## 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.

1. Loser66

Let $$z =i , |z |= 1, arg z = pi/2\\ log z = 1+ iarg (z) = 1 + i(pi/2)$$ Let $$w = -1+i , |w| = \sqrt2 , arg w= 3\pi/4\\w = \sqrt2 + i(3\pi/4)$$

2. Loser66

Hence $$logz + log w = 1 +\sqrt2 + i(\pi/2+3\pi/4) = 1+\sqrt2 + i (5\pi/4)$$ $$Real (zw) = 1 + \sqrt2$$

3. Loser66

Now 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 ??

4. Loser66

oh, I got it. Mistake at log z = log|z| + i arg z it is not log z = |z| + i arg z