Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2015 / Paper 1 / 11 / a

Table of contents
- i a
  - Solution i
- ii a
  - Solution ii

i

Solution

Direct substitution in the Möbius transformation gives

w (0) = \frac{- i}{i} = - 1, w (1) = \frac{1 - i}{1 + i} = \frac{( 1 - i ) ^{2}}{2} = - i .

Neither substitution meets the excluded denominator point

z = - i

.

ii

Solution

For real

λ

, the numerator and denominator have equal complex modulus:

∣ λ - i ∣ = λ^{2} + 1 = ∣ λ + i ∣.

The denominator is always nonzero. Hence

∣ w ∣ = \frac{∣ λ - i ∣}{∣ λ + i ∣} = 1.

Thus the real axis maps into the unit circle in the complex plane.

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