Topic: Math/Physic/Economic/Statistic Problems

May 19, 2021

  1. Let A = {z ∈ C : |Arg(z)| ≤ π
    6
    } and B = {z ∈ C : |z − 2i| < 7, |z − 2| ≤ |3i|}.
    (a) Make separate sketches of the sets A and B.
    [4 Marks]
    (b) Show that the set B is not open.
    [5 Marks]
  2. Show that the set C = {z ∈ C : |z + 1 + 2i| < 11} is open.
    [8 Marks]
  3. Determine ALL the values of (ieπ
    )
    i
    . Indicate the principal value.
    [6 Marks]
  4. Showing all your work, sketch the image of the following set under the principal logarithmic
    function
    D = {z ∈ C : |z| ≤ 4, |Arg(z)| ≤ π
    3
    }.
    [7 Marks]
  5. Let z ∈ C be arbitrary.
    (a) Using the definition of the complex trigonometric functions, show that cos(z +
    π
    2
    ) =
    − sin(z), and sin(z +
    π
    2
    ) = cos(z).
    [6 Marks]
    Page 1 of 3
    (b) Using (a), show that cos(z + π) = − cos z, and sin(z + π) = − sin(z).
    [4 Marks]
  6. Consider the following complex function
    g(z) = 1
    (z + i)
    2
    .
    (a) Compute the largest domain of g.
    [2 Marks]
    (b) Is g invertible in its largest domain? Justify.
    [5 Marks]
    (c) Show that g is differentiable in its domain.
    [6 Marks]
  7. Given x, y ∈ R, consider z and z as new variables given by z = x + iy, and z = x − iy.
    (a) Show that x =
    1
    2
    (z + z) and y =
    1
    2i
    (z + z).
    [2 Marks]
    (b) If ∂z denotes the partial derivative with respect to z keeping z fixed and ∂z denotes
    the partial derivative with respect to z keeping z fixed, show that
    i. ∂z =
    1
    2
    (∂x − i∂y)
    [4 Marks]
    ii. ∂z =
    1
    2
    (∂x + i∂y),
    [4 Marks]
    where ∂x denotes the partial derivative with respect to x keeping y fixed and ∂y
    denotes the partial derivative with respect to y keeping x fixed.
    (c) Write a complex valued functionf : C → C as
    f(z, z) = u(x, y) + iv(x, y)
    where u, v : R
    2 → R. Show that if f is differentiable then ∂zf = 0.
    [6 Marks]
  8. Justifying all your methods, calculate the following integrals, with γ = {z ∈ C : |z| =
    1
    2
    }
    Page 2 of 3
    (a)

    γ
    cos(2z) + sin(2z)
    (z − i)
    2
    dz
    [3 Marks]
    (b)

    γ
    cos(2z) + sin(2z)
    z(z − i)
    dz
    [4 Marks]
    (c)

    γ
    cos(2z) + sin(2z)
    z
    2
    .
    [6 Marks]
  9. Evaluate the following real integral, assuming that it exists, using complex analysis techniques
    ∫ +∞
    −∞
    3
    x
    2 + x + 1
    dx,
    justify your answer.
    [10 Marks]
  10. Let f be a complex function. If f is analytic on a connected open set A then, f has
    derivatives of all orders for all elements of A. Based on the results we looked at in
    lectures, explain why this is true. Does this also hold for real functions? Justify.
    [8 Marks]

Type of service: Math/Physic/Economic/Statistic Problems
Type of assignment: Calculation
Subject: Not defined
Pages/words: 10/2750
Number of sources: N/A
Academic level: Undergraduate
Paper format: N/A
Line spacing: Double
Language style: UK English

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