A calculation

Aug 2018
137
7
România
Hello all,

Calculate \(\displaystyle [\cos{(x^2)}+i\sin{(x^2)}]^x\).

All the best,

Integrator
 
Oct 2018
129
96
USA
$\displaystyle e^{i \theta} = \cos{(\theta)} + i \sin{(\theta)} $
 
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topsquark

Math Team
May 2013
2,518
1,049
The Astral plane
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
 

romsek

Math Team
Sep 2015
2,958
1,673
USA
$\cos(x^2)+i \sin(x^2) = e^{i x^2}$

$\left(e^{i x^2}\right)^x = e^{i x^3}$
 
Aug 2018
137
7
România
$\cos(x^2)+i \sin(x^2) = e^{i x^2}$

$\left(e^{i x^2}\right)^x = e^{i x^3}$
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