=6$, $\Arg(-6i)=-\dfrac{\pi}{2}$.

$(d)\;$ $|z|=\sqrt{4.1}$, $\Arg(z)=\pi-\arctan\left(\dfrac{19}{7}\right)$.

$(e)\;$ $|z|=\sqrt{50}$, $\Arg(z)=\arctan\left(\dfrac{1}{7}\right)$.

$(f)\;$ $|z|=\sqrt{5}$, $\Arg(z)=\arctan(2)-\pi$.

Question 2.5 $\;$ If $z_1=1+i\sqrt{3}$ and $z_2=2+2i$, find the modulus and principal argument of

$\;\;(a)^*\;\;z_1,\hspace{1.5cm}$ $(b)^*\;\;z_2,\hspace{1.5cm}$ $(c)\;\;\overline{z_1},\hspace{1.5cm}$ $(d)\;\;\overline{z_2},$

$\;\;(e)\;\;z_1z_2,\hspace{1cm}$ $(f)^*\;\;z_1/z_2,\hspace{1cm}$ $(g)\;\;z_2/z_1,\hspace{1cm}$ $(h)^*\;\;z_1+z_2.$

Quick Check

$(a)\;$ Tutorial question

$(b)\;$ Tutorial question

$(c)\;$ $\displaystyle |\overline{z_1}|=2\qquad\text{and}\qquad \Arg(\overline{z_1})=-\frac{\pi}{3}.$

$(d)\;$ $\displaystyle |\overline{z_2}|=2\sqrt{2}\qquad\text{and}\qquad \Arg(\overline{z_2})=-\frac{\pi}{4}.$

$(e)\;$ $\displaystyle |z_1z_2|=4\sqrt{2}\qquad\text{and}\qquad \Arg(z_1z_2)=\frac{7\pi}{12}.$

$(f)\;$ Tutorial question

$(g)\;$ $\displaystyle \left|\frac{z_2}{z_1}\right|=\sqrt{2}\qquad\text{and}\qquad \Arg\left(\frac{z_2}{z_1}\right)=-\frac{\pi}{12}.$

$(h)\;$ Tutorial question

Question 2.6 $\!{}^*\;$ Which of the following are true for all complex numbers $z$ and $w$? In each case, either prove it is true or give a counterexample.

$\;\;(a)\;\;\Re(z+w)=\Re(z)+\Re(w),\hspace{1.5cm}$ $(b)\;\;\Re(zw)=\Re(z)\Re(w),$

$\;\;(c)\;\;|z+w|=|z|+|w|,\hspace{3.3cm}$ $(d)\;\;\arg(z+w)=\arg(z)+\arg(w).$

Quick Check

$(a)\;$ True. $\hspace{1cm}$ $(b)\;$ False. $\hspace{1cm}$ $(c)\;$ False. $\hspace{1cm}$ $(d)\;$ False.

Question 2.7 $\;$ Suppose $a$, $b$ and $c$ are real numbers with $b^2<4ac$ and $z$ is a complex number satisfying \[az^2+bz+c=0.\] Find $|z|^2=z\overline{z}$ in terms of $a$, $b$ and $c$.

Quick Check

Hint: Use the quadratic formula to find $z$ and show that $z\overline{z}=c/a.$

Question 2.8 $\;$ Find the modulus and argument of $1+i$ and hence find the real and imaginary parts of $(1+i)^{10}$.

Quick Check

Find $|1+i|=\sqrt2,\;\;\arg(1+i)=\pi/4$ and $\Re((1+i)^{10})=0,\;\;\Im((1+i)^{10})=32$.

Week 5 Material

Question 2.9 $\;$ Find all possible values of

$\;\;(a)\;\; i^0, \hspace{1cm}$ $\;\;(b)\;\; i^1, \hspace{1cm}$ $\;\;(c)\;\; i^i, \hspace{1cm}$ $\;\;(d)\;\; (1+i)^i, \hspace{1cm}$ $\;\;(e)^*\;\; (1+i)^{2+i}.$

Quick Check

$(a)\;$ $i^0=1$. $\hspace{1cm}$ $(b)\;$ $i^1=i$.

$(c)\;$ $i^i=e^{-\pi/2-2n\pi} \quad\;\;$ for any $n\in\mathbb{Z}$.

