Find the bilinear transformation which maps the points z = −i, 1, i onto w = −1, 0, 1 respectively.

To find the bilinear (or Möbius) transformation that maps the points \( z = -i, 1, i \) onto \( w = -1, 0, 1 \), respectively, we use the general form of a bilinear transformation:

\[
w = \frac{az + b}{cz + d}
\]

The bilinear transformation is determined by four constants \(a\), \(b\), \(c\), and \(d\). To find these, we can substitute the corresponding points \( z \) and \( w \) into the equation and solve the system of equations.

### Step 1: Substitute the points into the transformation equation
For each point, substitute \(z\) and \(w\) into \( w = \frac{az + b}{cz + d} \):

1. For \( z = -i \), \( w = -1 \):
   \[
   -1 = \frac{a(-i) + b}{c(-i) + d}
   \]

2. For \( z = 1 \), \( w = 0 \):
   \[
   0 = \frac{a(1) + b}{c(1) + d}
   \]

3. For \( z = i \), \( w = 1 \):
   \[
   1 = \frac{a(i) + b}{c(i) + d}
   \]

### Step 2: Simplify each equation

1. \( -1 = \frac{-ai + b}{-ci + d} \)
   \[
   -1(-ci + d) = -ai + b
   \]
   Expanding:
   \[
   ci - d = -ai + b
   \quad \text{(Equation 1)}
   \]

2. \( 0 = \frac{a + b}{c + d} \) leads to:
   \[
   a + b = 0 \quad \text{(Equation 2)}
   \]

3. \( 1 = \frac{ai + b}{ci + d} \)
   \[
   1(ci + d) = ai + b
   \]
   Expanding:
   \[
   ci + d = ai + b
   \quad \text{(Equation 3)}
   \]

### Step 3: Solve the system of equations

From Equation 2, we know that \( a + b = 0 \), so \( b = -a \).

Substitute \( b = -a \) into Equations 1 and 3:

- Equation 1:
  \[
  ci - d = -ai - a
  \]
  Equating real and imaginary parts:
  \[
  c = -a, \quad -d = -a \implies d = a
  \]

- Equation 3:
  \[
  ci + d = ai - a
  \]
  Equating real and imaginary parts:
  \[
  c = a, \quad d = -a
  \]

### Step 4: Write the final transformation

Now we know that:
\[
c = -a, \quad d = a, \quad b = -a
\]

Thus, the bilinear transformation becomes:
\[
w = \frac{az - a}{-az + a}
\]

Factor out \( a \) from the numerator and denominator:
\[
w = \frac{a(z - 1)}{a(1 - z)} = \frac{z - 1}{1 - z}
\]

Thus, the bilinear transformation is:

\[
w = \frac{z - 1}{1 - z}
\]

### Final Answer:
The bilinear transformation that maps \( z = -i, 1, i \) onto \( w = -1, 0, 1 \) is:

\[
w = \frac{z - 1}{1 - z}
\]

Would you like any further details or clarification?

Here are 5 related questions:
1. What are the general properties of bilinear transformations?
2. How can we verify if a transformation is conformal?
3. What happens to circles under bilinear transformations?
4. How can we derive the inverse of a bilinear transformation?
5. How can bilinear transformations be used in complex function theory?

**Tip:** Bilinear transformations always map circles and straight lines in the complex plane to other circles or straight lines!

Explore how to find the bilinear (Möbius) transformation that maps z = -i, 1, i to w = -1, 0, 1. This detailed solution uses key concepts from complex analysis, including the Möbius transformation theorem, and is suitable for undergraduate students.

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