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    Complex number identities

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    Let z, w E C

    a) Prove the following identities:

    i) |z+w|^2 = |z|^2 + 2Re(zw) + |w|^2
    ii) |z - w|^2 = |z|^2 - 2Re(zw) + |w|^2

    b) Deduce that |z+w|^2 + |z-w|^2 = 2(|z|^2 + |w|^2).

    c) Use (a)(i) to prove that |z+w| < |z-| + |w| and give necessary conditions for equality to hold.

    d) Prove that [|z| - |w|] < |z-w|.

    © BrainMass Inc. brainmass.com October 10, 2019, 6:48 am ad1c9bdddf
    https://brainmass.com/math/calculus-and-analysis/complex-number-identities-557788

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    Solution Preview

    Let z = x1+i.y1, w = x2 + i.y2

    a.i.
    Left Hand Side (L.H.S.):
    |z+w|^2 = (x1+x2)^2 + (y1+y2)^2 = x1^2 + x2^2 + y1^2 +y2^2 + 2x1.x2 + 2y1.y2
    = x1^2 + y1^2 + x2^2 + y2^2 + 2(x1.x2+y1.y2)
    = |z|^2 + |w|^2 + 2Re(z.w_bar) = R.H.S.

    [Because, z.w_bar = (x1+i.y1).(x2-i.y2) = (x1.x2 + y1.y2) + i.(x2.y1 - x1.y2)]

    a.ii.
    L.H.S.:
    |z-w|^2 = (x1-x2)^2 + (y1-y2)^ = x1^2 + x2^2 + y1^2 +y2^2 - 2x1.x2 - 2y1.y2
    = x1^2 + y1^2 + x2^2 + y2^2 - 2(x1.x2+y1.y2)
    = |z|^2 + |w|^2 - 2Re(z.w_bar) = R.H.S.

    b.
    L.H.S.
    |z+w|^2 + |z-w|^2 = ...

    Solution Summary

    Some complex number identities are proven in the solution.

    $2.19