By Young N.
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Extra resources for An Introduction to Hilbert Space
39. Suppose a, b, c and d are real numbers. Find the conditions under which the quadratic equation x 2 + (a + ib)x + (c + id) = 0 has at least one real root. 40. Assume |αk | < 1 and λk ≥ 0, k = 1, 2, . , n, and λ1 + λ2 + · · · + λn = 1. 44 Complex Numbers Here, α1 , α2 , . , αn are complex numbers. Show that |λ1 α1 + λ2 α2 + · · · + λn αn | < 1. 41. Let w1 and w2 be two complex numbers. Denote w = w2 − w1 . Suppose w2 = rw1 , where r is real and non-zero. Show that 1 1 + cos(Arg w2 − Arg w) 2 = r 1 + cos(Arg w1 − Arg w) 1 when 0 < r < 1 when r > 1 .
W3 Hint: The given condition is equivalent to w 2 − w1 z 2 − z1 = . 33. If z1 , z2 and z3 represent the vertices of an equilateral triangle, show that z12 + z22 + z32 = z1 z2 + z2 z3 + z3 z1 . 34. Show that the area of the triangle whose vertices are z1 , z2 , z3 is given by the absolute value of z 3 − z1 1 |z3 − z2 |2 Im 2 z 3 − z2 . 35. Consider any triangle ABC whose sides are of length α, β, γ and for which the distances from the centroid to the vertices are λ, µ, ν. Show that α2 + β 2 + γ 2 = 3.
Show that if r1 eiθ1 + r2 eiθ2 = reiθ , then r 2 = r12 + 2r1 r2 cos(θ1 − θ2 ) + r22 r1 sin θ1 + r2 sin θ2 θ = tan−1 . r1 cos θ1 + r2 cos θ2 Generalize the result to the sum of n complex numbers. 15. One may find the square roots of a complex number using polar representation. First, we write formally z = r(cos θ + i sin θ ). Recalling the identities sin2 θ 1 − cos θ = 2 2 and cos2 θ 1 + cos θ = , 2 2 show that √ z1/2 = ± r 1 + cos θ +i 2 1 − cos θ 2 , 0 ≤ θ ≤ π. Explain why the above formula becomes invalid for −π < θ < 0.
An Introduction to Hilbert Space by Young N.