Question

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.

Answer 1

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

Answer 2

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

Answer 3

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 ??

Answer 4

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