### <Success>Complex numbers <br> lecture - 4</Success> <div style='float: left; width:50%;'> </div>

#### <Success>Polar representation of complex numbers</Success> <div style='float: left;text-align:left; width:50%;font-size:30px;'> <p>$ z=x+iy, \quad x, y \in R$</p> <p>Argand plane or complex plane</p> <p>$r$ - distance from the origin O to the point P.</p> <p>$θ$ -angle between OP and the +(ve) x-axis.</p> <p>$ \cos \theta = \frac {x}{r} $</p> <p>$ x = r \cos \theta$</p> <p>$ y = r \sin \theta$</p> </div> <div style='float: right; width:50%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/a8939933-7dbc-4b59-a7e9-e466f6511600/public"></p> </div> <div style='float: right; width:0%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/386503fd-379b-4397-afdf-21efff2ad600/public"></p> </div>

#### <Success>Polar representation of complex number</Success> <div style='float: left;text-align:left; width:100%;font-size:30px;'> <p>$ z = r \cos \theta + i\ r \sin \theta \quad r ≥ 0 , \theta \in [0, { \theta} {\pi}] $</p> <p>$ z = r (cos \theta + i\sin \theta) $</p> <p>This representation is called the polar representation of $z$.</p> <p>Polar coordinates system, $ (r, \theta) $</p> <p>Cartesian coordinates system, $ (x, y) $</p> <p>Given $(r, \theta) \mapsto x=r \cos \theta, y=r \sin \theta$</p> <p>$ r = \sqrt {x^2+y^2} \mapsto (x, y) $</p> <p>$ \tan \theta = \frac {y} {x}$</p> </div>

### <Success>Notations</Success> <div style='float: left;text-align:left; width:60%;font-size:30px;'> <p>$ z = r(\cos \theta + i \sin \theta) $</p> <p>$ z = r$ cis $\theta,\quad$ cis $\theta = \cos \theta +i \sin \theta$</p> <ul> <li>$\theta$ -argument of $z$</li> <li>$ r=|z| = \sqrt {x^2+y^2} $</li> </ul> </div> <div style='float: right; width:40%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/4b81cba0-489b-4bff-a1d2-c31fc6af2100/public"></p> </div> <div style='float: right; width:0%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/e6559fc5-d74e-47cf-0876-3d4cb523ac00/public"></p> </div>

### <Success>Remark</Success> <div style='float: left; width:60%;font-size:30px;'> <p><strong>(1)</strong> $ z=r (\cos\theta + i \sin\theta) $</p> <p>$ z=r [(\cos(\theta +2k\pi) +i \sin(\theta +2k\pi)] \ k\in \Z.$</p> <p>i.e., $(r, \theta +2k\pi) \mapsto x= r\cos\theta, \\ \quad \quad \quad \quad \quad \quad y=r\sin\theta$</p> </div> <div style='float: right; width:25%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/81062bbc-4ebf-457d-8b1a-9d63d8f25c00/public"></p> </div> <div style='float: right; width:25%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/3de2227a-d9ad-4eba-8e47-d963d0970b00/public"></p> </div> <div style='float: right; width:0%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/8bca4540-e6fa-4d1f-5ff6-4196fd771300/public"></p> </div>

### <Success>Remark</Success> <div style='float: left;text-align:left; width:50%;font-size:25px;'> <p><strong>(2)</strong> $ \tan\theta = \frac{1} {1} $ <br> $ \theta = \frac{\pi} {4} $</p> <p>$ z =1+i = {\sqrt 2}\biggl(\cos\frac {\pi} {4}+i \sin\frac {\pi} {4} \biggl) $</p> <p>consider ${\sqrt 2}\biggl[\cos(\frac {-\pi} {4})+i \sin(\frac {-\pi} {4}) \biggl] $</p> <p>$ = { \sqrt 2}\biggl[\cos\frac {\pi} {4}-i \sin\frac {\pi} {4} \biggl] $$ = \bar{z} $</p> </div> <div style='float: right; width:50%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/7642717d-7e38-4743-2a97-d6d3c6653800/public"></p> </div> <div style='float: right; width:0%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/8ff85a60-3eee-43e2-de35-da1f1a6b8300/public"></p> </div>