$(d)\;$ $(1+i)^i=e^{-\pi/4-2n\pi+i\ln\sqrt{2}} \quad\;\;$ for any $n\in\mathbb{Z}$.

$(e)\;$ $(1+i)^{2+i}=2e^{-\pi/4-2n\pi}e^{(\ln{2}+\pi)i/2} \quad\;\;$ for any $n\in\mathbb{Z}$.

Question 2.10 $\;$ For each of the following equations, find all complex solutions and plot them on the complex plane.

$\;\;(a)\;\; z^5-32=0, \hspace{1.7cm}$ $\;\;(b)\;\; z^5+32=0, \hspace{1.0cm}$ $\;\;(c)\;\; z^6+8i=0, \hspace{1cm}$

$\;\;(d)\;\; z^2+2z+2=0, \hspace{1cm}$ $\;\;(e)^*\;\; z^6-9z^3+8=0.$

Quick Check

$(a)\;$ $z=2e^{2n\pi i/5}=2\left(\cos\dfrac{2n\pi}{5}+i\sin\dfrac{2n\pi}{5}\right)$ for $n=-2,-1,0,1,2$.

$(b)\;$ $z=2e^{(2n+1)\pi i/5}= 2\left(\cos\dfrac{(2n+1)\pi}{5}+i\sin\dfrac{(2n+1)\pi}{5}\right)$ for $n=-2,-1,0,1,2$.

$(c)\;$ $z=\sqrt{2}e^{(4n-1)\pi i/12}= \sqrt2\left(\cos\dfrac{(4n-1)\pi}{12}+i\sin\dfrac{(4n-1)\pi}{12}\right)$ for $n=-2,-1,0,1,2$.

$(d)\;$ $z=-1\pm i$

$(e)\;$ $z=e^{2n\pi i/3}, \;2e^{2n\pi i/3} \qquad\text{for $n=-1,0,1$.}$ These six solutions can also be written as \[1,\;\dfrac{-1\pm i\sqrt3}{2}\quad\textrm{and} 2,\; -1\pm i\sqrt3.\]

Question 2.11 $\;$ Let $P(z)=a_nz^n+a_{n-1}z^{n-1}+...+a_1z+a_0$ be a polynomial with real coefficients $a_0,a_1,...,a_n$. If $P(z)=0$, show that $P(\overline{z})=0$ as well. In other words, a polynomial with real coefficients has roots which are real or occur in pairs of complex conjugates.

Quick Check

Hint: Take complex conjugates of $P(z)$.

Question 2.12 $\;$ Let $P(z)=z^4-z^3+z^2+2a$ where $a$ is a real number and suppose that $1+i$ is a root of $P(z)$.

$\;\;(a)\;$ Using the result of the previous question, find a real quadratic factor of $P(z)$.

$\;\;(b)\;$ Use $P(1+i)=0$ to find the value of $a$.

$\;\;(c)\;$ Express $P(z)$ as a product of two real quadratic factors and hence find all of its roots.

Quick Check

$(a)\;$ $(z-1-i)(z-1+i)=z^2-2z+2$ is a factor.

$(b)\;$ $a=1.$ $\hspace{1cm}$ $(c)\;$ The four roots are $1\pm i$, $\dfrac{-1\pm i\sqrt3}{2}$.

Question 2.13 $\;\;$ Find all complex solutions to the following equations

$\;\;(a)\;\; e^z=e, \hspace{1cm}$ $\;\;(b)\;\; e^z=3, \hspace{1cm}$ $\;\;(c)\;\; e^z=-3,\hspace{1cm}$ $\;\;(d)\;\; e^z=i.$

Quick Check

$(a)\;$ $z=1+2n\pi i\quad$ for $n\in\mathbb{Z}$. $\hspace{1cm}$ $(b)\;$ $z=\ln{3}+2n\pi i\quad$ for $n\in\mathbb{Z}$.

$(c)\;$ $z=\ln{3}+(2n+1)\pi i\quad$ for $n\in\mathbb{Z}$. $\hspace{1cm}$ $(d)\;$ $z=\left(\frac{\pi}{2}+2n\pi\right)i\quad$ for $n\in\mathbb{Z}$.

