Complex numbers techniques !

This note is for complex numbers lovers.

I have very keen interest in complex numbers and as many of you also are mad about using Complex Numbers in Various Section of Maths! Since It is Modern Mathematics Technique , And It Highly reduce our calculation .

So Here in this Note I want to share Some *Complex Number Techniques That I learnt till Now . And It is Humble Request to Post Related Complex number Techniques which You Had Learnt or Created at your Own ! So That Our Brilliant Community Learn from it.*

A few of them are :

$\bullet$ 1 ) - Find Summation of Series :

$C\quad =\quad \cos { \theta } +\cos { 2\theta } \quad +\cos { 3\theta } +\quad .\quad .\quad .\quad .\quad .\quad .\quad +\quad \cos { (n\theta ) } \\ \\ S\quad =\quad \sin { \theta } +\quad \sin { 2\theta } \quad +\sin { 3\theta } +\quad .\quad .\quad .\quad .\quad .\quad .\quad \quad +\quad \sin { (n\theta ) }$ .

By Using Euler's and Converting into :

$C\quad +\quad iS\quad =\quad { e }^{ i\theta }\quad +\quad { e }^{ i2\theta }\quad +\quad { e }^{ i3\theta }\quad +\quad .\quad .\quad .\quad .\quad .\quad .\quad .\quad { e }^{ i(n\theta ) }\quad \quad \quad \quad (\quad G.P\quad )\\ \\ \\ C\quad +\quad iS\quad =\frac { { e }^{ i\theta }({ e }^{ i(n\theta ) }\quad -\quad 1) }{ { e }^{ i\theta }\quad -\quad 1 }$ .

And Then Separate real and imaginary Part And Then Compare !

$\bullet$ 2 )- Find Integrals ${ I }_{ 1 }\quad =\int { { e }^{ ax }\cos { bx } } dx\quad \quad \& \quad { I }_{ 2 }\quad =\int { { e }^{ ax }\sin { bx } } dx\quad$ .

By Considering ${ I }_{ 1 }\quad +\quad i{ \cdot I }_{ 1 }\quad =\int { { e }^{ ax }(\cos { bx } +i\cdot \sin { bx) } } dx\quad =\quad \int { { e }^{ (a+ib)x } } dx\quad \\ \\ { I }_{ 1 }\quad +\quad i{ \cdot I }_{ 1 }\quad =\quad \cfrac { { e }^{ (a+ib)x } }{ a+ib }$ .

And then Separate Real and imaginary Part and Then Compare !

$\bullet$ 3 )- TPT : $\sin { \cfrac { \pi }{ 7 } } \cdot \sin { \cfrac { 2\pi }{ 7 } } \cdot \sin { \cfrac { 3\pi }{ 7 } } \quad =\quad \cfrac { \sqrt { 7 } }{ 8 }$ .

By Considering 7th Root's of unity :

$1\quad ,\quad { \alpha }_{ 1 }\quad ,\quad { \alpha }_{ 2 }\quad ,\quad \quad .\quad .\quad .\quad .\quad .\quad ,\quad { \alpha }_{ 6 }\quad \quad (\because \quad { \alpha }_{ k }\quad =\quad { e }^{ \cfrac { 2k\pi i }{ 7 } }\quad )$ .

And Using Property n'th roots of unity ( Here n = 7 ):

$\left| (1\quad -\quad { \alpha }_{ 1 })(1\quad -\quad { \alpha }_{ 2 })(1\quad -\quad { \alpha }_{ 3 })\quad .\quad .\quad .\quad .\quad .\quad (1\quad -\quad { \alpha }_{ 6 }) \right| \quad =\quad \left| \quad 7\quad \right|$ .

And Now : $1\quad -\quad { \alpha }_{ K }\quad =\quad 1\quad -\quad (\cos { \cfrac { 2\pi k }{ 7 } } \quad +\quad i\cdot \sin { \cfrac { 2\pi k }{ 7 } } )\\ \\ 1\quad -\quad { \alpha }_{ K }\quad =\quad 2\sin { \cfrac { \pi k }{ 7 } } (\cos { \cfrac { \pi k }{ 7 } } \quad -\quad i\cdot \sin { \cfrac { \pi k }{ 7 } } )\\ \\ \left| 1\quad -\quad { \alpha }_{ K } \right| =\quad 2\sin { \cfrac { \pi k }{ 7 } } \quad \quad \quad \quad \quad (II)\\ \\ \left| (1\quad -\quad { \alpha }_{ 1 })(1\quad -\quad { \alpha }_{ 2 })(1\quad -\quad { \alpha }_{ 3 })\quad .\quad .\quad .\quad .\quad .\quad (1\quad -\quad { \alpha }_{ 6 }) \right| =\quad 7\\ \\ { 2 }^{ 6 }\times { (\sin { \cfrac { \pi }{ 7 } } \cdot \sin { \cfrac { 2\pi }{ 7 } } \cdot \sin { \cfrac { 3\pi }{ 7 } } ) }^{ 2 }\quad =\quad 7\\ \\ \sin { \cfrac { \pi }{ 7 } } \cdot \sin { \cfrac { 2\pi }{ 7 } } \cdot \sin { \cfrac { 3\pi }{ 7 } } \quad =\quad \cfrac { \sqrt { 7 } }{ 8 }$ .

I'am Done ! Now It's Your Turn .

Please Post Your Complex Number Techniques Too !! :)

Enjoy Complex !! :) :)

Re-share This More And More So that it reaches to every complex numbers Lovers , So that we can learn new Techniques !

