Multiply the numerator & denominator of [itex]\displaystyle \frac{e^{i\theta}}{1-e^{i\theta}}[/itex] by the complex conjugate of [itex]\displaystyle 1-e^{i\theta}[/itex] which is [itex]\displaystyle 1-e^{-i\theta}\ .[/itex]zezima1 said:Homework Statement
Show that the function f(z) = z/(1-z) maps the unit circle to an infinite line.
Homework Equations
Polar form z = rexp(iθ)
The Attempt at a Solution
I've tried to see what happens, when we let f(z) = f(eiθ) and then get:
f(eiθ) = eiθ/(1-eiθ)
But I need some help on making this expression more illuminating. I want something for which I can take the real and imaginary part - what tricks can I use?
Write (eiθ-1) in terms of sin(θ) & cos(θ) .zezima1 said:Thanks, I'm still unsure how to do it though. Multiplying by the conjugate you get:
f(eiθ) = (eiθ-1)/(2-2cos(θ))
How do I show that this is the equation for a straight line?
A unit circle is a circle with a radius of 1 unit, centered at the origin (0,0) on a coordinate plane.
A function is a mathematical rule that assigns each input value to exactly one output value.
A function maps the unit circle onto a line by taking each point on the unit circle and assigning it a corresponding point on the line. This is achieved through the use of the function's rule or equation.
Showing that a function maps the unit circle onto a line is important because it provides evidence that the function is well-defined and that all points on the unit circle have been accounted for. It also helps to understand the behavior of the function and how it relates to the unit circle.
If a function does not map the unit circle onto a line, it means that there are points on the unit circle that are not being assigned a corresponding point on the line. This could indicate that the function is not well-defined or that the function's rule or equation needs to be revised.