Complex Analysis - Maths KU

7.2 Linear Fractional Transformations 409
15. T (z) =
–3
z +1
z − 2
y
Figure 7.17 Figure for Problem 15
2
x
16. T (z) =
−z − 1+i
z − 1+i
y
1 + i
1 – i
Figure 7.18 Figure for Problem 16
In Problems 17–20, use matrices to find (a) S −1 (z) and (b) S −1 (T (z)).
17. T (z) = z
iz +1
, S(z) =
iz − 1 z − 1
19. T (z) =
2z − 3 z − 2
, S(z) =
z − 3 z − 1
18. T (z) = iz
2z +1
, S(z) =
z − 2i z +1
20. T (z) =
z − 1+i (2 − i)z
, S(z) =
iz − 2 z − 1 − i
In Problems 21–26, construct a linear fractional transformation that takes the given
points z1, z2, and z3 onto the given points w1, w2, and w3, respectively.
21. z1 = −1, z2 =0, z3 =2; w1 =0, w2 =1, w3 = ∞
22. z1 = i, z2 =0, z3 = −i; w1 =0, w2 =1, w3 = ∞
23. z1 =0, z2 = i, z3 = ∞; w1 =0, w2 =1, w3 =2
24. z1 = −1, z2 =0, z3 =1; w1 = i, w2 =0, w3 = ∞
25. z1 =1, z2 = i, z3 = −i; w1 = −1, w2 =0, w3 =3
26. z1 =1, z2 = i, z3 = −i; w1 = −i, w2 = i, w3 = ∞
Focus on Concepts
27. Let a, b, c, and d be complex numbers such that ad − bc �= 0.
az + b
(a) Solve the equation w = for z.
cz + d
(b) Explain why (a) implies that the linear fractional transformation
T (z) =(az + b)/ (cz + d) is a one-to-one function.
28. Consider the equation
where λ is a positive real constant.
|z − a| = λ|z − b| (15)
(a) Show that the set of points satisfying (15) is a line if λ =1.
(b) Show that the set of points satisfying (15) is a circle if λ �= 1.
29. Let T (z) =(az + b)/ (cz + d) be a linear fractional transformation.
(a) If T (0) = 0, then what, if anything, can be said about the coefficients a, b,
c, and d?
x