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The Invariant Circles of a Bilinear Transformation

Published online by Cambridge University Press:  03 November 2016

M. P. Drazin
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Extract

We say that a bilinear transformation w=f(z) leaves a given locus C in the complex plane “invariant” if C coincides, as a point set, with its image f(C) under the transformation.

Type
Research Article
Information
The Mathematical Gazette , Volume 38 , Issue 323 , February 1954 , pp. 26 - 29
Copyright
Copyright © The Mathematical Association 1954

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References

page 26 note * It will be observed that 4ϒ2 =(a + d)2/(ad - bc) is an invariant of the transformation (1) under any simultaneous bilinear transformation of the z- and w-planes.

page 28 note * We may observe, in passing, that the relation r 2 =ρ2 - 1 (which must hold for ϒ2>0) means that all the invariant circles of the pencil are orthogonal to the unit (isometric) circle ∣ z ∣ = 1.