10.2 • BILINEAR TRANSFORMATIONS 409yFigure 10.8yaw
Images of Arg ( w) = c under the mapping z = --.
w-1-------.-EXERCISES FOR SECTION 10.2
- If w = 5 (z) =
: ;, find 5 -^1 (w).
2. If w = 5 (z) =~.find 5 -^1 (w).
y3. Find the image of the right half-plane Re (z) > 0 under w = i(;:;;>.
a= 0.14. Show that the bilinear transfor mation w = ;<;:;;> maps the portion of the disk
lzl < 1 t hat lies in the upper half-plane Im (z) > 0 onto the first quadrant u > 0,
v > 0.- Find the image of the upper half-plane Im (z) > 0 under the transformation
(1 -i) z + 2
w =.
(1+i)z+2- Find the bilinear transformation w = 5 (z) that maps the points z 1 = 0, z2 = i ,
and z3 = - i onto w1 = - 1, w2 = 1 , and w3 = 0, respectively. - Find the bilinear transformation w = S (z) that maps the points z1 = -i, z2 = 0,
and z3 = i onto w1 = - 1, W2 = i, and w3 = 1, respectively. - Find the bilinear transformation w = S (z) that maps the points z1 = 0, z2 = 1,
and z3 = 2 onto w 1 = 0, w 2 = 1, and W3 = oo, respectively. - Find the bilinear transformation w = 5 (z) that maps t he points z1 = 1, z2 = i ,
and z3 = -1 onto w1 = 0, w2 = 1, and W3 = oo, respectively.
- Show that the transformation w = ~ maps the unit disk lzl < 1 onto the right
half-plane Re (w) > 0. - Find the image of the lower half-plane Im (z) < 0 under w = ~-