-1,0,1 onto -i,-1,i using three points and their images given formula w-w1/w-w3 * w2-w3/w2-w1 = z-z1/z-z3 * z2-z3/z2-z1

To find the image of the points \(-1\), \(0\), and \(1\) under the given transformation, we can use the provided formula:

\[
\frac{w - w_1}{w - w_3} \cdot \frac{w_2 - w_3}{w_2 - w_1} = \frac{z - z_1}{z - z_3} \cdot \frac{z_2 - z_3}{z_2 - z_1}
\]

Here, let's assume:
- \(z_1 = -1\), \(z_2 = 0\), \(z_3 = 1\)
- \(w_1 = -i\), \(w_2 = -1\), \(w_3 = i\)

We need to find the corresponding image \(w\) for each \(z\).

### Step 1: Set up the equation for each \(z\)

1. **For \(z = -1\):**

   Substitute \(z = z_1 = -1\) into the equation:
   \[
   \frac{w - w_1}{w - w_3} \cdot \frac{w_2 - w_3}{w_2 - w_1} = \frac{-1 - z_1}{-1 - z_3} \cdot \frac{z_2 - z_3}{z_2 - z_1}
   \]

   Simplifying:
   \[
   \frac{w + i}{w - i} \cdot \frac{-1 - i}{-1 + i} = \frac{-1 + 1}{-1 - 1} \cdot \frac{0 - 1}{0 + 1}
   \]

2. **For \(z = 0\):**

   Substitute \(z = z_2 = 0\) into the equation:
   \[
   \frac{w - w_1}{w - w_3} \cdot \frac{w_2 - w_3}{w_2 - w_1} = \frac{0 - z_1}{0 - z_3} \cdot \frac{z_2 - z_3}{z_2 - z_1}
   \]

   Simplifying:
   \[
   \frac{w + i}{w - i} \cdot \frac{-1 - i}{-1 + i} = \frac{1}{-1} \cdot \frac{-1}{1}
   \]

3. **For \(z = 1\):**

   Substitute \(z = z_3 = 1\) into the equation:
   \[
   \frac{w - w_1}{w - w_3} \cdot \frac{w_2 - w_3}{w_2 - w_1} = \frac{1 - z_1}{1 - z_3} \cdot \frac{z_2 - z_3}{z_2 - z_1}
   \]

   Simplifying:
   \[
   \frac{w + i}{w - i} \cdot \frac{-1 - i}{-1 + i} = \frac{2}{0} \cdot \frac{-1}{2}
   \]

### Step 2: Solve each equation for \(w\)

Given that the transformation must map \(-1\) to \(-i\), \(0\) to \(-1\), and \(1\) to \(i\), solve the resulting simplified equations for \(w\).

### For \(z = -1\), solve:

\[
\frac{w + i}{w - i} \cdot \frac{-1 - i}{-1 + i} = \frac{0}{-2} \cdot \frac{-1}{2}
\]
This equation simplifies to \(w = -i\).

### For \(z = 0\), solve:

\[
\frac{w + i}{w - i} \cdot \frac{-1 - i}{-1 + i} = 1
\]
This equation simplifies to \(w = -1\).

### For \(z = 1\), solve:

\[
\frac{w + i}{w - i} \cdot \frac{-1 - i}{-1 + i} = -\infty
\]
This equation simplifies to \(w = i\).

So the image points are:
- \(z = -1\) maps to \(w = -i\)
- \(z = 0\) maps to \(w = -1\)
- \(z = 1\) maps to \(w = i\)

Would you like to see more details about these transformations, or do you have any questions? Here are five related questions to consider:

1. How do you solve for \(w\) in each transformation equation?
2. What is the significance of each term in the transformation formula?
3. How would the transformation change if the points \(w_1\), \(w_2\), and \(w_3\) were different?
4. Can this transformation be generalized to map other sets of points?
5. What type of transformation is this, geometrically?

**Tip:** Always check the consistency of each step when applying complex transformations, as errors can easily propagate.

Explore the mapping of points -1, 0, and 1 under a complex transformation using the Schwarz-Christoffel formula. Understand how this formula applies to complex analysis and conformal mappings. Suitable for advanced mathematics enthusiasts.

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