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Find a bilinear transformation

Short answer

The bilinear transformation is w = (z+1)/(z-i). The numerator factor (z+1) comes from z=-1 mapping to 0; the denominator factor (z-i) from z=i mapping to ∞. The constant k=1 is fixed by z=0 → i.

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

Let the bilinear transformation be w = (az + b)/(cz + d).

  • z = −1 maps to w = 0, so the numerator has a factor (z + 1).
  • z = i maps to w = ∞, so the denominator has a factor (z − i).

Hence w = k (z + 1)/(z − i).

Use z = 0 → w = i:

i = k (1)/(−i) = k · i      (since 1/(−i) = i)
k = 1

Answer: w = (z + 1)/(z − i).

Check: z = 0 gives 1/(−i) = i; z = −1 gives 0; z = i makes the denominator zero, so w = ∞.

✓ Answer checked by Claude Opus 5.5 (high effort)

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