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Find a bilinear transformation

Short answer

Zero at z=1 and pole at z=-1 give w = k(z-1)/(z+1). Substituting z=i, w=1 yields k=-i, so w = -i(z-1)/(z+1) = i(1-z)/(1+z).

Find the bilinear transform that maps the points 1, i, −1 in the z-plane into the points 0, 1, ∞ into the w plane.

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

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

Hence w = k (z − 1)/(z + 1).

Use z = i → w = 1:

1 = k (i − 1)/(i + 1)
(i − 1)/(i + 1) = (i − 1)(1 − i)/((1 + i)(1 − i)) = (i − i² − 1 + i)/2 = 2i/2 = i
so k·i = 1  → k = 1/i = −i

w = −i (z − 1)/(z + 1) = i (1 − z)/(1 + z)

Answer: w = i(1 − z)/(1 + z). Check: z = 1 → 0; z = −1 → ∞; z = i → i(1 − i)/(1 + i) = i(−i) = 1 ✓.

✓ Answer checked by Claude Opus 5.5 (high effort)

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