232sheet1

# 232sheet1 - that the statement holds for all n ∈ Z ....

This preview shows page 1. Sign up to view the full content.

B U Department of Mathematics Math 232 Introduction To Complex Analysis Spring 2008 Exercise Sheet 1 1 1. Express the following complex numbers in polar form. a) z = 1 - i b) z = - 3 + i c) z = (1 - i )( - 1 - i ) d) z = 3 + i 1 + i ! 2. Prove that for any z C , | z | ≤ | Re z | + | Im z | ≤ 2 | z | . When do the equalities hold? 3. Let z , w C with z 6 = w . Prove that Re ± w + z w - z ² = | w | 2 - | z | 2 | w - z | 2 . By setting z = r (cos θ + i sin θ ) and w = R (cos φ + i sin φ ), where 0 < r < R , show that Re ± w + z w - z ² = R 2 - r 2 R 2 - 2 Rr cos( θ - φ ) + r 2 4. Prove that C is not equipped with an order structure. 5. Sketch the regions determined by the following conditions. a) Re z > 2 b) 1 < Im z < 2 c) 1 < Im( z - i ) < 2 d) - 1 < Im( z + 2) < 1 e) | z | < 2 f) | z | > 1 g) 1 ≤ | z | < 2 h) | z - 1 | < 1 i) | z + 2 - i | < 2 j) | z - 1 | < | z + 1 | k) - 1 < Im z 0 l) 1 ≤ | Re z | ≤ 2 6. For λ R + , show that { z C || z | = λ | z - 1 |} is a circle unless λ takes one particular value. (What is it?) 7. Prove that (cos θ + i sin θ ) n = cos + i sin for all θ R , for all n N . Using this result, show
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: that the statement holds for all n ∈ Z . Finally generalize to the case where n is a rational number, namely, n = p/q with ( p,q ) = 1. 8. Find the moduli of the following numbers. a) z = (3-2 i )(-1+2 i ) (4-i ) 2 b) z = 2 i 3+8 i c) z = (1-2 i ) 4 (6-8 i ) 9. For z = 1 2 (1-i √ 3), ﬁnd | z | , arg z , Arg(-z ), arg ¯ z . 10. What is the locus of the points satisfying | z-z 1 | = | z-z 2 | ? 11. Justify that an equation for a line passing through the points z 1 ,z 2 ∈ C can be given as arg ³ z-z 1 z 2-z 1 ´ = kπ, k = 0 , 1 . 1 Please visit: http://www.math.boun.edu.tr/deptcourses/math232...
View Full Document

## This note was uploaded on 11/08/2011 for the course MATH 232 taught by Professor Gurel during the Spring '11 term at Boğaziçi University.

Ask a homework question - tutors are online