Exercise 3.4 (Conformal mappings, Transformation)

EXERCISE 3.4

Conformal mappings

1. What do you mean by conformal mappings ?

2. Define Critical point of a transformation.

3. Find the image of the circle | z | = a under the following transformations.

(i) W = 2 + 2 + 3i  [Ans. (u - 2)² + (v - 3)² = a²]

(ii) w = 2z [A.U N/D 2016 R-13] [ Ans. | w | = 2a ]

(iii) w = (1 + i) z +  2 – i [Ans. (u − 2)² + (v + 1)² = 2a²]

4. Find the image of the circle | z+1| = 1 in the complex plane under the mapping w = 1/z [ Ans. 2u + 1 = 0 which is a straight line] [A.U M/J 2014]

5. Find the image of | z- 3i| = 3 under the mapping w = 1/z [Ans. v = 1/6 ]

6. Consider the transformation w = ze^iπ/4 and determine the region in the w-plane corresponding to the triangular region bounded olno by x = 0, y = 0 and x + y = 1 in z plane.

7. Consider the transformation w = 3z, corresponding to the region R of z - plane bounded by x = 0, y = 0, x + y = 2.

8. Verify the transformation w = 1+ iz/1 + z maps the exterior of the circle | z | = 1 into the upper half plane v > 0. [Anna, May 1996]

9. Find the image of | z - 2i| = 3 under w = 1/z [Anna, May 1996]

10. Find the image of the following under w = 1/z

(i) the circle | z | =1

[A.U A/M 2019 R-17, N/D 2019 R-17]

 (ii) the circle | z - 2i | = 2

(iii) the strip 1 < x < 2

11. Show that by means of the inversion w = 1/z the circle given by | z-3| = 5 is mapped into the circle | w+ 3/16 | = 5/16

12. Show that the transformation w = iz + 1/z + i tranforms the exterior and interior regions of the circle | z | = 1 into the upper and lower half of the w plane respectively.

13. Show that w = z – i/z + i maps the real axis in the z plane onto | w | = 1 in the w plane. Show also that the upper half of the z plane, Im(z) ≥ 0, goes onto the circular disc | w | ≤ 1.

14. Prove that w = 1+ iz / 1 + z maps the line segment joining -1 and 1 onto a  semi circle in the w plane.

15. Show that the transformation w = z + i / z – i  maps the lower half plane

Im z ≤ 0 onto | w | ≤ 1.

16. Show that the transformation w = z – i/1- iz maps the circular disc | z❘ ≤ 1

onto the lower half of the w plane.

17. Show that the transformation w = z – i/1 – iz maps (i) the interior of the circle z = 1 onto the lower half of the w-plane and (ii) the upper half of the z plane onto the interior of the circle | w | = 1.

18. Prove that w = z/1 – z maps the upper half of the z plane onto the upper half of the w-plane. What is the image of the circle | z |= 1 beer under this transformation. [Ans. u = -1/2]

19. Show that the transformation w = i – z/i + z maps the circle | z | = 1 onto the imaginary axis of the w plane. Find also the images of the interior and exterior of the circle.

20. Find the image of the real axis of the z plane by the transformation w = 1/z + i [Ans. u² + v² + v = 0]

21. Show that the transformation w = z – 1/z + 1 maps the unit circle in the w-plane onto the imaginary axis in the z-plane. Find also the images of the interior and exterior of the unit circle.

22. Plot the image under the mapping w = z² of the rectangular  region bounded by (i) x = -1 , x = 2, y = 1 and y = 2

(ii) x = 1 , x = 3, y = 1 and y = 2

(iii) u = 1 , u = 3, v = 1 and v = 2

23. Under the mapping w = e^z discuss the transforms of the lines. (i) y = 0, (ii) y = π/2 , (iii) y = 2π.

24. Discuss the transformation w = i(1 − z)/(1 + z) and show that it maps the circle | z |= 1 into the real axis of the w-plane and the interior of the circle | z | < 1 into the upper half of the w-plane. [A.U A/M 2019 R-08]