### <Success>Remark</Success> <div style='float: left;text-align:left; width:40%;font-size:30px;'> <p>$ z^1= r[\cos(-\theta)+i \sin(-\theta)$</p> <p>$ = \bar{z} $</p> <p>$ (r,\theta + 2k{\pi}), \ k\in \Z \mapsto x= r\cos\theta, y=r\sin\theta $</p> <p>$ \theta_\circ \leq\theta \leq \theta_\circ +2 {\pi}$</p> <p>$ \theta_\circ =0 \quad 0\leq \theta <2 {\pi}$</p> </div> <div style='float: right; width:60%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/7750789e-6d94-4e93-da70-45439592dc00/public"></p> </div> <div style='float: right; width:0%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/4a5ccdf7-61b6-4ad7-7269-19900fe4ec00/public"></p> </div>

### <Success>Remark</Success> <div style='float: left;text-align:left; width:40%;font-size:30px;'> <p>$ {\pi} < \theta \leq {\pi}$</p> <p>Principal argument of $z= \theta \in (-\pi,\ \pi]$</p> </div> <div style='float: right; width:60%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/4a0cf9bc-4474-4294-dc47-a742ee26a500/public"></p> </div> <div style='float: right; width:0%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/a41ebbad-6158-4272-b4f3-b5d3af8fed00/public"></p> </div>

### <Success>Remark</Success> <div style='float: left;text-align:left; width:30%;font-size:30px;'> <p><strong>(3)</strong> $ r=0 \quad \theta \in[0,2\pi] $</p> <p>$ (0,\theta)$ represents the origin.</p> <p>It means that we do not have well defined polar representation for the origin.</p> </div> <div style='float: right; width:70%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/bcdbd69c-03f8-44aa-a1af-badcb9b90900/public"></p> </div> <div style='float: right; width:0%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/0daa941e-0b60-4e91-ef11-92efcb76d000/public"></p> </div>

### <Success>Formula for arg(z)</Success> <div style='float: lefttext-align:left;width:100%;font-size:25px;'> <p>Observation $\ \ \tan(x+\pi) = \tan x $ <br> $\tan(x+k\pi) = \tan x, \ $ for $k\in \Z $ <br> Find $\theta \in (-\pi,\pi)$ such that $ \ \ \tan(\theta) =\frac {y} {x} $ <br> $ (\theta) = \tan^{-1}\biggl(\frac {y} {x}\biggl) + k\pi, \quad k\in \Z $</p> <p>Restrict $\tan^{-1}\biggl(\frac {y} {x}\biggl)$ in the interval $(\frac {-\pi} {2}, \frac {\pi} {2})$</p> <p>which we call it as arc $\tan\frac {y} {x} $</p> <p>$ \theta = $ arc $\tan\frac {y} {x}+k_+\pi $</p> </div>

