Question

Find the bilinear transformation which maps the points z=1,-i,-1 and w=0,1,infinity.Find the fixed points

Answer 1

Answer:

Step-by-step explanation:

Let the transformation be w=az+bcz+d

Now putting the value of z and w we get

1) w = 0, z = 1 0 = a(1)+bc(1)+d∴a+b=0 i.e. a = -b

2) w = 1, z = I 1 = a(i)+bc(i)+d∴ci+d=ai+b∴(a−c)i=d−b

3) w = ∞, z = -1 ∴∞=a(−1)+bc(−1)+d∴−c+d=0 i.e. c = d

Hence we have a = -b (1)

(a - c)i = d - b (2)

c = d (3)

∴ substitute (1) and (3) in equation (2)

(a - c)i = c + a

∴ ai - a = c + ci

a(i - 1) = c(i + 1)

ac=i+1i−1∗i+1i+1=(i+1)2−1−1=i2+2i+1−2

∴ a = -ci

i.e.

c=−1ia∴c=ai∴c−d=ai

w=az+bcz+dw=az−a(ai)z+ai

Now

w=a(z−1)ai(z+1)∴w=−iz−1z+1