For the transformation w=(z+i)/(z-i)
show that as z moves along the real axis, w moves along a circle centre O and radius 1
I do not know where to start
changing it to polar form will help
w=magnitude @ (arctan 1/z - arctan -1/z)
where magnitude is sqrt (z^2+1)/(z^2+1)
or
w= 1 @ (2 arctan 1/z)
Is that a circle?
check my thinking.
I am really sorry but I do not understand your notation
please I need further help
No problem, let me explain step by step.
1. Start by substituting z = x (assuming z is a real number) into the expression for w:
w = (z + i) / (z - i)
w = (x + i) / (x - i)
2. Now, let's multiply the numerator and denominator by the conjugate of the denominator, which is (x + i):
w = (x + i) * (x + i) / [(x - i) * (x + i)]
w = (x^2 + 2ix - 1) / (x^2 - i^2)
Since i^2 = -1, we can simplify further:
w = (x^2 + 2ix - 1) / (x^2 + 1)
3. To express w in polar form, we can write w as:
w = r * e^(iθ), where r is the magnitude and θ is the argument.
4. Let's find the magnitude (r) of w:
Magnitude (r) = sqrt(Re(w)^2 + Im(w)^2)
Here, Re(w) is the real part of w, and Im(w) is the imaginary part of w.
Re(w) = x^2 - 1
Im(w) = 2x
Magnitude (r) = sqrt((x^2 - 1)^2 + (2x)^2)
Magnitude (r) = sqrt(x^4 - 2x^2 + 1 + 4x^2)
Magnitude (r) = sqrt(x^4 + 2x^2 + 1)
5. Next, let's find the argument (θ) of w:
Argument (θ) = arg(w) = arctan(Im(w) / Re(w))
Argument (θ) = arctan((2x) / (x^2 - 1))
6. Now, we have w in the form w = r * e^(iθ), where r = sqrt(x^4 + 2x^2 + 1) and θ = arctan((2x) / (x^2 - 1)).
7. To show that w moves along a circle, we can rewrite w using Euler's formula:
w = r * e^(iθ) = r * (cos(θ) + i * sin(θ))
Here, cos(θ) and sin(θ) represent the x-coordinate and y-coordinate, respectively, on the unit circle.
Therefore, we can see that as x (or z) moves along the real axis, both the x and y coordinates of w (cos(θ) and sin(θ)) change, indicating that w moves along a circle centered at the origin (O) in the complex plane, with a radius of r = sqrt(x^4 + 2x^2 + 1).
I hope this explanation helps you. Let me know if you have any further questions!