The problem involves finding the locus of points that satisfy the equation 2π|z - 1| = Arg(z - 1) within the constraint |z - 1| ≤ 2 in the context of complex numbers represented in an Argand diagram.
Participants are exploring different interpretations of the equations and their implications. Some have proposed using a substitution to analyze the problem further, while others are questioning the correctness of transformations and the resulting plots. There is no explicit consensus on the final approach yet, but the discussion is progressing with clarifications being made.
There is an ongoing discussion about the assumptions related to the argument of complex numbers and how shifting the origin affects the equations. Participants are also considering the implications of the spiral nature of the locus derived from the equations.
Gone said:Homework Statement
Find the locus of points which satisfy
2π*|z −1| = Arg(z − 1) where |z −1| ≤ 2.
Homework Equations
n/a
The Attempt at a Solution
I know that |z-1| ≤ 2 is the 'inside' bits of a circle center (1,0) with a radius 2
After that I get confused surely with |z-1|≤2 then Arg(z-1) has to be between 0 and 4π... but then that's all space?
Thanks in advance
Pranav-Arora said:Since everything is with reference to ##1+0i##, it would be good idea to shift the origin here. Shifting the origin, your equations transform to:
$$2\pi |z|=\arg(z)$$
$$|z|\leq 2$$
Use the substitution ##z=re^{i\theta}## in the first equation, do you see where that leads to?
Yes.Gone said:So that means that ##2\pi r=\theta## which is a spiral beginning at (0,0)
Well, no. I don't think that transformation is correct. Once you plot the graph, you need to move everything by 1 unit towards right.so to get the answer is it just
$$2\pi (r-1)=\theta$$