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Loser66
 one year ago
I need help!! Please
Prove \(log (zw ) \neq log z + log w \) if \(Re z, Re w \leq 0\)
I do counter example but not get the real part right.
Loser66
 one year ago
I need help!! Please Prove \(log (zw ) \neq log z + log w \) if \(Re z, Re w \leq 0\) I do counter example but not get the real part right.

This Question is Closed

Loser66
 one year ago
Best ResponseYou've already chosen the best response.0Let \(z =i , z = 1, arg z = pi/2\\ log z = 1+ iarg (z) = 1 + i(pi/2)\) Let \(w = 1+i , w = \sqrt2 , arg w= 3\pi/4\\w = \sqrt2 + i(3\pi/4)\)

Loser66
 one year ago
Best ResponseYou've already chosen the best response.0Hence \(logz + log w = 1 +\sqrt2 + i(\pi/2+3\pi/4) = 1+\sqrt2 + i (5\pi/4)\) \(Real (zw) = 1 + \sqrt2\)

Loser66
 one year ago
Best ResponseYou've already chosen the best response.0Now zw \(zw = i (1+i) = 1 i, zw = \sqrt 2, arg (zw) = 3\pi/4\) Hence \( log (zw) = \sqrt 2 + i (3\pi/4)\) The Imaginary parts are different and it is correct. But the real parts must be same. What is wrong ??

Loser66
 one year ago
Best ResponseYou've already chosen the best response.0oh, I got it. Mistake at log z = logz + i arg z it is not log z = z + i arg z
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