- #1
Bacat
- 151
- 1
Not really homework, but part of a homework problem I am working on.
I know that [tex]e^{i\pi}+1=0[/tex] (Euler's Identity)
And also that [tex]e^{i \pi} = e^{-i \pi}[/tex]
But I'm having trouble understanding [tex]e^{\frac{-i \pi}{4}}[/tex]
In the complex plane this is a clockwise rotation around the origin of [tex]\frac{\pi}{2}[/tex] radians. But I think it should reduce to some real constant which I am having trouble finding.
In Mathematica, I get two different answers...
[tex]N[e^{\frac{-i \pi}{4}}] = 0.707107 - 0.707107 i[/tex]
This implies that [tex]e^{\frac{-i \pi}{4}} = \frac{1}{\sqrt{2}}(1-i)[/tex] which seems wrong to me.
The other answer given is:
Simplify[tex][e^{\frac{-i \pi}{4}}] = -(-1)^{\frac{3}{4}}[/tex]
But this reduces to 1, which I believe is probably the correct answer.
Is the first result just spurious rounding?
Can I just write the following identity as true?
[tex]e^{\frac{-i \pi}{4}} = 1[/tex]
I know that [tex]e^{i\pi}+1=0[/tex] (Euler's Identity)
And also that [tex]e^{i \pi} = e^{-i \pi}[/tex]
But I'm having trouble understanding [tex]e^{\frac{-i \pi}{4}}[/tex]
In the complex plane this is a clockwise rotation around the origin of [tex]\frac{\pi}{2}[/tex] radians. But I think it should reduce to some real constant which I am having trouble finding.
In Mathematica, I get two different answers...
[tex]N[e^{\frac{-i \pi}{4}}] = 0.707107 - 0.707107 i[/tex]
This implies that [tex]e^{\frac{-i \pi}{4}} = \frac{1}{\sqrt{2}}(1-i)[/tex] which seems wrong to me.
The other answer given is:
Simplify[tex][e^{\frac{-i \pi}{4}}] = -(-1)^{\frac{3}{4}}[/tex]
But this reduces to 1, which I believe is probably the correct answer.
Is the first result just spurious rounding?
Can I just write the following identity as true?
[tex]e^{\frac{-i \pi}{4}} = 1[/tex]