MHB Can Complex Equations Be Solved More Efficiently?

[Markov2]

Solve ${{\left( \dfrac{1+i}{\sqrt{2}} \right)}^{x}}+{{\left( \dfrac{1-i}{\sqrt{2}} \right)}^{x}}=\sqrt{2}.$

Is there a faster way to solve this?

 

[ThePerfectHacker]

Solve ${{\left( \dfrac{1+i}{\sqrt{2}} \right)}^{x}}+{{\left( \dfrac{1-i}{\sqrt{2}} \right)}^{x}}=\sqrt{2}.$

Is there a faster way to solve this?

You can check that $e^{\pi i/4} = \frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2}$ and $e^{-\pi i/4} = \frac{\sqrt{2}}{2} - i \frac{\sqrt{2}}{2}$.

Therefore, the equation becomes,
$$ e^{ \pi i x/4} + e^{-\pi i x/4} = \sqrt{2} \implies \cos \left( \frac{\pi i x}{4} \right) = \frac{\sqrt{2}}{2} $$

Do you see how we got that?

 

[Markov2]

Oh yes, you just used Euler's formula, so the solutions are $\dfrac{\pi }{4}ix = \dfrac{\pi }{4} \pm 2k\pi ,{\text{ }}k \in \mathbb{Z}$ and $\dfrac{\pi }{4}ix = \dfrac{{7\pi }}{4} \pm 2k\pi ,{\text{ }}k \in \mathbb{Z},$ are those correct?

Thanks for the help!

 

[ThePerfectHacker]

Oh yes, you just used Euler's formula, so the solutions are $\dfrac{\pi }{4}ix = \dfrac{\pi }{4} \pm 2k\pi ,{\text{ }}k \in \mathbb{Z}$ and $\dfrac{\pi }{4}ix = \dfrac{{7\pi }}{4} \pm 2k\pi ,{\text{ }}k \in \mathbb{Z},$ are those correct?

Thanks for the help!

Careful, those are not all of the solutions? Because $\cos -\frac{\pi}{4} = \frac{\sqrt{2}}{2}$ also!

 

[Markov2]

Oh yes! So the we have these sets also: $ - \displaystyle\frac{\pi }{4}ix = \frac{\pi }{4} \pm 2k\pi ,{\text{ }}k \in \mathbb{Z},{\text{ }} - \frac{\pi }{4}ix = \frac{{7\pi }}{4} \pm 2k\pi ,{\text{ }}k \in \mathbb{Z},$

 

[Plato2]

Solve ${{\left( \dfrac{1+i}{\sqrt{2}} \right)}^{x}}+{{\left( \dfrac{1-i}{\sqrt{2}} \right)}^{x}}=\sqrt{2}.$

I am a bit confused by the replies.
It clear that $x=\pm 1$ are solutions.
Was $x$ suppose to be complex also?

Moreover $\exp(i\theta)+\exp(-i\theta)=2\cos(\theta)$ not $2\cos(i\theta)$.

 

[Markov2]

No, $x$ is supposed to be real, thanks for the catch Plato!

 

[Plato2]

No, $x$ is supposed to be real, thanks for the catch Plato!

Then might look at $x=\pm 7,~\pm 9,~\pm 23,~\pm 25,\cdots$.
What is going on?

 

[Markov2]

Mmm you mean would have to rephrase the condition for $x$ ?

 

[Markov2]

What are the solutions then? Do we have to reformule the problem?