exsheet1

# exsheet1 - z with itself 7 Suppose n is a positive integer...

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Sivakumar M407 Example Sheet 1 1. Find a polar representation ( i.e., modulus–amplitude form) of the complex number z = 3 - i . What is Arg( z ) ? 2. Show that | z | = 1 if and only if 1 /z = z . 3. Suppose z and w are nonzero complex numbers. Show that zw is nonzero. 4. Suppose A is a real number and B is a complex number. Show that | z | 2 + A 2 = | z + A | 2 - 2Re( Az ) and | z | 2 + 2Re( Bz ) = | z + B | 2 - | B | 2 for any complex number z . 5. Show that | z + w | 2 - | z - w | 2 = 4Re( z w ) for every pair of complex numbers z and w . 6. (a) Suppose n is a positive integer and z 1 , . . . , z n are complex numbers. Use mathematical induction to prove the following: (i) n X k =1 z k n X k =1 | z k | . (ii) n X k =1 z k ! = n X k =1 z k . (iii) ( z 1 · · · z n ) = z 1 · · · z n . (iv) | z 1 · · · z n | = | z 1 | · · · | z n | . (b) Deduce from (iii) and (iv) above that z n = z n and | z n | = | z | n for every complex number z (where z n denotes the n -fold product of
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Unformatted text preview: z with itself). 7. Suppose n is a positive integer and a ,...,a n are real numbers. Let P ( z ) := ∑ n k =0 a k z k , z ∈ C (where z := 1). Prove that P ( z ) = 0 for some complex number z if and only if P ( z ) = 0. 8. Give an example to show that Arg( zw ) need not equal Arg( z ) + Arg( w ). 9. Suppose | z | = 1. Show that Arg ² z-1 z + 1 ³ = ´ π/ 2 , if Im( z ) > 0;-π/ 2 , if Im( z ) < 0. 10. Show that De Moivre’s formula holds for negative integers n as well. 11. Suppose z = x + iy is a complex number. Prove that | z | ≤ | x | + | y | ≤ √ 2 | z | . 1...
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