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Math Help - Solve iz + z = 0

  1. #1
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    Solve iz + z = 0

    Hey guys, quick question

    (b) Find all complex solutions of iz + z = 0 and sketch them in the complex plane.

    Well I let z = x + iy, simplified and obtained x - y + i(x - y) = 0

    But as for sketching them I'm not entirely sure how to proceed. The answers have up and until the answer I obtained but then immediately proceeds to sketching y = x

    Can someone please explain why y = x is the solution?
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  2. #2
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    Re: Differential Calculus - Complex solutions

    Your simplification is wrong

    \displaystyle \begin{align*} iz + z &= 0 \\ i(x + iy) + x + iy &= 0 + 0i \\ ix - y + x + iy &= 0 + 0i \\ x - y + i(x + y) &= 0 + 0i \end{align*}

    Now equating real and imaginary parts gives

    \displaystyle x - y = 0 and \displaystyle x + y = 0.

    So the solutions are \displaystyle y = x and \displaystyle y = -x.
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  3. #3
    MHF Contributor chisigma's Avatar
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    Re: Differential Calculus - Complex solutions

    Quote Originally Posted by madbandlt View Post
    Hey guys, quick question

    (b) Find all complex solutions of iz + z = 0 and sketch them in the complex plane.

    Well I let z = x + iy, simplified and obtained x - y + i(x - y) = 0

    But as for sketching them I'm not entirely sure how to proceed. The answers have up and until the answer I obtained but then immediately proceeds to sketching y = x

    Can someone please explain why y = x is the solution?
    The equation i z + z = (1+i) z = 0 is first degree algebrical and its only solution is z=0...

    Kind regards

    \chi \sigma
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  4. #4
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    Re: Differential Calculus - Complex solutions

    Quote Originally Posted by Prove It View Post
    Your simplification is wrong

    \displaystyle \begin{align*} iz + z &= 0 \\ i(x + iy) + x + iy &= 0 + 0i \\ ix - y + x + iy &= 0 + 0i \\ x - y + i(x + y) &= 0 + 0i \end{align*}

    Now equating real and imaginary parts gives

    \displaystyle x - y = 0 and \displaystyle x + y = 0.

    So the solutions are \displaystyle y = x and \displaystyle y = -x.
    The equations \displaystyle x - y = 0 and \displaystyle x + y = 0 have to be true simultaneously. Therefore the only solution is x = y = 0.

    But the OP has not actually typed the correct question. The correct question is probably the following:

    Find all complex solutions of iz + \overline{z} = 0 and sketch them in the complex plane.

    In which case, you end up with the two equations x - y = 0 and x - y = 0 and the solution is obviously y = x.
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