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Real parameter given complex variable modulus

by jubbles
Tags: complex, modulus, ratio, real
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jubbles
#1
Jan17-11, 12:39 PM
P: 1
1. The problem statement, all variables and given/known data
Suppose z is complex number with |z| = 1 (also assume that z is not 1 + 0*i)
Let z = a + b*i where a and b are real numbers.

Find a real parameter, t, such that

z = (t-i)/(t+i), where i = sqrt(-1)


2. Relevant equations


|z| = sqrt(a^2+b^2)



3. The attempt at a solution
Since we are given |z| = 1, this implies that a^2 + b^2 = (a+b*i)*(a-b*i) = 1.
Also, (t-i)/(t+i) can be expressed as [1/(t^2+1)]*[(t^2-1)+(-2*t)*i].

Despite my efforts, I end up at two deadends: I reduce an equation to a tautology like 0 = 0 or I reduce to an expression for t that involves i (thereby making t imaginary instead of real).

Any help is greatly appreciated.
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hunt_mat
#2
Jan17-11, 04:09 PM
HW Helper
P: 1,583
First off find c(t),d(t) such that:
[tex]
c(t)+d(t)i=\frac{t-i}{t+i}
[/tex]
Then use the fact that c(t)^{2}+d(t)^{2}=1
SammyS
#3
Jan17-11, 05:11 PM
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P: 7,799
Quote Quote by jubbles View Post
3. The attempt at a solution
Since we are given |z| = 1, this implies that a^2 + b^2 = (a+b*i)*(a-b*i) = 1.
Also, (t-i)/(t+i) can be expressed as [1/(t^2+1)]*[(t^2-1)+(-2*t)*i].

Despite my efforts, I end up at two deadends: I reduce an equation to a tautology like 0 = 0 or I reduce to an expression for t that involves i (thereby making t imaginary instead of real).

Any help is greatly appreciated.
Write

[tex]\frac{1}{t^2+1}\,\left[(t^2-1)+(-2t)i\right][/tex]

as

[tex]\frac{(t^2-1)}{t^2+1}-\frac{2t}{t^2+1}\,i\right][/tex]


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