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  1. #1
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    please help me

    Without using cauchy-Riemann equations prove that if v is a harmonic conjugate of u on a domain D then u is a harmonic conjugate of -v on D .

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    for $\displaystyle z,w \in C prove-- that :
    \mid z+w \mid = \mid z \mid + \mid w\mid $ iff
    $\displaystyle z \overline {w} \in R with z \overline {w} \geq 0 $
    proof:
    $\displaystyle \Rightarrow $ i don't know how i prove it but $\displaystyle \Leftarrow $ easy "i prove it "
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  2. #2
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    Quote Originally Posted by flower3 View Post
    Without using cauchy-Riemann equations prove that if v is a harmonic conjugate of u on a domain D then u is a harmonic conjugate of -v on D .
    If v is a harmonic conjugate of u then there is an analytic function $\displaystyle f(z)$ with real part u and imaginary part v. Now look at the real and imaginary parts of the analytic function $\displaystyle if(z)$.

    Quote Originally Posted by flower3 View Post
    for $\displaystyle z,w \in C$ prove-- that :
    $\displaystyle \mid z+w \mid = \mid z \mid + \mid w\mid $ iff
    $\displaystyle z \overline {w} \in R$ with $\displaystyle z \overline {w} \geq 0 $
    $\displaystyle \begin{aligned}|z+w| = |z|+|w|\:&\Leftrightarrow\:|z+w|^2 = (|z|+|w|)^2\\ &\Leftrightarrow\: (z+w)(\overline {z} + \overline {w}) = |z|^2 + 2|z||w| + |w|^2\\&\Leftrightarrow\: z\overline {z} + 2\text{Re}(z\overline {w}) + w\overline {w}) = |z|^2 + 2|z||w| + |w|^2\\ &\Leftrightarrow\: \text{Re}(z\overline {w}) = |z||w| = |z\overline {w}|\end{aligned}$

    You should be able to deduce that the last of those conditions is equivalent to $\displaystyle z\overline {w}$ being real and ≥0.
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