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Complex numbers arguments

  1. May 28, 2013 #1
    draw on a argand diagram [tex] |arg(z + 1)| = \dfrac{\pi}{2} [/tex]

    I got the correct drawing... but I'm not sure why it's correct.

    What I thought was [itex] arg(z + 1) = \dfrac{\pi}{2} [/itex] and that's a half line from the point (-1,0) going upwards, and [itex] arg(z + 1) = -\dfrac{\pi}{2} [/itex] and that's a half life from the point (-1,0) going downwards. Joining these we get a straight line passing through the point (-1,0).

    this gives the correct sketch according to my book, but what I don't understand is how we can just "add" these two graphs together - and perhaps if someone could explain a better way of thinking about it?
     
  2. jcsd
  3. May 28, 2013 #2

    cepheid

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    Hello converting1,

    The idea of this question is to find all points in the complex plane that satisfy this equation |arg(1+z)| = pi/2. Every point in the complex plane corresponds to a complex number. So, stated another way, the goal is to find all complex numbers z that satisfy the equation.

    First of all, to simplify, write w = 1+z, so that we have some complex number w. It has magnitude |w| and argument arg(w). On the Argand diagram, these correspond to the *polar coordinates* r and theta respectively. I have put an explanation for this below**. Recall that when you express the coordinates of a point in polar coordinates, the r coordinate is the distance between that point and the origin. The theta coordinate is the angle made by a line joining the origin to that point. This angle is measured counterclockwise from the positive x-axis. As you correctly stated, if |arg(w)| = pi/2, then arg(w) = +pi/2 OR -pi/2. So, what this amounts to saying is that theta = +pi/2 or -pi/2.

    In polar coordinates, the curve given by the equation theta = constant is just a straight line at angle theta from the x-axis. The reason is that r can be anything, and so the distance is not constrained. The set of all points that have the same theta coordinate, but varying r coordinate, lies along this straight line. In our specific case, the line corresponding to theta = +pi/2 is a line going straight vertically up from the origin. The line corresponding to theta = -pi/2 is a line going straight vertically down from the origin.

    Doing a coordinate transformation from w back to z corresponds to shifting the whole curve left along the real axis by 1. For example, when w = 0, it follows that z = w - 1 = -1. The vertical line going through the origin is given by the equation w = 0 + yi, where y is a variable (it can be any real number). Therefore z = -1 + yi, which is a vertical line passing through (-1,0) instead of (0,0).


    **Why |w| corresponds to a radial coordinate r, and arg(w) corresponds to an angular coordinate θ:

    In polar form, the complex number w is expressed as |w|eiarg(w). Therefore, from the Euler relation for complex exponentials, it follows that:

    w = |w|cos(arg(w)) + i*|w|sin(arg(w))

    where the first term is the horizontal (real) component, and the second term is the vertical (imaginary) component. So, for this point on the complex plane, the distance "r" from the origin to the point is just the following (by Pythagoras):

    r = √[ |w|2(cos2(arg(w)) + sin2(arg(w)) ]

    But we have the trig identity that cos2x + sin2x = 1, so this just becomes:

    r = √(|w|2) = |w|

    The magnitude of the complex number tells you the distance from the origin of the point on an Argand diagram corresponding to that complex number. It is the radial coordinate r.

    Similarly, the angle θ between a position vector going to that point and the positive x-axis is just going to be arctan(vertical component / horizontal component) = arctan(Im(w)/Re(w)):

    θ = arctan[|w|sin(arg(w)) / |w|cos(arg(w))]

    Of course, the |w|'s cancel and we just have sin(arg(w))/cos(arg(w)) = tan(arg(w)). Therefore:

    θ = arctan[tan(arg(w))] = arg(w)

    The argument of the complex number tells you the angle between the real axis and a line connecting the origin to the point on an Argand diagram corresponding to that complex number. It is the angular coordinate θ.
     
  4. May 28, 2013 #3

    Dick

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    I would let u=z+1. So now you've got arg(u)=pi/2, the positive y axis or arg(u)=(-pi/2), the negative y axis. Since z=u-1, you just move the y axis (without the origin) one unit left. But you seem to have figured it out without that. Still you 'add' them because it's either arg(z+1)=pi/2 OR arc(z+1)=(-pi/2). So you union the two graphs.
     
  5. May 28, 2013 #4
    Thank you very much to the both of you!
     
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