Recall that a Mobius transformation f has an equation of the form $f(z) = (az+b)/(cz+d) where a, b, c, and d are complex numbers.
Suppose that f is a Mobius transformation such that f(1)=i, f(i)=-1, and f(-1)=1. Find the value of f(-i).
To find the value of f(-i), we need to determine the specific Mobius transformation that satisfies the given conditions.
Let's first express the equations f(1) = i, f(i) = -1, and f(-1) = 1 in terms of the equation for Mobius transformations: f(z) = (az + b)/(cz + d).
Plugging in z = 1 into the equation, we have:
f(1) = (a*1 + b)/(c*1 + d) = i
Plugging in z = i into the equation, we have:
f(i) = (a*i + b)/(c*i + d) = -1
Plugging in z = -1 into the equation, we have:
f(-1) = (a*-1 + b)/(c*-1 + d) = 1
Now we have three equations with three unknowns (a, b, c, d).
Let's solve this system of equations.
Starting with the first equation, we can cross-multiply to eliminate the denominator:
(a + b) = i(c + d)
Now let's simplify the second equation:
(ai + b) = -1(ci + d)
Similarly, simplifying the third equation:
(-a + b) = (c + d)
Now we have a system of three linear equations:
a + b = i(c + d) ---> Equation 1
ai + b = -ci - d ---> Equation 2
-a + b = c + d ---> Equation 3
Let's solve this system of equations.
First, let's isolate b in Equation 3:
b = a + c + d
Now substitute b in Equation 1 and Equation 2:
a + a + c + d = i(c + d) ---> Equation 1'
ai + a + c + d = -ci - d ---> Equation 2'
Simplifying Equation 1':
2a + (c + d) = i(c + d)
Simplifying Equation 2':
(a - c) + (a - d)i = 0
Equating the real and imaginary parts of Equation 2' separately, we have:
a - c = 0 ---> Equation 2a
a - d = 0 ---> Equation 2b
From Equation 2a, we have:
a = c
From Equation 2b, we have:
a = d
Now substituting these results back into Equation 1', we have:
2a + (c + d) = i(c + d)
Since a = c and a = d, we can write:
2a + 2a = i(2a)
Simplifying, we have:
4a = 2ai
Dividing both sides by 2a (noting that a ≠ 0), we get:
4 = 2i
Dividing by 2, we get:
2 = i
However, this leads to a contradiction, because if 2 = i, then we would have 4 = i^2. However, i^2 = -1, not 4.
Therefore, there is no Mobius transformation f that satisfies the given conditions.