### <Success>Formula for arg(z)</Success> <div style='float: left;text-align:left; width:50%;font-size:30px;'> <p>If $x \neq 0$</p> <p>arg (z) = arc $\tan \left(\frac{y}{x}\right)+k_{+} \pi $</p> <p>Where $k_{+} =$</p> <math xmlns="http://www.w3.org/1998/Math/MathML" display="block"> <mrow data-mjx-texclass="INNER"> <mo data-mjx-texclass="OPEN">{</mo> <mtable columnalign="left left" columnspacing="1em" rowspacing=".2em"> <mtr> <mtd> <mn>0</mn> <mstyle scriptlevel="0"> <mspace width="1em"></mspace> </mstyle> <mi>x</mi> <mo>></mo> <mn>0</mn> <mtext> </mtext> <mi>a</mi> <mi>n</mi> <mi>d</mi> <mtext> </mtext> <mi>y</mi> <mo>≥</mo> <mn>0</mn> <mo stretchy="false">(</mo> <mi>I</mi> <mrow data-mjx-texclass="ORD"> </mrow> <mi>e</mi> <mi>q</mi> <mi>u</mi> <mi>a</mi> <mi>t</mi> <mi>i</mi> <mi>o</mi> <mi>n</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> <mstyle scriptlevel="0"> <mspace width="1em"></mspace> </mstyle> <mi>x</mi> <mo>></mo> <mn>0</mn> <mtext> </mtext> <mi>a</mi> <mi>n</mi> <mi>d</mi> <mtext> </mtext> <mi>y</mi> <mo>≤</mo> <mn>0</mn> <mo stretchy="false">(</mo> <mi>I</mi> <mi>V</mi> <mi>e</mi> <mi>q</mi> <mi>u</mi> <mi>a</mi> <mi>t</mi> <mi>i</mi> <mi>o</mi> <mi>n</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> <mstyle scriptlevel="0"> <mspace width="1em"></mspace> </mstyle> <mi>x</mi> <mo><</mo> <mn>0</mn> <mtext> </mtext> <mi>a</mi> <mi>n</mi> <mi>d</mi> <mtext> </mtext> <mi>y</mi> <mo>≥</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> <mstyle scriptlevel="0"> <mspace width="1em"></mspace> </mstyle> <mi>x</mi> <mo><</mo> <mn>0</mn> <mtext> </mtext> <mi>a</mi> <mi>n</mi> <mi>d</mi> <mtext> </mtext> <mi>y</mi> <mo><</mo> <mn>0</mn> </mtd> </mtr> </mtable> <mo data-mjx-texclass="CLOSE" fence="true" stretchy="true" symmetric="true"></mo> </mrow> </math> If $x = 0$ <math xmlns="http://www.w3.org/1998/Math/MathML" display="block"> <mi>a</mi> <mi>r</mi> <mi>g</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace linebreak="newline"></mspace> <mrow data-mjx-texclass="INNER"> <mo data-mjx-texclass="OPEN">{</mo> <mtable columnalign="left left" columnspacing="1em" rowspacing=".2em"> <mtr> <mtd> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> <mi>i</mi> <mi>f</mi> <mstyle scriptlevel="0"> <mspace width="1em"></mspace> </mstyle> <mi>y</mi> <mo>></mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mfrac> <mrow> <mo>−</mo> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> <mstyle scriptlevel="0"> <mspace width="1em"></mspace> </mstyle> <mi>y</mi> <mo><</mo> <mn>0</mn> <mstyle scriptlevel="0"> <mspace width="1em"></mspace> </mstyle> </mtd> </mtr> </mtable> <mo data-mjx-texclass="CLOSE" fence="true" stretchy="true" symmetric="true"></mo> </mrow> </math> </div>

### <Success>Example</Success> <div style='float: left;text-align:left; width:50%;font-size:30px;'> <p>$ z=-1 + i $</p> <p>$ r= \sqrt{1 +1} =\sqrt2 $</p> <p>$ \theta=$ arc $\tan\frac{1}{-1} +k_+\pi $</p> <p>$ \theta = \frac {-\pi} {4} +\pi $</p> <p>$\theta = \frac {3} {4}\pi $</p> <p>Exercise: $Z= \sqrt{2} + 2 \sqrt{3}i $</p> </div> <div style='float: right; width:50%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/0e3422ae-b669-422d-aee2-b32e1e9c5e00/public"></p> </div> <div style='float: right; width:0%;'> <p><img alt="alt text" src="https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/8f42c6f0-c318-4936-2c91-71887d530900/public"></p> </div>

