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Math Help - please help me

  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 .

    __________________________________________________ __________
    for  z,w \in C prove-- that :<br />
\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 "
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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 f(z) with real part u and imaginary part v. Now look at the real and imaginary parts of the analytic function if(z).

    Quote Originally Posted by flower3 View Post
    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.
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