Recent content by yamata1

I added it in the 8th post .

##(e^{3i\pi}+e^{\frac{i\pi}{3}})=-1+e^{\frac{i\pi}{3}}= -1 +1/2 + \frac{(i \sqrt(3))}{2}=-1/2 + \frac{(i \sqrt(3))}{2} =e^{\frac{2i\pi}{3}}##

Yes the mistake is in the exponent##(e^{3i\pi}+e^{\frac{\pi}{3}})##. It's ##(e^{3i\pi}+e^{\frac{i\pi}{3}})##. I just plugged ##\sqrt[3\,]{e^{\frac {2i\pi}{3}}} +\sqrt[3\,]{-\sqrt 3 e^{i\pi/6}}= 3^{1/6} e^{-5i\pi/18}+e^{\frac {4i\pi}{18}}## into ##\sqrt[3\,]...

I think I've clocked it. ## \sqrt[3\,] {e^{3i\pi}+e^{\frac{\pi}{3}}}= \sqrt[3\,]{e^{\frac {2i\pi}{3}}}## ##\sqrt[3\,]{e^{\frac {2i\pi}{3}}} +\sqrt[3\,]{-\sqrt 3 e^{i\pi/6}}= 3^{1/6} e^{-5i\pi/18}+e^{\frac {4i\pi}{18}}## ##=e^{\frac {4i\pi}{18}}(3^{1/6}e^{\frac {-9i\pi}{18}}+1)## ##=e^{\frac...

If I write ##e^{3i\pi}=(e^{i\pi})^3=(-1)^3=-1## Then ##z=-\frac {3}{2} - i \frac{\sqrt 3}{2}=-\sqrt 3 e^{i\pi/6} ##

##(-\frac 32 - i \frac{\sqrt 3}{2})^{1/3}= \sqrt[3\,] {e^{3i\pi}-e^{\frac{\pi}{3}}}## and ##(-1)^{2\over 9}=\sqrt[3\,] {e^{3i\pi}+e^{\frac{\pi}{3}}}## When trying to solve ##(-3/2 - \frac{i}{2} \sqrt{3})=(a+bi)^3## I get ##a(a^2-b-2b^2)=\frac{3}{2}## and ##b(2a^2+a-b^2)=\frac{\sqrt{3}}{2}##. How...

How do I find the real part of ## (-3/2 - \frac{i}{2} \sqrt{3})^{(1/3)}## ? The cubic root makes it hard for me to see what to do here.

My attempt : multiplying numerator and denominator by ## (-1)^{2/9} + (-3/2 - \frac{i}{2} \sqrt{3})^{(1/3)} wehave \frac{(-1)^{4/9} +2(-1)^{2/9}(-3/2 - \frac{i}{2} \sqrt{3})^{(1/3)}+ (-3/2 - \frac{i}{2} \sqrt{3})^{(2/3)}}{(-1)^{4/9}+ (-3/2 - \frac{i}{2} \sqrt{3})^{(2/3)}} ##

I don't know how to start with the factorization. $$\frac{(-1)^{2/9} + (-3/2 - \frac{i}{2} \sqrt{3})^{(1/3)}}{(-1)^{2/9}- (-3/2 - \frac{i}{2} \sqrt{3})^{(1/3)}}$$ Any hints would be nice. Thank you.

So giving a different metric gives us the right topology to make H a Lie group. Thank you.

How can we write the topology change for H as a topological group ?

"The group can, however, be given a different topology, in which the distance between two points is defined as the length of the shortest path in the group joining to " What does this give us in formal math ?

How do we change the topology to make H a Lie group ?

I can't find a proper homeomorphism in ##U_2##.

1-a finite-dimensional real smooth manifold 2- the group operations of multiplication and inversion are smooth maps.