Thread: solutions to relative simple expression

I could need a hint on how to find all solutions to this equation



where C is a real constant, N is a positive integer, and and z is the complex variable. I'm not sure I can just write

. It should probably look more like , for .

But then I'm unsure as how to proceed. I have tried writing z in polar form but it didn't give me a break through.

Originally Posted by niaren

I could need a hint on how to find all solutions to this equation



where C is a real constant, N is a positive integer, and and z is the complex variable. I'm not sure I can just write

. It should probably look more like , for .

But then I'm unsure as how to proceed. I have tried writing z in polar form but it didn't give me a break through.

Dammit Don't know how I could miss that. Thanks.

Originally Posted by niaren

Dammit Don't know how I could miss that. Thanks.

I expect there are other solutions too though...

Do you mean that there is another way to arrive at the N solutions or do you mean that there are more than N solutions?

Originally Posted by niaren

.

Let p = 1/N
[1 + z^(-1)] / (1 + z) = C^p

(z + 1) / z = C^p + zC^p

z^2 C^p + z(C^p - 1) - 1 = 0

z = {1 + C^p +- SQRT[C^(2p) + 2C^p + 1]} / (2C^p)

I'm sure you can see your way to wrap that up, getting same as Prove It

Originally Posted by Wilmer

Let p = 1/N

z = {1 + C^p +- SQRT[C^(2p) + 2C^p + 1]} / (2C^p)

I actually had that expression written down but gave up because it looked too complicated. I will look at it tomorrow.

Originally Posted by niaren

I actually had that expression written down but gave up because it looked too complicated. I will look at it tomorrow.

Why?
C^(2p) + 2C^p + 1 = (C^p + 1)^2

Do something similar with this one, Bob? :

[(1 + 1/a) / (1 + a)]^(1/m) + [(1 + b) / (1 + 1/b)]^n = k

Solve for b.

b = [(ka^(1/m) - 1) / a^(1/m)]^(1/n) ... unless I goofed!