Question 1

(a) Find a linear fractional transformation S(w) deined on the extended complex plane C* which maps the points

w1  = a ,    w2  = bi ,    w3  = ∞

to the points 0, 1, ∞ respectively.

(b) Find a linear fractional transformation w = F(z) which maps the open unit disc |z| < 1 to the open half plane

 +  > 1 .

Give a detailed explanation for your answer.

Solution.  Using ideas from lectures, we can immediately write

down

bi — a

which satisies the given conditions. We now choose three points — 1, i, 1 on the boundary of the circle and map them to the points a, bi, ∞ on the boundary of the half plane.  The points on the circle are mapped to 0, 1, ∞ by the transformation

z + 1 i — 1

T(z) =

while the points on the line are mapped to 0, 1, ∞ by the transfor- mation S(w) above. So the transformation which maps the three points on the circle to the three points on the line is given by

w = S — 1 (T(z))    ⇔   S(w) = T(z)

(a — b — ai)z — (a + b + ai)

z — 1                  ,

and this is the required F(z).

Explanation.  A linear fractional transformation maps a circle to a line or circle in the extended complex plane.  So the image of the unit circle under w  = F(z) is a line or circle containing the points F( — 1), F(i), F(1). Since one of these points is ∞, the image is a line; and since the other two points are a and bi, it is the line through these two points. Thus the boundary |z| = 1 of the unit disc is mapped to the boundary of the given half plane; and so the interior of the circle is mapped to one side or the other of the line.  If we go around the circle from — 1 to i to 1 in that order, the given region is on our right; therefore if we go from a to bi to ∞ along the line, the image is on our right. Therefore it

is the required region.

Comments.

. Each student was given a question with speciic randomised values of a, b.

. You could have chosen diferent points on the line and circle to get a diferent correct answer.

. The question clearly asked for a detailed explanation.  An-swers giving a correct transformation without adequate ex- planation did not receive full marks.