Question 2.14 $\;$ Find the principal values of the logarithms

$\;\;(a)\;\; \Ln{e},\hspace{1cm}$ $\;\;(b)\;\; \Ln{3},\hspace{1cm}$ $\;\;(c)\;\; \Ln(-3),\hspace{1cm}$ $\;\;(d)\;\; \Ln{i}.$

Quick Check

$(a)\;\; 1,\hspace{1cm}$ $(b)\;\; \ln{3},\hspace{1cm}$ $(c)\;\; \ln{3}+\pi i,\hspace{1cm}$ $(d)\;\; \frac{\pi}{2}i$.

Question 2.15 $\;$ Find all complex numbers $z$ for which

$\;\;(a)\;\; \cos{z}=2,\hspace{1cm}$ $\;\;(b)^*\;\; \sin{z}=-1,\hspace{1cm}$ $\;\;(c)\;\; \cosh{z}=-2.\hspace{1cm}$

Quick Check

$(a)\;$ $z=2n\pi-i\ln(2\pm\sqrt3).$ or $z=2n\pi\pm i\ln(2+\sqrt3)\qquad\text{for $n\in\mathbb{Z}$.}$

$(b)\;$ $z=-\frac{\pi}{2}+2n\pi\qquad\text{for $n\in\mathbb{Z}$.}$

$(c)\;$ $z=\ln(2\pm\sqrt3)+(2n+1)\pi i\qquad\text{for $n\in\mathbb{Z}$.}$

Question 2.16 $\;$ For real $\theta$, use de Moivre’s theorem to express

$\;\;(a)\;\; \cos(3\theta)$ as a polynomial in $\cos\theta,\hspace{1cm}$ $\;\;(b)\;\; \sin(5\theta)$ as a polynomial in $\sin\theta$.

Quick Check

$(a)\;$ $\cos(3\theta)=4\cos^3\theta-3\cos\theta.$

$(b)\;$ $\sin(5\theta)=16\sin^5\theta-20\sin^3\theta+5\sin\theta.$

Question 2.17 $\;$ Show that $\cos{4\theta}=8\cos^4\theta-8\cos^2\theta+1$ and deduce that \[\cos\frac{\pi}{8}=\frac{1}{2}\sqrt{2+\sqrt2}.\]

Quick Check

Hint: What are the four solutions of $8c^4-8c^2+1=0$? Explain why one of these values must be $\cos(\pi/8)$ and decide which one.

Question 2.18 $\;$ For real $\theta$, use de Moivre’s theorem to show that

$\;\;(a)\;\; \cos^3\theta=\dfrac{\cos(3\theta)+3\cos\theta}{4}, \hspace{1cm}$ $\;\;(b)^*\;\; \cos^4\theta=\dfrac{\cos(4\theta)+4\cos(2\theta)+3}{8}.$

Quick Check

$(a)\;$ Express $\cos(3\theta)$ in terms of $\cos\theta$ and rearrange. Alternatively, expand $\cos^3\theta=\left(\dfrac{e^{i\theta}+e^{-i\theta}}{2}\right)^3$ and recombine.

$(b)\;$ similar but with $\sin$.

Question 2.19 $\;$ For real $\theta$, use de Moivre’s theorem to express

$\;\;(a)\;\; \sin^3\theta$ as a combination of $\sin(3\theta)$ and $\sin\theta$,

$\;\;(b)\;\; \sin^4\theta$ as a combination of $\cos(4\theta)$ and $\cos(2\theta)$.

Quick Check

$(a)\;$ $\sin^3(\theta)=\dfrac{3\sin\theta-\sin(3\theta)}{4}$, $\hspace{1cm}$ $(b)\;$ $\sin^4(\theta)=\dfrac{\cos(4\theta)-4\cos(2\theta)+3}{8}$.

Question 2.20 $\;$ Given that \[1+z+z^2+z^3+...=\sum_{n=0}^{\infty}z^n=\frac{1}{1-z}\quad\text{for $|z|<1$}\] use de Moivre’s theorem to calculate the following infinite sums for real $\theta$:

$\;\;(a)\;\; \displaystyle\sum_{n=0}^{\infty}\left(\frac{1}{2}\right)^n\cos(n\theta),\hspace{1cm}$ $\;\;(b)\;\; \displaystyle\sum_{n=0}^{\infty}\left(\frac{1}{2}\right)^n\sin(n\theta).$

Quick Check

Hint: do both at once by using $e^{in\theta}=\cos(n\theta)+i\sin(n\theta)$.

