Probability and complex function: Unit III: Analytic functions : Exercise 3.5
EXERCISE
3.5
Bilinear
transformation
1.
Find the fixed points of the following mappings
(i)
w = z – 1 – i/z+2 Ans. z = -1 ± i√3
+ 4i/2
ii)
w = 2z -5/z + 4 Ans. z = -1 ± 2i
(iii)
w = z -2/z + 3 Ans. z = -1± i
(iv)w
= 1/z – 2i Ans. z = i
(v) w = 5z + 4/2 + 5 Ans. z = ± 2
(vi)
w = z2 Ans. z = 0, 1 [ A.U
M/J 2014]
2.
Define bilinear transformation.
3.
Find the most general bilinear transformation that maps the upper half of the
z-plane onto the interior of the unit circle in the w-plane.
4.
Find the bilinear transformation which maps the points z into w Answers
(1)
2, i, -2; 1, i, −1 w= 3z + 2i/ iz + 6
(2)
-i, 0, i ; -1, i, 1 [AU N/D 2015 R-13] w
= -i (z – 1/z + 1 )
(3)
0, -i, 2i; 5i, ∞ , 1/3 w = 3z - 5i/iz - 1
(4)
1, -1, ∞ ; 1+i, 1-i, 1 w = z + i/z
(5)
0, 1, ∞ ; i, 1, -i [A.U M/J 2012, 2013]
[A.U M/J 2005, 2005][A.U A/M 2008] A.U A/M 2019 R13] w
= z + i/1 + zi
(6)
1, i, -1; 2, i, -2 [AU N/D 2004, 2005]
[AU M/J 2016 R-13] w = -(6z – 2i/iz – 3)
(7)
0, 1, ∞ ; -5, -1, 3 [AU D15/J16 R-08] w
= 3z - 5/z + 1
(8)
i, -1, 1; 0, 1, ∞ w =2z - 2i /(1 + i) (z - 1)
(9)
∞, i, 0 ; 0, -i, ∞ w = 1/z
(10)
0, -i, -1; i, 1, 0 [AU M/J 2008, 2009] w = i (1 + z)/1 - z
(11) -i, 0, i; ∞, -1, 0 w = z – 1/z + 1
(12) 0, 1, ∞ ; i, -1; -i w = -i( z + i /z – i)
(13)
-1,0,1; -1, -2, 1 [AU N/D 2016 R-08] [AU
A/M 2017 R-13] w = z – i / 1- iz
(14)
0, 1, -1; -1,0, ∞ [AU N/D 2016 R-13] w
= z -1/z + 1
(15)-2,
0,2; 0, i, -1 [A.U May 2002] w = (-i)z + (-2i)/3z + (-2)
(16)
1, -1, ∞ ; -1, -i, i [AU N/D 2019 (R17)]
w = z + (1 + 2i)/(-i)z + (-2-1)
Probability and complex function: Unit III: Analytic functions : Tag: : Problems with Answer | Analytic functions - Exercise 3.5 (Bilinear transformation)
Probability and complex function
MA3303 3rd Semester EEE Dept | 2021 Regulation | 3rd Semester EEE Dept 2021 Regulation