--- author: Alvie Rahman date: \today title: MMME1026 // Mathematics for Engineering tags: [ uni, nottingham, mmme1026, maths, complex_numbers ] --- # Lecture 1 // Complex Numbers (2021-10-04) ## Complex Numbers - $i$ is the unit imaginary number, which is defined by: $$ i^2 = -1 $$ - An arbritary complex number is written in the form $$z = x + iy$$ Where: - $x$ is the real part of $z$ (which you may seen written as $\Re(z) = x$ or Re$(z) = x$) - $y$ is the imaginary part of $z$ (which you may seen written as $\Im(z) = y$ or Im$(z) = y$) - Two complex numbers are equal if both their real and imaginary parts are equal e.g. $$(3 + 4i) + (1 -2i) = (3+1) + i(4-2) = 2 + 2i$$ ### Complex Conjugate Given complex number $z$: $$z = z + iy$$ The complex conjugate of z, $\bar z$ is: $$\bar{z} = z -iy$$ ### Division of Complex Numbers - Multiply numerator and denominator by the conjugate of the denominator
#### Example > \begin{align*} z_1 &= 5 + i \\ z_2 &= 1 -i \\ \\ \frac{z_1}{z_2} &= \frac{5+i}{1-i} \cdot \frac{\bar{z_2}}{\bar{z_2}} = \frac{(5+i)(1+i)}{(1-i)(1+i)} \\ &= \frac{5 + i + 5i -1}{1 + 1} \\ &= \frac{4 + 6i}{2} = 2 + 3i > \end{align*}
### Algebra and Conjugation When taking complex conjugate of an algebraic expresion, we can replace $i$ by $-i$ before or after doing the algebraic operations: \begin{align*} \overline{z_1+z_2} &= \bar{z_1}+\bar{z_2} \\ \overline{z_1z_2} &= \bar{z_1}\bar{z_2} \\ \overline{z_1/z_2} &= \frac{\bar{z_1}}{\bar{z_2}} \end{align*} The conjugate of a real number is the same as that number. #### Application If $z$ is a root of the polynomial equation $$0 = az^2 + bz + c$$ with **real** coefficients $a$, $b$, and $c$, then $\bar{z}$ is also a root because \begin{align*} 0 &= \overline{az^2 + bz + c} \\ &= \bar{a}\bar{z}^2 + \bar{b}\bar{z} + \bar{c} \\ &= a\bar{z}^2 + b\bar{z} + c \end{align*} ### The Argand Diagram A general complex number $z = x + iy$ has two components so it can can be represented as a point in the plane with Cartesion coordinates $(x, y)$. \begin{align*} 4-2i &\leftrightarrow (4, -2) \\ -i &\leftrightarrow (0, -1) \\ z &\leftrightarrow (x, y) \\ \bar z &\leftrightarrow (x, -y) \end{align*} ### Plotting on a Polar Graph We can also describe points in the complex plain with polar coordinates $(r, \theta)$: \begin{align*} z &= r(\cos\theta + i\sin\theta) &\text{ polar form of $z$} \\ r &= \sqrt{x^2+y^2} &\text{(modulus)}\\ \theta &= \arg z,\text{ where} \tan \theta = \frac y x &\text{(argument)} \\ x &= r\cos \theta \\ y &= r\sin \theta \end{align*} Be careful when turning $(x, y)$ into $(r, \theta)$ form as $\tan^{-1} \frac y x = \theta$ does not always hold true as there are many solutions. #### Choosing $\theta$ Correctly 1. Determine which quadrant the point is in (draw a picture). 2. Find a value of $\theta$ such that $\tan \theta = \frac y x$ and check that it is consistent. If it puts you in the wrong quadrant, add or subtract $\pi$. # Lecture 2 // Complex Numbers (2021-10-12) ## Exponential Functions - The exponential function $f(x) = \exp x$ may be wirtten as an infinite series: $$\exp x = e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ... $$ - The function $f(x) = e^{-x}$ is just $\frac 1 {e^x}$ - Note the important properties: \begin{align*} e^{a+b} &= e^a e^b \\ (e^a)^b &= e^{ab} \end{align*} ## Euler's Formula $$e^{i\theta} = \cos\theta + i\sin\theta$$ - Properties of $e^{i\theta}$: For any real angle $\theta$ we have $$|e^{i\theta}| = |\cos\theta + i\sin\theta| = \sqrt{\cos^2\theta + \sin^2\theta} = 1$$ and $$ \arg {e^{i\theta}} = \theta $$ - A complex number in *polar form*, where $r = |z|$, and $\theta = \arg z$, may alternatively be written in its *exponential form*: $$z = re^{i\theta}$$ **Note**: $$\bar z = r\cos\theta - ir\sin\theta = re^{-i\theta}$$
### Example 1 Write $z = -1 + i$ in exponential form > $\arg z = \frac {3\pi} 4$ > $|z| = \sqrt 2$ > > So $z = \sqrt2e^{i\frac{3\pi} 4}$
### Example 2 The equations for a mechanical vibration problem are found to have the following mathematical solution: $$z(t) = \frac{e^{i\omega t}}{\omega_0^2-\omega^2 + i\gamma}$$ where $t$ represents time and $\omega$, $\omega_0$ and $\gamma$ are all positive real physical constants. Although $z(t)$ is complex and cannot directly represent a physical solution, it turns out that the real and imaginary parts $x(t)$ and $y(t)$ in $z(t) = x(t) + iy(t)$ can. Polar notation can be used to extract this physical information efficiently as follows: a. Put the denominator in the form $$ae^{i\delta}$$ where you should give explicit expressions for $a$ and $\delta$ in terms of $\gamma$, $\gamma_0$, and $\gamma$. > \begin{align*} a &= \sqrt{\gamma^2 + (\omega_0^2 - \omega^2)^2} \\ \delta &= \tan^{-1}\frac \delta {\omega_0^2 - \omega^2} > \end{align*} b. Hence find the constants $b$ and $\varphi$ such that $$x(t) = b\cos(\omega t + \varphi)$$ and write a similar expression for $y(t)$. > \begin{align*} z &= \frac{e^i\omega t}{ae^{i\delta}} = \frac 1 a e^{i (\omega t - \delta)} \\ x + iy &= \frac 1 a \cos(\omega t - \delta) + \frac 1 a \sin(\omega t - \delta) \\ \therefore \Re z &= x = \frac 1 a \cos(\omega t - \delta), \\ \Im z &= y = \frac 1 a \sin(\omega t - \delta) \\ \\ b &= \frac 1 a = \frac 1 {\sqrt{\gamma^2 + (\omega_0^2 - \omega^2)^2}} \\ \varphi &= -\delta = \tan^{-1}\frac \delta {\omega_0^2 - \omega^2}\\ \\ y(t) &= \frac 1 a \sin(\omega t - \delta) \\ > \end{align*}
## Products of Complex Numbers Suppose we have 2 complex numbers: $$z_1 = x_1 + iy_1 = r_1e^{i\theta_1}$$ $$z_2 = x_2 + iy_2 = r_2e^{i\theta_2}$$ Using $e^a e^b = e^{a+b}$, the product is: \begin{align*} z_3 = z_1 z_2 &= (r_1e^{i\theta_1})(r_2e^{i\theta_2}) \\ &= r_1r_2e^{i\theta_1}e^{i\theta_2} \\ &= r_1r_2e^{i(\theta_1+\theta_2)} \\ \\ |z_1z_2| &= |z_1|\times|z_2| \\ \arg z_1z_2 &= \arg z_1 \times \arg z_2 \end{align*} ## de Moivre's Theorem Let $z = re^{i\theta}$. Consider $z^n$. Since $z = r(\cos\theta + i\sin\theta)$, \begin{align*} z^n &= r^n(\cos\theta + i\sin\theta)^n &\text{(1)} \\ \end{align*} But also \begin{align*} z^n &= (re^{i\theta})^n \\ &= r^n(e^{i\theta})^n \\ &= r^ne^{in\theta} \\ &= r^n(\cos{n\theta} + i\sin{n\theta}) &\text{(2)} \\ \end{align*} By equating (1) and (2), we find de Moivre's theorem: \begin{align*} r^n(\cos\theta +i\sin\theta)^n &= r^n(\cos{n\theta} + i\sin{n\theta}) \\ (\cos\theta +i\sin\theta)^n &= (\cos{n\theta} + i\sin{n\theta}) \end{align*}
### Example 1 Write $1+i$ in polar form and use de Moivre's theorem to calculate $(1+i)^{15}$. > \begin{align*} r &= |1+i| = \sqrt2 \\ \theta &= \arg{1+i} = \frac \pi 4 \\ \\ \text{So } 1 + i &= \sqrt2(\cos{\frac pi 4} + i\sin{\frac \pi 4}) = \sqrt2e^{i\frac \pi 4} \text{ and}\\ (i+i)^{15} &= (\sqrt2)^{15}\left(\cos{\frac \pi 4} + i\sin{\frac \pi 4}\right)^{15} \\ &= 2^{\frac 15 2} \left(\cos{\frac{15\pi} 4} + i\sin{\frac{15\pi} 4}\right) \\ &= 2^{\frac 15 2} \left(\frac 1 {\sqrt2} - \frac i {\sqrt2}\right) \\ &= 2^7 (1 - i) \\ &= 128 - 128i > \end{align*}
### Example 2 Use de Moivre's theorem to show that \begin{align*} \cos{2\theta} &= \cos^2\theta-\sin^2\theta \\ \text{and} \\ \sin{2\theta} &= 2\sin\theta\cos\theta \end{align*} > Let $n=2$: > \begin{align*} (\cos\theta+i\sin\theta)^2 &= \cos^2\theta + 2i\sin\theta\cos\theta - \sin^2\theta \\ \text{Real part: } \cos^2\theta - \sin^2\theta &= \cos{2\theta}\\ \text{Imaginary part: } 2\sin\theta\cos\theta &= \sin{2\theta} > \end{align*}
