exsheet4b - Sivakumar Example Sheet 4b M407 1. Suppose that...

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Unformatted text preview: Sivakumar Example Sheet 4b M407 1. Suppose that n is a fixed positive integer. Show that cos(nθ) can be expressed as a polynomial n in cos θ; more precisely, there exist real numbers a0 , . . . , an such that cos(nθ) = k=0 ak (cos θ)k . (Use strong induction: The assertion is clear for n = 1. Assume that it holds for every 1 ≤ n ≤ k and prove it for n = k + 1.) 2. (i) Let a and b be fixed real numbers with a < b. Find the image of the set {z = x + iy : a ≤ x ≤ b} under the exponential function z → exp(z ). (ii) Let 0 < α < π be a fixed number. Find the image of the set {z = x + iy : −α < y < α} under the exponential function z → exp(z ). 3. Find all values of z such that cos z = √ 2 i. 4. Find all values of z such that sin z = 0. 5. Prove parts (i)–(v) of Theorem 2.2.7. 6. (i) Let z = x + iy . Use the addition formula for the cosine and sine functions (Theorem 2.2.7(iv),(v)) to show that cos(z ) = cos x cosh y − i sin x sinh y and sin(z ) = sin x cosh y + i cos x sinh y. (ii) Let z = x + iy . Show that | cos z |2 = cos2 x + sinh2 y and | sin z |2 = sin2 x + sinh2 y . (iii) Let z = x + iy . Show that | sinh y | ≤ | cos z | ≤ cosh y and | sin hy | ≤ | sin z | ≤ cosh y . 1 ...
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