A calculation

Integrator

Hello all,

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

All the best,

Integrator

Greens

$\displaystyle e^{i \theta} = \cos{(\theta)} + i \sin{(\theta)}$

1 person

topsquark

Math Team
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
$\cos(x^2)+i \sin(x^2) = e^{i x^2}$

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

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$\left(e^{i x^2}\right)^x = e^{i x^3}$
I'm afraid complex exponentiation doesn't work like that.

1 person

topsquark

Math Team
I'm afraid complex exponentiation doesn't work like that.
Okay you got me, too. Why doesn't it?

-Dan

1 person

romsek

Math Team
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.

2 people

Integrator

$\cos(x^2)+i \sin(x^2) = e^{i x^2}$

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