Then apply the geometric sum with $z=\dfrac12e^{i\theta}$ and rewrite in terms of $\cos$ and $\sin$. You should find

$(a)\;$ $\dfrac{4-2\cos\theta}{5-4\cos\theta}$, $\hspace{1cm}$ $(b)\;$ $\dfrac{2\sin\theta}{5-4\cos\theta}$.

Question 2.21 $\;$ Use the definitions \[\cos{z}=\dfrac{e^{iz}+e^{-iz}}{2} \quad\text{and}\quad \sin{z}=\dfrac{e^{iz}-e^{-iz}}{2i}\] to show that

$\;\;(a)\;\; \sin(z+w)=\sin{z}\cos{w}+\cos{z}\sin{w},$

$\;\;(b)\;\; \cos(z+w)=\cos{z}\cos{w}-\sin{z}\sin{w}.$

In other words, prove the usual double angle formulae hold for all complex $z$ and $w$.

Quick Check

Hint: start from the (more complicated) right-hand sides and simplify.

Question 2.22 $\;$ Recall the hyperbolic functions: \[\cosh{y}=\dfrac{e^y+e^{-y}}{2} \quad\text{and}\quad \sinh{y}=\dfrac{e^y-e^{-y}}{2}.\] Show that for an arbitrary complex number $z=x+iy$,

$\;\;(a)\;\; \sin{z}=\sin{x}\cosh{y}+i\cos{x}\sinh{y},\hspace{1cm}$

$\;\;(b)\;\; |\sin{z}|=\sqrt{\sin^2{x}+\sinh^2{y}}$.

Find similar formulae for $\cos{z}$ and $|\cos{z}|$.

Quick Check

Hint: start from the (more complicated) right-hand sides and simplify.

Week 6 Material

Question 2.23 $\;$ Suppose the points $A$, $B$, $C$ and $D$ represent the complex numbers $z_1$, $z_2$, $z_3$ and $z_4$ respectively, and $O$ is the origin $z=0$.

$\;\;(a)\;\;$ If $OABC$ is a parallelogram and $z_1=1+i$ and $z_2=4+5i$, find $z_3$.

$\;\;(b)\;\;$ If $ABCD$ is a square, find $z_2$ and $z_4$ in the cases

$\hspace{1.5cm}(i)\;\; z_1=2+i$ and $z_3=6+7i\hspace{1cm}$ $(ii)\;\; z_1=6-2i$ and $z_3=6i$.

Quick Check

$(a)\;$ $z_3=3+4i.$

$(b)\;(i)\;$ $z_2=7+2i, \; z_4= 1+6i$ and $\;\;\;\;(ii)\;$ $z_2=7+5i, \; z_4=-1-i$.

Question 2.24 $\;$ Three points $z_1$, $z_2$ and $z_3$ form an equilateral triangle in the complex plane. Given that $z_1=2+i$ and $z_2=3+i(1+\sqrt{3})$,

$\;\;(a)\;\;$ Find the modulus and argument of $z_2-z_1$.

$\;\;(b)\;\;$ Find the modulus of $z_3-z_1$.

$\;\;(c)\;\;$ Find the possible arguments of $z_3-z_1$.

$\;\;(d)\;\;$ Hence find the possible values of $z_3$.

Quick Check

$(a)\;$ $|z_2-z_1|=2 \quad\text{and}\quad \operatorname{Arg}(z_2-z_1)=\frac{\pi}{3}.$

$(b)\;$ $|z_3-z_1|=2$.

$(c)\;$ $\arg(z_3-z_1)=0 \quad\text{or}\quad 2\pi/3$.

$(d)\;$ $z_3=4+i\quad\text{or}\quad 1+i(1+\sqrt3)$.

Question 2.25 $\;$ A regular hexagon in the complex plane has centre $z_0=3+i$ and one of the vertices at $z_1=3+3i$. Find the other vertices of the hexagon.

Quick Check

The six vertices are $3 + 3i,\; 3-i, \; 3\pm\sqrt3,\; 3\pm\sqrt3+2i$.

