Complex Variable's Equation

1. Jun 25, 2013

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 zm = |z|m(cos(θ+ 2∏) + isin(θ+2∏)
therefore z and zm are parallel and differ by some scalar

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

zM+1 =[|z|M]z

It seems to only work when it has rotated around once but I can change that

Last edited: Jun 25, 2013
2. Jun 25, 2013

Bacle2

3. Jun 25, 2013

Miike012

4. Jun 25, 2013

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

5. Jun 25, 2013

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.

Last edited: Jun 25, 2013
6. Jun 26, 2013

Miike012

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