How Does Exponentiation with Complex Variables Influence Vector Orientation?

[Miike012]

I've been reading a book on complex variables and I came up with an equation which may or may not be useful but I thought it was interesting

Explanation
Given a complex vector z = a + bi I can calculate z raised to the M + 1 power where M = 360/ArcTan(b/a)

Side Note
Sorry I didn't give any reason to the alterations to the equations below but basically the logic behind the reason is that say z = |z|(cos(θ) + isin(θ)) then I know that zⁿ = |z|ⁿ(cos(nθ) + isin(nθ)) and because I know trig functions repeat I know that the product will eventually rotate 360 degrees measured from vector z.

For example given z = |z|(cos(θ) + isin(θ)) and say z^m = |z|^m(cos(θ+ 2∏) + isin(θ+2∏)
therefore z and z^m are parallel and differ by some scalar

Equation
z^M+1 = [|z|^M+1/|Z|^M+1]z^M+1 = [|z|^M+1/|z|]z =[|z|^M]z

∴
z^M+1 =[|z|^M]z
It seems to only work when it has rotated around once but I can change that

 

[Bacle2]

Maybe you're thinking about De Moivre's formula:

http://en.wikipedia.org/wiki/De_Moivre's_formula

 

[Miike012]

Maybe you're thinking about De Moivre's formula:

http://en.wikipedia.org/wiki/De_Moivre's_formula

Nope. I just came up with it

 

[Miike012]

For Example

(101+17i)^{360/ArcTan(17/101) + 1} = (101 + 17i)*√(101^2+17^2)^{360/ArcTan(17/101)}

Notice that the left side is raised to a power greater than 1 while the right side is raised to the first power then multiplied by some constant

 

[Bacle2]

I wasn't implying you came up with it yourself, I was just suggesting something along the lines of what I thought you were doing. Is your exponentiation defined only for that specific value with ArcTan? And is that ArcTan a value in [0,2pi) ( since ArcTan is multiple-valued)?

I don't know if I understood correctly, but z and z^n are not always parallel; just take

z=x+ix ; then z^2=i2x , which is not parallel to x+ix.

 

[Miike012]

I wasn't implying you came up with it yourself, I was just suggesting something along the lines of what I thought you were doing. Is your exponentiation defined only for that specific value with ArcTan? And is that ArcTan a value in [0,2pi) ( since ArcTan is multiple-valued)?

I don't know if I understood correctly, but z and z^n are not always parallel; just take

z=x+ix ; then z^2=i2x , which is not parallel to x+ix.

if zⁿ = z^M+1 where M = 360/45 then zⁿ is parallel to z where z = (x + ix)