Markdown

Appears as

*italics* or _italics_

italics

**bold** or __bold__

bold


- bulleted
- list

bulleted
list


1. numbered
2. list

numbered
list

Note: you must add a full line of space before and after lists for them to show up correctly

paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)

example link

> This is a quote

This is a quote

    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"

# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"

Math

Appears as

Remember to wrap math in $ ... $ or \[ ... \] to ensure proper formatting.

2 \times 3

2 \times 3

2^{34}

2^{34}

a_{i-1}

a_{i-1}

\frac{2}{3}

\frac{2}{3}

\sqrt{2}

\sqrt{2}

\sum_{i=1}^3

\sum_{i=1}^3

\sin \theta

\sin \theta

\boxed{123}

\boxed{123}

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Expanding $cos^{7}x$ in terms a series of cosines of multiples of x.

$y = cosx + isinx$

$y^{n} + \dfrac{1}{y^{n}} = 2cosnx$ , ( Since $y={ e }^{ ix }$ )

$y + \dfrac{1}{y} = 2cosx$

$cos^{7}x = ( y + \dfrac{1}{y})^{7}$

$= y^{7} + 7y^{5} + 21y^{3} + 35y + 35\dfrac{1}{y} + 21\dfrac{1}{y^{3}} + 7\dfrac{1}{y^{5}} + \dfrac{1}{y^{7}}$

$= ( y^{7} + \dfrac{1}{y^{7}}) + 7(y^{5} + \dfrac{1}{y^{5}}) + 21(y^{3} + \dfrac{1}{y^{3}}) + 35(y + \dfrac{1}{y})$

$= 2cos7x + 7.2cos5x + 21.2cos3x + 35.2cosx$

$cos^{7}x = \dfrac{1}{64}(cos7x + 7cos5x + 21cos3x + 35cosx)$

U Z - 6 years, 6 months ago

$\lim _{ n\rightarrow \infty }{ \prod _{ r=1 }^{ n }{ { (Very\quad Nice) }^{ r } } }$ .

Deepanshu Gupta - 6 years, 6 months ago

Awsome @megh choksi thanks for sharing

Aman Sharma - 6 years, 6 months ago

Well, there are a lot of applications of Euler's theorem.

For example :

1) $\displaystyle \sum_{r=0}^{n} {n \choose r} \cos (r x) = \Re \bigg(\sum_{r=0}^{n} {n \choose r} e^{i r x}\bigg)$

2) $\displaystyle \sum_{r=0}^{\infty} \dfrac{\cos rx}{2^r} = \Re \bigg(\sum_{r=0}^{\infty} \bigg(\dfrac {e^{ix}}{2} \bigg)^r \bigg)$

3) Expressing $\sin rx$ and $\cos rx$ in terms of $\sin x$ and $\cos x$

jatin yadav - 6 years, 6 months ago

Did a silly thing , sorry i am posting it here

U Z - 6 years, 5 months ago

Lol ! :)

You r creative :) :)

Deepanshu Gupta - 6 years, 5 months ago

@Deepanshu Gupta awsone post............i need your help how to solve following problems:-

(1)..Let $z_1,z_2,z_3$ be three complex numbers such that:- $|z_1|=|z_2|=|z_3|=|\frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}|=1$ Find the value of $z_1z_2z_3$

(2)let z be a complex number such that:- $|z+\frac{4}{z}|=2$ Find maximum value of |z|

(3)let z be a complex number then find maximum value of |z|+|z-1|

(4)if $z^2+z+1=0$ where z is a complex number then find tje value of:- $(z+\frac{1}{z})^2 + (z^2+\frac{1}{z^2})^2+...........+(z^6+\frac{1}{z^6})^2$

Please help me how to solve these problems...i know it is off topic i am sorry for that......i am a beginer so please post solutions...... also sorry to desturb you

Hints:

Sol 4 ) - Note Roots of quadratic are $\quad \omega \quad ,\quad { \omega }^{ 2 }\quad \longrightarrow \quad Cube\quad root\quad of\quad unity$

And Use Following Properties To get Answer :

$\quad \omega \quad +\quad { \omega }^{ 2 }\quad +\quad { \omega }^{ 3 }\quad =\quad 0\quad \quad \quad (\therefore \quad \omega \quad +\quad { \omega }^{ 2 }\quad =\quad -1\quad )\\ \quad \quad \quad \quad \quad \& \quad \quad \omega \quad =\quad \cfrac { 1 }{ { \omega }^{ 2 } } \quad$ .

Sol 2)- Let $\left| z \right| \quad =\quad r$ .

use Triangle inequality :

$\\ \left| r-\cfrac { 4 }{ r } \right| \quad \le \quad \left| z\quad +\quad \cfrac { 4 }{ z } \right| \quad \le \quad r\quad +\quad \cfrac { 4 }{ r } \\ \\ \left| r-\cfrac { 4 }{ r } \right| \quad \le \quad 2\quad \quad \quad .\quad .\quad .(I)\\ \quad r\quad +\quad \cfrac { 4 }{ r } \quad \ge \quad 2\quad \quad \quad \quad (Useless\quad \because \quad True\quad \forall \quad r\quad \quad by\quad AM-GM)$ .

Now Solve $I$ equation and get required maximum value of "r" .

I will Post rest of two later !

3) I think there is something wrong with it as when $a,b$ increases where $z=a+ib$ $a,b$ belongs to R, then |z| increases and also |z-1| so it will grow to infinity.

So the answer is infinity or a typo.

Gautam Sharma - 6 years, 6 months ago

Just one word; Brilliant! :)

sanat mishra - 5 years, 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$