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MATH 555 : PROBLEMS - 1 1. Prove the following identities : (A) Im(iz) = Re(z) for any z ∈ C . (B) Re(iz) = −Im(z) for any z ∈ C . (C) |z|2 + a2 = |z + a|2 − 2Re(az) for any z ∈ C where a ∈ R . (D) |z|2 + 2Re(bz) = |z + b̄|2 − |b|2 for any b, z ∈ C . (E) |1 − āz|2 − |z − a|2 = (1 − |z|2 )(1 − |a|2 ) for any iff z, a ∈ C . 2. Compute (1 + i)n − (1 − i)n where n ∈ Z , n ≥ 0 . 3. Given c ∈ C with |c| = 1 and c 6= ±1 prove that z−c =1 cz − 1 iff z ∈ R . 4. Given z, a ∈ C with |a| < 1, prove that z−a < 1, = 1 , > 1 1 − āz iff |z| < 1, = 1 , > 1 respectively. 5. Consider C with its standard structure as a vector space over the field R. The set {1, i} constitutes the standard basis of C as vector space over R. (A) Prove that each R-linear map ϕ : C −→ C is of the form ϕ(z) = az + bz for some a, b ∈ C. (B) Compute tr(ϕ), det(ϕ) and ϕ−1 if det(ϕ) 6= 0 in terms of a, b ∈ C. (C) Prove that an R-linear ϕ : C −→ C is C-linear iff ϕ ◦ J = J ◦ ϕ where J : C −→ C is an R-linear map which is represented by the matrix 0 −1 1 0 with respect to the standard basis of C over R . 6. Let n ∈ N with n ≥ 2 and ω = eπ i/n . (A) Show that n−1 Y xn − 1 = x − ω 2k for every x 6= 1 . x−1 k=1 (B) Show that n−1 Y 1 − ω 2k = n . k=1 (C) Show that ω n(n−1) 2 = in−1 . (D) Establish the Gauss Formula n−1 Y sin kπ n k=1 = n . 2n−1 (E) Employ the Gauss Formula to show that Z π log sin x dx = −π log 2 0 (F) Prove that n−1 Y k=1 cos kπ n = 0 (−1) 2 2n−1 if n is even . n−1 if n is odd 7. (A) Given A, B, C ∈ R, prove that the minimum and maximum values of the function f (θ) = A + B cos 2θ + C sin 2θ are A − p p B 2 + C 2 and A + B 2 + C 2 . (B) Prove that |z|2 + |w|2 − |z 2 + w2 | ≤ 2 | az + bw |2 a2 + b2 for any z, w ∈ C and a, b ∈ R with a2 + b2 6= 0. 1 Put cos θ = a b and sin θ = 2 . a2 + b2 a + b2 1 ≤ |z|2 + |w|2 + |z 2 + w2 | 2