185hw2-1.pdf - Math 185 Homework 2 \u2013 Due 09\/11 Peter Koroteev 1 Let f(z = 1\/z Describe what f does to the inside and outside of the unit circle and

185hw2-1.pdf - Math 185 Homework 2 – Due 09/11 Peter...

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Math 185 Homework 2 – Due 09/11 Peter Koroteev 1. Let f ( z ) = 1 /z . Describe what f does to the inside and outside of the unit circle, and also what it does to points on the unit circle. This map is called inversion through the unit circle. 2. Let g ( z ) = 1 / ¯ z . Describe what g does to the inside and outside of the unit circle, and also what it does to points on the unit circle. This map is called reflection through the unit circle. 3. Let f ( z ) = e z . Describe the image under f of the following sets: (a) The set of z = x + iy such that x 1 and 0 y π (b) The set of z = x + iy such that 0
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Unformatted text preview: ≤ y≤ π with no restriction on x . 4. Describe how to construct Riemann surfaces for the functions z( w ) given by the solu- tions of the following equations (a) w= z 4 (b) w = ( z- 1)5 / 2 5. Show that e z =e ¯ z. 6. Sketch the image under the map w = log z of each of the following figures (a) half disk | z | < 1, Re z > 0 , (b) vertical line y = e . 7. How would you make a branch cut to define a single-valued branch of the function log( z + i- 1). How about log( z- z )?...
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