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764 CHAPTER 23 Conformal Mappings and Applications

The next section is devoted to the construction of conformal mappings between
domains.

SECTION 23.1 PROBLEMS

1. In each of parts (a) through (e), find the image (a) Determine the image of D under the mapping
of the rectangle under the mapping w = e . Sketch w = cos(z). Sketch D and its image.
z
the rectangle in the z-plane and its image in the (b) Determine the image of D under the mapping
w-plane. w = sin(z). Sketch this image.
(a) 0 ≤ x ≤ π,0 ≤ y ≤ π 9. Determine the image of D under the mapping w=2z .
2
(b) −1 ≤ x ≤ 1,−π/2 ≤ y ≤ π/2 Sketch this image.
(c) 0 ≤ x ≤ 1,0 ≤ y ≤ π/4
10. Determine the image of the infinite strip 0 ≤ Im ≤ 2π
(d) 1 ≤ x ≤ 2,0 ≤ y ≤ π
z
under the mapping w = e .
(e) −1 ≤ x ≤ 2,−π/2 ≤ y ≤ π/2
2. In each of parts (a) through (e), find the image of In each of Problems 11 through 16, find the image of the
the rectangle under the mapping w = cos(z). Sketch given circle or line under the bilinear transformation.
the rectangle in the z-plane and its image in the
11. w = 2i/z;Re(z) =−4
w-plane.
(a) 0 ≤ x ≤ 1,1 ≤ y ≤ 2 12. w = 2iz − 4;Re(z) = 5
(b) π/2 ≤ x ≤ π,1 ≤ y ≤ 3 13. w = (z − i)/iz;(z + z)/2 + (z − z)/2i = 4
(c) 0 ≤ x ≤ π,π/2 ≤ y ≤ π
14. w = (z − 1 + i)/(2z + 1);|z|= 4
(d) π ≤ x ≤ 2π,1 ≤ y ≤ 2
(e) 0 ≤ x ≤ π/2,0 ≤ y ≤ 1 15. w = (2z − 5)(z + i); z + z − (3/2i)(z − z) − 5 = 0
3. In each of parts (a) through (e), find the image of 16. w = ((1 + 3i)z − 2)/z;|z − i|= 1
the rectangle under the mapping w = 4sin(z). Sketch
In each of Problems 17 through 21, find a bilinear transfor-
the rectangle in the z-plane and its image in the
mation taking the given points to the indicated images.
w-plane.
(a) 0 ≤ x ≤ π/2,0 ≤ y ≤ π/2 17. 1 → 1,2 →−i,3 → 1 + i
(b) π/4 ≤ x ≤ π/2,0 ≤ y ≤ π/2 18. i → i,1 →−i,2 → 0
(c) 0 ≤ x ≤ 1,0 ≤ y ≤ π/6
(d) π/2 ≤ x ≤ 3π/2,0 ≤ y ≤ π/2 19. 1 → 1 + i,2i → 3 − i,4 →∞
(e) 1 ≤ x ≤ 2,1 ≤ y ≤ 2 20. −5 + 2i → 1,3i → 0,−1 →∞
4. Determine the image of the sector π/4 ≤ θ ≤ 5π/4 21. 6 + i → 2 − i,i → 3i,4 →−i
2
under the mapping w = z . Sketch the sector and its
image. 22. Prove that the composition of two conformal map-
pings is conformal.
5. Determine the image of the sector π/6 ≤ θ ≤ π/3
3
under the mapping w = z . Sketch the sector and its 23. Show that the mapping w = T (z)= z is not conformal.
image. 24. Suppose T is a bilinear transformation that is not the
identity mapping or a translation. Show that T must
6. Show that the mapping
have either one or two fixed points. Why does this fail
1 1 for translations?
w = z +
2 z
25. A point z 0 is a fixed point of a bilinear transforma-
maps the circle |z|= r onto an ellipse with foci ±1in tion T if T (z 0 )=z 0 . Suppose a bilinear transformation
the w-plane. Sketch a typical circle and its image. T has three fixed points. Show that T must be the
identity mapping, sending each z to itself.
7. Show that the mapping of Problem 6 maps a half-line
θ = k onto a hyperbola with foci ±1inthe w-plane, 26. Let T and S be bilinear mappings that agree at three
assuming that sin(k) = 0and cos(k) = 0. Sketch a points. Show that T = S.
typical half-line and its image.
27. Define the cross ratio of four complex numbers
8. Let D consist of all z in the rectangle having vertices z 1 , z 2 , z 3 ,and z 4 to be the image of z 1 under the
±αi and π ± αi, with α a positive number. bilinear transformation that maps

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