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3.9. Fix a ? C with ?a? < 1 and consider

¦_a(z) = z – a / 1 – az

(a) Show that ¦_a(z) is a Mobius transformation.

(b) Show that ¦^-1_a(z) = ¦_-a(z).

(c) Prove that ¦_a(z) maps the unit disk D[0, 1] to itself in a bijective fashion.

3.13. Let ¦(z) = 2x/x + 2. Draw two graph, one showing the following six sets in the z- plane and the other showing their images in the w-plane. Label the sets. (You should only need to calculate the images of 0, ±2, ±(1 + i), and ¥; remember that Mobius transformations preserve angles.)

(a) The x-axis plus ¥

(b) The y-axis plus ¥

(c) The line x = y plus ¥

(d) The circle with radius 2 centered at 0

(e) The circle with radius 1 centered at 1

(f) The circle with radius 1 centered at -1

3.18. Find a Mobius transformation that maps the unit disk to {x + iy ? C : x + y > 0}.

3.21. Find each Mobius transformation¦:

(a) ¦ Maps 0 ® 1, 1 ® ¥, ¥ ® 0.

(b) ¦ Maps 1 ® 1, -1 ® i, -i ® -1.

(c) ¦ Maps the x-axis to y = x, the y-axis to y = -x, and the unit circle to itself.

3.24. Suppose z₁, z₂ and z₃ are distinct points in C. Show that z is on the circle passing through z₁, z₂ and z₃ if and only if [z, z₁, z_2, z₃] is real or ¥.

3.46. Find the image of the annulus 1 < ?z? < e under the principal value of the logarithm.

3.47. Use Exercise 2.24 to give an alternative proof that Log is holomorphic in C\R_£0.

3.49. Show that ?a^z? a^{Re z} if a is a positive real constant.

3.53. For this problem, ¦(z) = z².

(a) Show that the image under ¦ of a circle centered at the origin is a circle centered at the origin.

(b) Show that the image under ¦ of a ray starting at the origin is a ray starting at the origin.

(c) Let T be the figure formed by the horizontal segment from 0 to 2, the circular are from 2 to 2i, and then the vertical segment from 2i to 0. Draw T and ¦(T).

(d) Is the right angle at the origin in part (c) preserved? Is something wrong here?