### Example 3 Given that $n \in \mathbb{N}$ and $\omega = -1 + i$, show that $w^n + \bar{w}^n = 2^{\frac n 2 + 1}\cos{\frac{3n\pi} 4}$ with Euler's formula. > \begin{align*} r &= \sqrt{2} \\ \arg \omega = \theta &= \frac 3 4 \pi \\ \\ \omega^n &= r^n(cos{n\theta} + i\sin{n\theta}) \\ \bar\omega^n &= r^n(cos{n\theta} - i\sin{n\theta}) \\ \omega^n + \bar\omega^n &= r^n(2\cos{n\theta}) \\ &= 2^{\frac n 2 + 1}\cos{\frac {3n\pi} 4} > \end{align*}
## Complex Roots of Polynomials
### Example Find which complex numbers $z$ satisfy $$z^3 = 8i$$ > 1. Write $8i$ in exponential form, > > $|8i| = 8$ and $\arg{8i} = \frac \pi 2$ > > $\therefore 8i = 8e^{i\frac \pi 2}$ > > > 2. Let the solution be $r = re^{i\theta}$. > > Then $z^3 = r^3e^{3i\theta}$. > > 3. $z^3 = r^3e^{3i\theta} = 8e^{i\frac \pi 2}$ > > i. Compare modulus: > > $r^3 = 8 \rightarrow r = 2$ > > ii. Compare argument: > > $$3\theta = \frac \pi 2$$ > > is a solution but there are others since > > $$e^{i\frac \pi 2} = e^{i \frac \pi 2 + 2n\pi}$$ > > so we get a solution whenever > > $$3\theta = \frac \pi 2 + 2n\pi$$ > > for any integer `n` > > - $n = 0 \rightarrow z = \sqrt3 + i$ > - $n = 1 \rightarrow z = -\sqrt3 + i$ > - $n = 2 \rightarrow z = -2i$ > - $n = 3 \rightarrow z = \sqrt3 + i$ > - $n = 4 \rightarrow z = -\sqrt3 + i$ > - The solutions start repeating as you can see > > In general, an $n$-th order polynomial has exactly $n$ complex roots. > Some of these complex roots may be real numbers. > > 4. There are three solutions
# Matrices (and Simultaneous Equations) ## Gaussian Elimination Gaussian eliminiation can be used when the number of unknown variables you have is equal to the number of equations you are given. I'm pretty sure it's the name for the method you use to solve simultaneous equations in school. For example if you have 1 equation and 1 unknown: \begin{align*} 2x &= 6 \\ x &= 3 \end{align*} ### Number of Solutions Let's generalise the example above to $$ax = b$$ There are 3 possible cases: \begin{align*} a \ne 0 &\rightarrow x = \frac b a \\ a = 0, b \ne 0 &\rightarrow \text{no solution for $x$} \\ a = 0, b = 0 &\rightarrow \text{infinite solutions for $x$} \end{align*} ### 2x2 System A 2x2 system is one with 2 equations and 2 unknown variables.
#### Example 1 \begin{align*} 3x_1 + 4x_2 &= 2 &\text{(1)} \\ x_1 + 2x_2 &= 0 &\text{(2)} \\ \end{align*} \begin{align*} 3\times\text{(2)} = 3x_1 + 6x_2 &= 0 &\text{(3)} \\ \text{(3)} - \text{(1)} = 0x_1 + 2x_2 &= -2 \\ x_2 &= -1 \end{align*} We've essentially created a 1x1 system for $x_2$ and now that's solved we can back substitute it into equation (1) (or equation (2), it doesn't matter) to work out the value of $x_1$: \begin{align*} 3x_1 + 4x_2 &= 2 \\ 3x_1 - 1 &= 2 \\ 3x_1 &= 6 \\ x_1 &= 2 \end{align*} You can check the values for $x_1$ and $x_2$ are correct by substituting them into equation (2).
### 3x3 System A 3x3 system is one with 3 equations and 3 unknown variables.
#### Example 1 \begin{align*} 2x_1 + 3x_2 - x_3 &= 5 &\text{(1)} \\ 4x_1 + 4x_2 - 3x_3 &= 5 &\text{(2)} \\ 2x_1 - 3x_2 + x_3 &= 5 &\text{(3)} \\ \end{align*} The first step is to eliminate $x_1$ from (2) and (3) using (1): \begin{align*} \text{(2)}-2\times\text{(1)} = -2x_2 -x_3 &= -1 &\text{(4)} \\ \text{(3)}-\text{(1)} = -6x_2 + 3x_3 &= -6 &\text{(5)} \\ \end{align*} This has created a 2x2 system of $x_2$ and $x_3$ which can be solved as any other 2x2 system. I'm too lazy to type up the working, but it is solved like any other 2x2 system. \begin{align*} x_2 &= -2 x_3 &= 5 \end{align*} These values can be back-substituted into any of the first 3 equations to find out $x_1$: \begin{align*} -2x_1 + 3x_2 - x_3 = 2x_1 + 6 - 3 = 5 \rightarrow x_1 = 1 \end{align*}