complex trigonometry??

Prove that -i2\frac{i}{2}Ln(a+ixaix\frac{a+ix}{a-ix}) = arctan(xa\frac{x}{a})

i = 1\sqrt{-1}

Post the solution if you have solved it

Note by Abdulmuttalib Lokhandwala
5 years, 5 months ago

No vote yet
1 vote

  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.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. 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"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

Sort by:

Top Newest

Differentiate both the sides individually. Doing that shall yield aa2+x2\frac{a}{a^{2}+x^{2}} on both sides,

Thus, your conjecture is proved.

Shaan Vaidya - 5 years, 5 months ago

Log in to reply

Note that you still have to show that they are equal at one point, otherwise the graphs could be vertical shifts of each other.

Don't forget your constant +C + C when integrating!

Calvin Lin Staff - 5 years, 5 months ago

Log in to reply

Yes sir I have taken care of the constant c while integrating the function in two ways and they are equal at one point.

abdulmuttalib lokhandwala - 5 years, 5 months ago

Log in to reply

TheexponentialformofTan(z)isieizeizeiz+eizIfz=i2Log(a+ixaix),thenTan(z)isi((a+ixaix)12(a+ixaix)12)((a+ixaix)12+(a+ixaix)12)1SimplifyingthisreducesittoxaThe\quad exponential\quad form\quad of\quad Tan(z)\quad is\\ i\frac { { e }^{ -iz }-{ e }^{ iz } }{ { e }^{ -iz }+{ e }^{ iz } } \\ If\quad z=-\frac { i }{ 2 } Log(\frac { a+ix }{ a-ix } ),\quad then\quad Tan(z)\quad is\\ i({ (\frac { a+ix }{ a-ix } ) }^{ -\frac { 1 }{ 2 } }-{ (\frac { a+ix }{ a-ix } ) }^{ \frac { 1 }{ 2 } })({ (\frac { a+ix }{ a-ix } ) }^{ -\frac { 1 }{ 2 } }+{ (\frac { a+ix }{ a-ix } ) }^{ \frac { 1 }{ 2 } })^{ -1 }\\ Simplifying\quad this\quad reduces\quad it\quad to\quad \frac { x }{ a }

Michael Mendrin - 5 years, 5 months ago

Log in to reply

Thanks Michael for the solution I got these result by integrating the function 1x2+a2\frac{1}{x^2+a^2} by two different methods

abdulmuttalib lokhandwala - 5 years, 5 months ago

Log in to reply

Vaidya already provided the other method, so I thought I'd include the exponential form route. You know, the brute force way.

Michael Mendrin - 5 years, 5 months ago

Log in to reply

As I have mentioned below, it should be ax2+a2\frac{a}{x^2+a^2} and not 1x2+a2\frac{1}{x^2+a^2}

Shaan Vaidya - 5 years, 5 months ago

Log in to reply

@Shaan Vaidya Ya both are true but finally when you do integration a will Already get cancelled and will get the same result

abdulmuttalib lokhandwala - 5 years, 5 months ago

Log in to reply

Beautiful method!!

Shaan Vaidya - 5 years, 5 months ago

Log in to reply

please do read my post at : https://brilliant.org/discussions/thread/math-is-getting-broken/ it is related to this question. thanks

Soham Zemse - 5 years, 3 months ago

Log in to reply

nice post

abdulmuttalib lokhandwala - 5 years, 3 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...