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 June 16th, 2019, 09:42 AM #1 Senior Member   Joined: Aug 2018 From: România Posts: 112 Thanks: 7 A calculation Hello all, Calculate $\displaystyle [\cos{(x^2)}+i\sin{(x^2)}]^x$. All the best, Integrator June 16th, 2019, 11:14 AM #2 Senior Member   Joined: Oct 2018 From: USA Posts: 102 Thanks: 77 Math Focus: Algebraic Geometry $\displaystyle e^{i \theta} = \cos{(\theta)} + i \sin{(\theta)}$ Thanks from topsquark June 16th, 2019, 01:52 PM   #3
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 Originally Posted by Integrator Hello all, Calculate $\displaystyle [\cos{(x^2)}+i\sin{(x^2)}]^x$. All the best, Integrator
We need some restrictions here. I'm presuming that x is a real number so we have to be careful about $\displaystyle cos( x^2 ) \leq 0$ for various values of x.

-Dan June 16th, 2019, 02:00 PM #4 Senior Member   Joined: Sep 2015 From: USA Posts: 2,646 Thanks: 1476 $\cos(x^2)+i \sin(x^2) = e^{i x^2}$ $\left(e^{i x^2}\right)^x = e^{i x^3}$ June 16th, 2019, 03:18 PM   #5
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 Originally Posted by romsek $\left(e^{i x^2}\right)^x = e^{i x^3}$
I'm afraid complex exponentiation doesn't work like that. June 16th, 2019, 05:38 PM   #6
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 Originally Posted by Micrm@ss I'm afraid complex exponentiation doesn't work like that.
Okay you got me, too. Why doesn't it?

-Dan June 16th, 2019, 07:36 PM   #7
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 Originally Posted by Micrm@ss I'm afraid complex exponentiation doesn't work like that.
and you're just going to leave it at that?

How about being useful and showing us how it works then.

Sheesh. June 16th, 2019, 09:42 PM   #8
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Quote:
 Originally Posted by romsek $\cos(x^2)+i \sin(x^2) = e^{i x^2}$ $\left(e^{i x^2}\right)^x = e^{i x^3}$
Hello,

I do not understand!I think that $\displaystyle \left(e^{i x^2}\right)^{x} =e^{i^x\cdot x^{2x}}$ where $\displaystyle x\in \mathbb R , x>0$ is an identity and so $\displaystyle \left(e^{i x^2}\right)^{x} = e^{i x^3}$ is an equation.
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How do we calculate $\displaystyle [\cos(x^2)+i \sin(x^2)]^x$?

All the best,

Integrator Tags calculation Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post MMath Elementary Math 3 July 7th, 2016 10:13 PM joskevermeulen Calculus 1 December 29th, 2015 06:02 AM Dacu Elementary Math 9 November 21st, 2014 05:34 PM diegosened Algebra 2 April 7th, 2010 05:53 AM r-soy Calculus 1 December 31st, 1969 04:00 PM

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