Question 2.26 $\;$ Sketch the set of numbers $z$ in the complex plane satisfying:

$\;\;(a)\;\;|z|=2,\hspace{3.4cm}$ $\;\;(b)\;\;|z-2|=|z+2|,\hspace{1.5cm}$ $\;\;(c)^*\;\;\sqrt{3}\;|z-2|=|z+2|,$

$\;\;(d)\;\;|z-2|+|z+2|=6,\hspace{1cm}$ $\;\;(e)^*\;\;\arg\left(\dfrac{z-2}{z+2}\right)=\dfrac{\pi}{2},\hspace{1cm}$ $\;\;(f)\;\;\arg\left(\dfrac{z-2}{z+2}\right)=\dfrac{\pi}{4}$.

Quick Check

Hint: Either work geometrically in terms of lengths and angles, or set $z=x+iy$ and rewrite the conditions in terms of $x$ and $y$.

Question 2.27 $\;$ Find the unique complex number $z$ satisfying \[|z-1-3i|=|z-1-i|\qquad\text{and}\qquad\Arg(z)=\frac{\pi}{4}.\] (You can do this algebraically, but there is a quick geometric way too.)

Quick Check

You should find $z=2+2i$.

Question 2.28 $\;$ Describe the sets of complex numbers such that

$\;\;(a)\;\;$ $w=(z-i)/(z-1)$ is a real number,

$\;\;(b)\;\;$ $w=(z-i)/(z-1)$ is purely imaginary.

$\;\;$(Hint: $w$ is real exactly when $w=\overline{w}$ and is purely imaginary when $w=-\overline{w}$.

$\;\;$ Use this to find a relationship between $x=\Re(z)=\dfrac{z+\overline{z}}{2}$ and $y=\Im(z)=\dfrac{z-\overline{z}}{2i}$.)

Quick Check

This is quite difficult, but can be done using the hint and some careful algebra.

Question 2.29 $\;$ In a series RLC circuit, the complex impedance $Z$ depends on the (real-valued) frequency $\omega$ via the formula \[Z(\omega)=R+i\omega L+\frac{1}{i\omega C}\] where the capacitance $C$, inductance $L$ and resistance $R$ are real constants.

$\;\;(a)\;\;$ Find, in terms of $L$ and $C$, the resonant frequency of the circuit,

$\qquad\;\,$ i.e. the frequency $\omega_0$ at which the impedance $Z(\omega_0)$ has minimal modulus.

$\;\;(b)\;\;$ Show that \[\frac{Z(\omega)}{Z(\omega_0)}= 1+iQ\left(\frac{\omega}{\omega_0}-\frac{\omega_0}{\omega}\right)\] $\qquad\;\,$ for some real constant $Q$ (this is called the ``$Q$ factor” of the circuit).

$\;\;(c)\;\;$ Find, in terms of $\omega_0$ and $Q$, the values of $\omega$ for which the ratio in part $(b)$ has modulus $\sqrt{2}$.

$\;\;(d)\;\;$ Given that the complex voltage $\widehat{V}$ and current $\widehat{I}$ are related by $\widehat{V}=\widehat{I}Z(\omega)$, find the

$\qquad\;\,$ phase difference $\Arg(\widehat{V})-\Arg(\widehat{I})$ in terms of $Q$, $\omega$ and $\omega_0$.

Quick Check

$(a)\;$ $\omega_0=\dfrac{1}{\sqrt{LC}}$.

$(b)\;$ You should find $Q=\dfrac{1}{R}\sqrt{\dfrac{L}{C}}$

$(c)\;$ $\omega=\dfrac{\omega_0}{2Q}\left(\pm 1\pm\sqrt{1+4Q^2}\right).$

$(d)\;$ $\operatorname{Arg}(V)-\operatorname{Arg}(I)=\arctan\left(Q\left(\dfrac{\omega}{\omega_0}-\dfrac{\omega_0}{\omega}\right)\right).$

Question 2.30 $\!{}^*\;$ In a particular RLC circuit, the complex impedance $Z$ satisfies \[Z^{-1}=(R+i\omega L)^{-1}+i\omega C\] where the capacitance $C$, inductance $L$ and resistance $R$ are real constants and $\omega$ is the (real-valued) frequency. Show that \[|Z|^2=\frac{R^2+\omega^2L^2}{(1-CL\omega^2)^2+C^2R^2\omega^2}\] and \[\Arg(Z)=\arctan\left[\frac{\omega}{R}\Big(L-C(R^2+\omega^2L^2)\Big)\right].\]

(Hint: make use of $|\alpha/\beta|=|\alpha|/|\beta|$, etc.)

Quick Check

Hint: Use the hint or it’s a mess!