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Complex numbers help

  1. May 10, 2009 #1
    1. The problem statement, all variables and given/known data

    1.
    (a) Express -1 + √3i in modulus-argument form. Evaluate (-1 + √3i)^8
    expressing your answer in (a + ib) form.

    Find also the square roots of -1 + √3i in (a + ib) form.
    (b) Use complex numbers to find

    (intergral is between 0 and ∞) ∫ e^-x cos2x dx.

    2.
    (a) Find the modulus and argument of -1 - i√3. Hence find (-1 - i√3) ^ 10 in a+ib
    form. Find also the square roots of -1 - i√3 in a + ib form.

    (b) Use complex numbers to find

    ∫ e^-x sin3x dx

    3.
    (a) Find the square roots of 1 - i√3. Find also
    [(1- i√3)/ (1+i√3)] ^ 8

    (b) Use complex numbers to find

    ∫ e^kx cosx dx

    where k is a constant.

    2. Relevant equations



    3. The attempt at a solution

    For q1. but nt sure u gta check it for me q2 seems similar but q3 lost on it.

    q1.
    1. Modulus = √ ((-1)&2 + (√3)^2)
    = √(1 + 3) = 2
    Argument = arctan(√3/-1) = 4 pi /3 (draw a picture to make sure you have the angle in the correct quadrant)

    The 8th power has modulus 2^8 = 256 and argument 8 x (4 pi/3) = 32 pi /3 = 2pi/3 (subtract multiples of 2 pi).

    modulus 2 pi /3 = cos (2pi/3) + i sin (2pi/3) = -0.5 + √3/2 i

    (-1 + √3i)^8 = 256 (-0.5 + √3/2 i) = -128 + 128√3 i

    The square root is similar - modulus √2, amplitude 2 pi / 3

    (b) ∫ e^-x sin3x dx = Im ∫ e^-x(cos 3x + i sin 3x) dx
    = Im ∫ e^x e^3ix dx
    = Im ∫ e^(-1+3i) x dx
    = Im e^{-1+3i)x / (-1 + 3i)
    = Im (-1 - 3i) e^{1+3i)x / (-1 + 3i)(-1 - 3i)
    = Im (-1- 3i) e^-x (cos3x + i sin 3x) / 10
    = e^-x(-sin 3x - 3 cos 3x) / 10
     
  2. jcsd
  3. May 10, 2009 #2

    Cyosis

    User Avatar
    Homework Helper

    Haven't looked at q2 and q3 yet, but the argument you calculated for q1 is wrong. Draw z in the complex plane and calculate the angle using normal trigonometry. As a result the rest of a and b is also wrong, although you used the correct method.
     
    Last edited: May 10, 2009
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