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# expand on equations 1 and 2

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see attached please expand on and explain the attached work

ie how does one make the transformations shown in equation 1 and 2 and 3...please show this work in complete, painful detail

I have given this as a new post as I believe it is outside the original post

https://brainmass.com/math/complex-analysis/expand-on-equations-453610

## SOLUTION This solution is FREE courtesy of BrainMass!

Problem:

There are 2 complex numbers Z that satisfy Z^n = 1 and {(Z+1)^n} =1 (n not equal to 0) for some positive integer n. Find Z.

Solution:

Every complex number z can be expressed in the exponential form
( 1)
which comes from the common algebraic form
( 2)
If we represent the complex number z in the xOy plane, we observe that
( 3)
so that the expression (2) becomes:
( 4)
Using Euler's formula:
( 5)
the exponential representation of the complex number will be found:
( 6)
where
( 7)

If we want to compute , then we will have:
( 8)
From (5) one derives
( 9)
and therefore, it follows that
( 10)
Introducing the last result in (8), we will get:
( 11)
Let's denote z1 and z2 the required complex numbers.
The number z1 satisfies the equation
( 12)
From (11) one derives:
( 13)
Using the expression (1), one finds that
( 14)
On the other hand:
where k  Z.
 where k = 0, 1, ..., n-1 ( 15)
(there are n distinct values for z1). The notation (z1)k means the value of order k for z1.
It can be checked that for k = n we have
( 16)
so that, for k  n, the values of z1 satisfying (1) are included in the set of values given by (2).

The same judgment will be used for z2, if instead of z1 we will put z2+1:
( 17)
 where k = 0, 1, ..., n-1 ( 18)
The above expression can be processed as follows:
( 19)
By applying Euler's formula:
( 20)
and putting , we will get
( 21)
where k = 0, 1, ..., n-1.

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