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Math Help - please help me

  1. #1
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    please help me

    show that :
    if Re (z1) >0 and Re( z2) >0 then :
     Arg (z1z2) =
      Arg (z1)+ Arg (z2)
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  2. #2
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    Fact: If z=r\mathrm e^{\mathrm i\theta} where r>0 and -\pi<\theta\leq\pi then r=|z| and \theta=\mathop{arg} z.

    If \mathop{arg} z_1=\theta and \mathop{arg} z_2=\phi then z_1=|z_1|\mathrm e^{\mathrm i\theta} and z_2=|z_2|\mathrm e^{\mathrm i\phi}.

    Therefore z_1z_2=|z_1||z_2|\mathrm e^{\mathrm i\theta}\mathrm e^{\mathrm i\phi}=|z_1z_2|\mathrm e^{\mathrm i(\theta+\phi)}.

    If \mathop{Re}(z_1)>0 then -{\textstyle\frac\pi2}<\theta<{\textstyle\frac\pi2}, and if \mathop{Re}(z_2)>0 then -{\textstyle\frac\pi2}<\phi<{\textstyle\frac\pi2}. Therefore -\pi<\theta+\phi<\pi.

    Thus \mathop{arg}(z_1z_2)=\theta+\phi=\mathop{arg} z_1+\mathop{arg} z_2.
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  3. #3
    Super Member malaygoel's Avatar
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    Let z_1=a_1+ib_1

    and z_2=a_2+ib_2

    be two complex numbers where a_1,a_2>0

    arg(z_1)=tan^{-1}\frac{b_1}{a_1}

    arg(z_2)=tan^{-1}\frac{b_2}{a_2}

    Now, z_1z_2=(a_1a_2-b_1b_2)+i(a_1b_2+a_2b_1)

    arg(z_1z_2)=tan^{-1}\frac{a_1b_2+a_2b_1}{a_1a_2-b_1b_2}

    arg(z_1z_2)=tan^{-1}\frac{b_1}{a_1}+tan^{-1}\frac{b_2}{a_2}

    I don't know how the property a_1,a_2>0 is used.
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  4. #4
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    i Noted that  Arg({Z\scriptstyle1} {Z\scriptstyle2}) \leq Arg({Z\scriptstyle1  }) +Arg({Z\scriptstyle2}  )
    this true or not ??
    i take some examples like :
    { Z\scriptstyle1} = -5
    { Z\scriptstyle2} = -1-i ,
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  5. #5
    Super Member malaygoel's Avatar
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    z=a+ib=|z|e^{i\theta}

    There are four cases:
    (1) a\geq 0,b\geq 0,
    then we have, 0 \leq \theta \leq \frac{\pi}{2}

    (2) a\leq 0,b\geq 0
    then we have, \frac{\pi}{2} \leq \theta \leq {\pi}

    (3) a\leq 0,b\leq 0
    then we have, \pi \leq \theta \leq \frac{3\pi}{2}

    (4) a\geq 0,b\leq 0
    then we have, \frac{3\pi}{2} \leq \theta \leq 2\pi

    Now,
    if sum of arguments( \theta_1,\theta_2) is less than 2\pi,
    then
    <br />
Arg({Z\scriptstyle1} {Z\scriptstyle2}) = Arg({Z\scriptstyle1 }) +Arg({Z\scriptstyle2} )<br />

    And,
    if sum of arguments( \theta_1,\theta_2) is equal to or more than 2\pi,
    then
    <br />
Arg({Z\scriptstyle1} {Z\scriptstyle2}) = Arg({Z\scriptstyle1 }) +Arg({Z\scriptstyle2} )-2\pi<br />
    <br />
Arg({Z\scriptstyle1} {Z\scriptstyle2}) < Arg({Z\scriptstyle1 }) +Arg({Z\scriptstyle2} )<br />

    Quote Originally Posted by flower3
    <br />
{ Z\scriptstyle1} = -5<br />
    <br />
{ Z\scriptstyle2} = -1-i<br />
    in the example you provided,
    \theta_1=\pi
    \theta_2=\frac{5\pi}{4}
    arg(Z_1Z_2)=\pi +\frac{5\pi}{4}-2\pi=\frac{\pi}{4}
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  6. #6
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    (3)
    then we have,

    (4)
    then we have,
    these cases are not true because :
     Arg Z=\theta and -\pi \prec \theta \leq \pi
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  7. #7
    Super Member malaygoel's Avatar
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    Quote Originally Posted by flower3 View Post
    these cases are not true because :
     Arg Z=\theta and -\pi \prec \theta \leq \pi
    there is no condtion like this.

    it varies from 0 to 2\pi

    Are you familiar with Argand Plane?

    Then plot -1-i on it and find \theta.

    EDIT: you and me are saying the same thing...don't know which is standard.
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