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Thread: Describing graphically an inequality (COMPLEX VARIABLES)

  1. #1
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    Describing graphically an inequality (COMPLEX VARIABLES)

    Hello everyone, hope you're all doing well.
    I'm having difficulty trying to describe graphically the set of points in the complex plane defined by the inequality:
    Im(z^4) > 0

    I assume it is referring to all points in the first and second quadrant of a complex plane, but is there any way to prove that this is true. If it is not true and I am just picturing it incorrect please let me know.

    Thank you
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  2. #2
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    Re: Describing graphically an inequality (COMPLEX VARIABLES)

    Quote Originally Posted by elijah123 View Post
    Hello everyone, hope you're all doing well.
    I'm having difficulty trying to describe graphically the set of points in the complex plane defined by the inequality:
    Im(z^4) > 0

    I assume it is referring to all points in the first and second quadrant of a complex plane, but is there any way to prove that this is true. If it is not true and I am just picturing it incorrect please let me know.
    Be careful. Let z=2\exp\left(\dfrac{3\pi\bf{i}}{4}\right) then \Im(z)=\sqrt{2}>0. BUT \Im(z^4)=0
    Thanks from elijah123
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  3. #3
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    Re: Describing graphically an inequality (COMPLEX VARIABLES)

    Quote Originally Posted by elijah123 View Post
    Hello everyone, hope you're all doing well.
    I'm having difficulty trying to describe graphically the set of points in the complex plane defined by the inequality:
    Im(z^4) > 0

    I assume it is referring to all points in the first and second quadrant of a complex plane, but is there any way to prove that this is true. If it is not true and I am just picturing it incorrect please let me know.

    Thank you
    let $z = r e^{i \theta},~r\geq, ~0 \leq \theta < 2 \pi$

    $z^4 = r^4 e^{i 4 \theta}$

    $\Im(z^4) = r^4 \sin(4 \theta)$

    This holds true if $r=0$

    else if $r > 0$

    $\Im(z^4)>0 \Rightarrow \sin(4 \theta) > 0$

    $ 4 \theta \pmod{2 \pi} \in [0, \pi)$

    so the set of points $z$ we are looking for is

    $r>0 \text{ AND }\theta \in \left[0, \dfrac \pi 4\right ) \cup \left[\dfrac \pi 2, \dfrac {3\pi}{4}\right) \cup \left[\pi, \dfrac {5\pi}{4}\right) \cup \left[\dfrac{3\pi}{2}, \dfrac{7\pi}{4}\right)$

    OR

    $r=0$
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    Re: Describing graphically an inequality (COMPLEX VARIABLES)

    Quote Originally Posted by romsek View Post
    $r=0$
    r=|z|\ge 0 therefore r=0\iff z=0 so \Im(0)=0\not>0
    Thanks from romsek and elijah123
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    Re: Describing graphically an inequality (COMPLEX VARIABLES)

    Quote Originally Posted by Plato View Post
    r=|z|\ge 0 therefore r=0\iff z=0 so \Im(0)=0\not>0
    once again my eyesight fails me thank you
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    Re: Describing graphically an inequality (COMPLEX VARIABLES)

    Describing graphically an inequality (COMPLEX VARIABLES)-img.jpg
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  7. #7
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    Re: Describing graphically an inequality (COMPLEX VARIABLES)

    Thank you all for your replies, sorry I couldn't reply back to all of them, it was late at night while I was working on this and fell asleep.
    Once again thank you.
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