Complex number and power series

[rainwyz0706]

Homework Statement

Let ω be the complex number e^(2πi/3)， Find the power series for e^z + e^(ωz) + e^((ω^2) z).

Homework Equations

The Attempt at a Solution

I can show that 1+w+w^2=0, don't know if it would help. Could anyone please give me some hints? Any input is appreciated!

 

[Dick]

Write out the three individual power series and sum them. Collect like powers of z. Use your equation for w to try and simplify the coefficients.

 

[rainwyz0706]

I can write it as the sum of (z^n)*(1+w^n+w^2n)/n!, n from 0 to infinity. But I'm still not sure how to simplify 1+w^n+w^2n from 1+w+w^2=0. Could you explain it in a bit more details? Thanks a lot!

 

[Dick]

I can write it as the sum of (z^n)*(1+w^n+w^2n)/n!, n from 0 to infinity. But I'm still not sure how to simplify 1+w^n+w^2n from 1+w+w^2=0. Could you explain it in a bit more details? Thanks a lot!

Look at the n=0,1,2,3 cases. Hint, what's w^3? Once you've got those you should find it pretty easy to generalize.

 

[rainwyz0706]

thanks, I got it!