What is the Power Series for exp(z) + exp(w*z) + exp(z*w^2)?

[gertrudethegr]

We have already shown 1+ w+ w^2 =0

If w is the complex number exp(2*Pi*i/3) , find the power series for;
exp(z) +exp(w*z) + exp (z*w^2)

We have already shown 1+ w+ w^2 =0

 

[Dick]

If 1+w+w^2=0, can you tell me what are w^3, w^4, w^5 ...? Now do a power series expansion of the exponentials and combine like powers of z. Do you see a pattern?

 

[gertrudethegr]

Thanks!

So w^3=1
w^4= w^1
w^5=w^2 etc

so
exp(z) +exp(w*z) + exp (z*w^2)= 3 + 0 +0 + 3*z^3/3! + 0 + 0 +3*z^6/6! etc,

But where do i go from here?
Thankyou for your time

 

[Dick]

Thanks!

So w^3=1
w^4= w^1
w^5=w^2 etc

so
exp(z) +exp(w*z) + exp (z*w^2)= 3 + 0 +0 + 3*z^3/3! + 0 + 0 +3*z^6/6! etc,

But where do i go from here?
Thankyou for your time

You are done, aren't you? The problem asked for a simple form of the power series, and you just showed it to me.

 

[gertrudethegr]

That is a valid point!

Thanks dick