### <Success>Formula for arg(z)</Success> <div style='float: left;text-align:left; width:50%;font-size:30px;'> <table> <thead> <tr> <th>z = x + i y</th> <th>arg(z)</th> </tr> </thead> <tbody> <tr> <td>$I^{st}$ & IV quadrant</td> <td>arc $\tan$ (y/x)</td> </tr> <tr> <td>II quadrant</td> <td>arc $\tan$ (y/x) + $\pi$</td> </tr> <tr> <td>III quadrant</td> <td>arc $\tan$ (y/x)- $ \pi $</td> </tr> </tbody> </table> </div>

### <Success>Example</Success> <div style='float: left;text-align:left; width:100%;font-size:30px;'> <p>$ Z= 1+\cos a +i \sin a,a\in (0,2\pi) $</p> <p>$ r= \sqrt{(1+\cos c)^2 + sin^2 c} = \sqrt {2+2\cos a} $</p> <p>$ r= \sqrt {2(1+\cos a)} = \sqrt {2.2 cos^2 \frac {a} {2}} $</p> <p>$ r= 2|\cos \frac {a} {2}| $</p> <p>For $a\in(0,\pi),\quad z$ is in the $1^{st}$ quadrant</p> <p>$ \theta =$ arc $\tan \biggl(\frac {\sin a} {1+\cos a}\biggl) =$ arc $\tan \biggl(\frac {2\sin\frac {a} {2} \cos \frac {a}{2}} {2\cos^2\frac{a} {2}}\biggl) $</p> <p>arc $\tan (\tan (\frac {a} {2})) = \frac {a} {2} $</p> </div>

### <Success>EExercise </Success> <div style='float: left;text-align:left; width:100%;font-size:30px;'> <p>$a\in (\pi, 2\pi) $</p> <p>Verify that $\theta =\frac {a} {2} -\pi$</p> <math xmlns="http://www.w3.org/1998/Math/MathML" display="block"> <mi>Z</mi> <mo>=</mo> <mrow data-mjx-texclass="INNER"> <mo data-mjx-texclass="OPEN">{</mo> <mtable columnalign="left left" columnspacing="1em" rowspacing=".2em"> <mtr> <mtd> <mn>2</mn> <mrow data-mjx-texclass="ORD"> <mo stretchy="false">|</mo> </mrow> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mfrac> <mi>a</mi> <mn>2</mn> </mfrac> <mrow data-mjx-texclass="ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">[</mo> <mi>c</mi> <mi>i</mi> <mi>s</mi> <mo stretchy="false">(</mo> <mfrac> <mi>a</mi> <mn>2</mn> </mfrac> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>,</mo> <mstyle scriptlevel="0"> <mspace width="1em"></mspace> </mstyle> <mi>a</mi> <mi>ϵ</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>π</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <mrow data-mjx-texclass="ORD"> <mo stretchy="false">|</mo> </mrow> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mfrac> <mi>a</mi> <mn>2</mn> </mfrac> <mrow data-mjx-texclass="ORD"> <mo stretchy="false">|</mo> </mrow> <mi>c</mi> <mi>i</mi> <mi>s</mi> <mo stretchy="false">(</mo> <mfrac> <mi>a</mi> <mn>2</mn> </mfrac> <mo>−</mo> <mi>π</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mstyle scriptlevel="0"> <mspace width="1em"></mspace> </mstyle> <mi>a</mi> <mi>ϵ</mi> <mo stretchy="false">(</mo> <mi>π</mi> <mo>,</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo data-mjx-texclass="CLOSE" fence="true" stretchy="true" symmetric="true"></mo> </mrow> </math> <p>$ a=\pi, \quad z= 1+\cos\pi+i\sin\pi $</p> <p>$ = 0 $</p> </div>

