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Math Help - [SOLVED] Argument of complex number

  1. #1
    Moo
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    [SOLVED] Argument of complex number

    Hi !

    Sorry if it isn't the correct subforum...


    The question is :
    Find the argument of \frac{(z_1+z_2)^2}{z_1z_2}, given that |z_1|=|z_2|=1

    I found it to be 2arg(z_1+z_2)-arg(z_1)-arg(z_2), but couldn't go any further.. Did I miss something ?


    Thanks in advance
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  2. #2
    Super Member flyingsquirrel's Avatar
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    Hi
    Quote Originally Posted by Moo View Post
    [...] given that |z_1|=|z_2|=1
    This gives us z_1=\exp(i\theta) and z_2=\exp(i\varphi).
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  3. #3
    Moo
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    Quote Originally Posted by flyingsquirrel View Post
    Hi

    This gives us z_1=\exp(i\theta) and z_2=\exp(i\varphi).
    I know, but well, the z_1+z_2 is disturbing...

    I got to expressions like \cos \frac{\theta_1-\theta_2}{2} but couldn't simplify :/
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  4. #4
    Super Member flyingsquirrel's Avatar
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    Quote Originally Posted by Moo View Post
    I know, but well, the z_1+z_2 is disturbing...

    I got to expressions like \cos \frac{\theta_1-\theta_2}{2} but couldn't simplify :/
    \frac{(z_1+z_2)^2}{z_1z_2}=\frac{\mathrm{e}^{i2\th  eta_1}+2\mathrm{e}^{i(\theta_1+\theta2)}+\mathrm{e  }^{i2\theta_2}}{\mathrm{e}^{ i(\theta_1+\theta_2)}}=\mathrm{e}^{i(\theta_1-\theta_2)}+2+\mathrm{e}^{i(\theta_2-\theta_1)}

    If this is the expression you got to, you have the answer since this is a real number.
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  5. #5
    Moo
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    Quote Originally Posted by flyingsquirrel View Post
    \frac{(z_1+z_2)^2}{z_1z_2}=\frac{\mathrm{e}^{i2\th  eta_1}+2\mathrm{e}^{i(\theta_1+\theta2)}+\mathrm{e  }^{i2\theta_2}}{\mathrm{e}^{ i(\theta_1+\theta_2)}}=\mathrm{e}^{i(\theta_1-\theta_2)}+2+\mathrm{e}^{i(\theta_2-\theta_1)}

    If this is the expression you got to, you have the answer since this is a real number.
    I got it, but what I need is the argument... ?
    I'm not sure I'm getting what you're saying :/
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  6. #6
    Super Member flyingsquirrel's Avatar
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    Quote Originally Posted by Moo View Post
    I got it, but what I need is the argument... ?
    I'm not sure I'm getting what you're saying :/
    I should have added that \frac{(z_1+z_2)^2}{z_1z_2}=\mathrm{e}^{i(\theta_1-\theta_2)}+2+\mathrm{e}^{i(\theta_2-\theta_1)}=2\cos(\theta_1-\theta_2)+2\geq0

    This tells us that the argument is 0\mod 2\pi except when z_1=-z_2.
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  7. #7
    Moo
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    Quote Originally Posted by flyingsquirrel View Post
    I should have added that \frac{(z_1+z_2)^2}{z_1z_2}=\mathrm{e}^{i(\theta_1-\theta_2)}+2+\mathrm{e}^{i(\theta_2-\theta_1)}=2\cos(\theta_1-\theta_2)+2\geq0

    This tells us that the argument is 0\mod 2\pi except when \theta_1=-\theta_2.
    Oh yeah, that's better

    Hmm \theta_1=-\theta_2 ? ^^

    I'd say "arg=0 mod 2pi and theta_1-theta_2=-pi mod 2pi"


    Edit : it's ok now
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  8. #8
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    Hey Moo

    I just have to post the geometric approach to this. hope you like my handwriting!

    Bobak

    Edit: I wrote equilateral when I meant isosceles, Thanks for spotting that Moo.
    Attached Thumbnails Attached Thumbnails [SOLVED] Argument of complex number-complex.jpg  
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  9. #9
    Moo
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    Nice handwriting !


    (should be isoscele though lol)


    Thank you both of you !
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