complex trigonometry??

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

i = $\sqrt{-1}$

Post the solution if you have solved it

#InverseTrigonometricFunctions #Complex

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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

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

Thus, your conjecture is proved.

Shaan Vaidya - 7 years, 1 month ago

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 $+ C$ when integrating!

Calvin Lin Staff - 7 years, 1 month ago

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 - 7 years, 1 month ago

$The\quad exponential\quad form\quad of\quad Tan(z)\quad is\\ i\frac { { e }^{ -iz }-{ e }^{ iz } }{ { e }^{ -iz }+{ e }^{ iz } } \\ If\quad z=-\frac { i }{ 2 } Log(\frac { a+ix }{ a-ix } ),\quad then\quad Tan(z)\quad is\\ i({ (\frac { a+ix }{ a-ix } ) }^{ -\frac { 1 }{ 2 } }-{ (\frac { a+ix }{ a-ix } ) }^{ \frac { 1 }{ 2 } })({ (\frac { a+ix }{ a-ix } ) }^{ -\frac { 1 }{ 2 } }+{ (\frac { a+ix }{ a-ix } ) }^{ \frac { 1 }{ 2 } })^{ -1 }\\ Simplifying\quad this\quad reduces\quad it\quad to\quad \frac { x }{ a }$

Michael Mendrin - 7 years, 1 month ago

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

abdulmuttalib lokhandwala - 7 years, 1 month ago

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

Michael Mendrin - 7 years, 1 month ago

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

Shaan Vaidya - 7 years, 1 month ago

@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 - 7 years, 1 month ago

Beautiful method!!

Shaan Vaidya - 7 years, 1 month ago

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

Soham Zemse - 6 years, 11 months ago

nice post

abdulmuttalib lokhandwala - 6 years, 11 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}`	$\boxed{123}$