Elementary Functions 249
2. Let w = l/z with z = x + iy, w = u +iv. Show that x = u/(u^2 + v^2 ),
y = -v /( u^2 + v^2 ), and prove analytically the properties of the function
w = 1 / z concerning the transformation of straight lines and circles by
substitution of these formulas into the general equation of a circle in
the xy-plane, namely,a(x^2 + y^2 ) +bx+ cy + d = 0
which contains the equation of a straight line for a = 0.- Show that the function w = r^2 /z, with r > 0, defines a pure inversion
with respect to the circle lzl = r. - Find the bilinear transfo~mation that maps the points 3, 1, 2 - i into
the points 0, -2, -i, respectively.
5. Find the bilinear transformation that maps the points 1, i, -1 into the
points 0, 1, oo, respectively. Show that this transformation maps the
disk lzl .:S 1 onto the half-plane Im w 2:: 0.
*6. Show that w = ~:t~ maps Im z 2:: 0 onto Im w 2:: 0 iff the coefficients
a, b, c, d, are all real and ad - be > 0.
*7. Show that w = ei°'[(z - /3)/(/3z - 1)], a real, 1/31 < 1, maps the disk
lzl .:S 1 onto lwl .:S 1.- Find the fixed points of each of the following transformations.
z+2
(a) w = -z + 4i (b) w =
z
( c) w = 2z z + - 1 2 (d) w = 3z 4z + 2 - 1
z 3z-4
( e) w = 2z - 1 ( f) w = z - 1 - Find the multiplier for the transformations in problem 8, and express
those transformations in terms of the multiplier. Hint: First normalize
each transformation so that ad - be = 1. - Classify each of the following transformations.
z z+i
(a) w= 2z+l (b) w= 3z+(1+3i)
(c)w= (2+i)z+l (d)w= (1/^2 +i)z+l
4z + (2 - i)^1 / 4 z + (% - i)
(e) w= (l+i)z+l
z + (1-i) - A transformation T is said to be an involution when T^2 = I, I being
the identical transformation. Find the condition to be satisfied by the
coefficients of w = ( az + b) / ( cz + d), with ad - be = 1, in order for the
transformation to be an involution. - The cross ratio of four points in the complex plane can be written in 24
different ways, depending on the order in which the points are taken.