Any two, non zero, complex numbers.

Quote:

Originally Posted by universalsandbox

Log is the principal branch of the log function.
If two non-zero complex numbers $z_{1},z_{2}$
Then show the following:
Log( $z_{1}z_{2}$ ) = Log( $z_{1}$ ) + Log( $z_{2}$ ) - 2N*pi*i, for N = 0 or N = 1

Thanks. Your help is appreciated.

Let $z_1 = r_1e^{i\theta_1},z_2=r_2e^{i\theta_2}$ where $\theta_1,\theta_2 \in (-\pi,\pi]$ .
Then, $z_1z_2 = r_1r_2 e^{i(\theta_1+\theta_2)}$

Therefore, $\arg (z_1z_2) = \theta_1 + \theta_2 + 2\pi N$ where $N =0\text{ or }-1 \text{ or }1$ .

But, $\log z_1 = \ln r_1 + i \theta_1 \text{ and }\log_2 = \ln z_2 + i \theta_2$ .
And, $\log z_1z_2 = \ln r_1r_2 + i (\theta_1+\theta_2) + 2i\pi N$

From here it follows that $\log z_1z_2 = \log z_1 + \log z_2 + 2\pi i N$