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 $z,w \in C prove-- that :
\mid z+w \mid = \mid z \mid + \mid w\mid$
iff
$z \overline {w} \in R with z \overline {w} \geq 0$
proof:
$\Rightarrow$ i don't know how i prove it but $\Leftarrow$ easy "i prove it "

2. Originally Posted by flower3
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 $f(z)$ with real part u and imaginary part v. Now look at the real and imaginary parts of the analytic function $if(z)$.

Originally Posted by flower3
for $z,w \in C$ prove-- that :
$\mid z+w \mid = \mid z \mid + \mid w\mid$ iff
$z \overline {w} \in R$ with $z \overline {w} \geq 0$
\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 $z\overline {w}$ being real and ≥0.