HW1 - equation | z + c | = | 1 + cz | . 6. (problem 20,...

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P ROBLEM SET 1 due in class on Friday, January 16. Complex Variables MAT334 Remark: Please try to write clearly. In particular every statement that you make in your solution should follow from the assumptions of the problem, some of the material that was a prerequisite for the course (high-school algebra, calculus), was done in class, or some of the statements that you have already established in your solution. If you are using some of the theorems that have a name, mention that (e.g. ”By Mean Value Theorem. ..”). 1. Let z = - 3 - 2 i and w = 2 - i . Compute zw and w 2 /z 2. Find the polar representation of the complex numbers 1 - i and 2 / (1 + i ) . 3. Find all fourth roots of i and fifth roots of 1 . 4. Prove the Parallelogram law: For any two complex numbers w, z , | z + w | 2 + | z - w | 2 = 2 | z | 2 + 2 | w | 2 5. Let c be a complex number such that | c | < 1 . Solve (that is find all solutions) the
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Unformatted text preview: equation | z + c | = | 1 + cz | . 6. (problem 20, Sec. 1.1) Let B and C be nonnegative real numbers and A a complex number. Suppose that B-2 Re ( A )+ | | 2 C for all complex numbers . Show that | A | 2 BC . (Hint: If C=0, show that A = 0 . If C 6 = 0 , then choose = A/C .) 7. (problem 21, Sec. 1.1) Let a 1 , . . . , a n and b 1 , . . . , b n be complex numbers. Prove the Schwartz inequality : n X j =1 a j b j 2 n X j =1 | a j | 2 ! n X j =1 | b j | 2 ! . (Hint: For all complex numbers , one has that n j =1 | a j-b j | 2 . Expand this and apply the problem 6. with A = n j =1 a j b j , B = n j =1 | a j | 2 , and c = n j =1 | b j | 2 .)...
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