### <Success>Multiplication</Success> <div style='float: left;text-align:left; width:50%;font-size:30px;'> <p>$z_1=r_1$ cis $\theta_1 \quad z_2 = r_2$ cis $\theta_2$</p> <p>$z_1. z_2 = r_1r_2[\cos\theta_1 -\sin\theta_1, \sin\theta_2+i(\cos\theta_1\sin\theta_2+\sin\theta_1\cos\theta_2)]$</p> <p>$ z_1. z_2 =r_1r_2$ cis $(\theta_1+\theta_2) $</p> </div> ====== ### <Success>Multiplication</Success> <div style='float: left;text-align:left; width:50%;font-size:30px;'> <p><strong>Remark</strong>: $ |z_1. z_2| =r_1r_2 =|z_1| |z_2| $</p> <p>arg $(z_1 z_2) =$ arg $(z_1) +$ arg $(z_2) + k_+2\pi $</p> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="block"> <msub> <mi>k</mi> <mo>+</mo> </msub> <mo>=</mo> <mrow data-mjx-texclass="INNER"> <mo data-mjx-texclass="OPEN">{</mo> <mtable columnalign="left left" columnspacing="1em" rowspacing=".2em"> <mtr> <mtd> <mn>0</mn> <mstyle scriptlevel="0"> <mspace width="1em"></mspace> </mstyle> <mo>−</mo> <mi>π</mi> <mo><</mo> <mi>a</mi> <mi>r</mi> <mi>g</mi> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mi>a</mi> <mi>r</mi> <mi>g</mi> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>π</mi> </mtd> <mtd></mtd> </mtr> <mtr> <mtd> <mn>1</mn> <mstyle scriptlevel="0"> <mspace width="1em"></mspace> </mstyle> <mi>a</mi> <mi>r</mi> <mi>g</mi> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mi>a</mi> <mi>r</mi> <mi>g</mi> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> <mo>≤</mo> <mo>−</mo> <mi>π</mi> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> <mstyle scriptlevel="0"> <mspace width="1em"></mspace> </mstyle> <mi>a</mi> <mi>r</mi> <mi>g</mi> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mi>a</mi> <mi>r</mi> <mi>g</mi> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> <mo>></mo> <mi>π</mi> </mtd> </mtr> </mtable> <mo data-mjx-texclass="CLOSE" fence="true" stretchy="true" symmetric="true"></mo> </mrow> </math></p> </div>

### <Success>Example</Success> <div style='float: left;text-align:left; width:100%;font-size:30px;'> <p>$z_1=1+i \quad z_2 = \sqrt{3} +i $</p> <p>$z_1=\sqrt{2}$ cis $\biggl(\frac {\pi} {4}\biggl), z_2 = 2$ cis $\biggl(\frac {\pi} {6}\biggl) $</p> <p>$z_1 z_2 = 2\sqrt{2}$ cis $\biggl(\frac {5\pi} {12}\biggl) $</p> <p><strong>De Moivre’s Formula</strong></p> <p>$z=r(\cos\theta + i\sin\theta)$</p> <p>$z^n=r^n(\cos n\theta + i \sin n\theta)\quad n\geq1$</p> <p>$z^2=z.z =r^2(\cos2\theta + i\sin 2\theta)$</p> </div>

### <Success>Example</Success> <div style='float: left;text-align:left; width:100%;font-size:30px;'> <p>$z_1=1+i $</p> <p>Calculate $z^{1000}$</p> <p>$(1+i)^{1000} = [\sqrt{2}$ cis $(\frac {\pi} {4})]^{1000}$</p> <p>$(1+i)^{1000}=2^{500}$ cis $(250\pi)$</p> <p>$(1+i)^{1000}=2^{500}$</p> <p><strong>Exercise:</strong> prove that</p> <p>$\sin 5\theta=16 \sin^5\theta-20 \sin^3\theta +5\sin\theta$</p> <p>$\cos 5\theta=16 \cos^5\theta-20 \cos^3\theta +5\cos\theta$</p> <p><strong>Hint:</strong> cis $5\theta=($cis $\theta)^5$</p> </div>

<div> <h2><Success>Thank you</Success></h2> </div>