Proving Logarithm Identity for Nonzero Complex Numbers

[bigplanet401]

Homework Statement

Show that, for any two nonzero complex numbers z_1 and z_2,
<br /> \text{Log } (z_1 z_2) = \text{Log } z_1 + \text{Log } z_2 + 2 N \pi i \, ,<br />

where N has one of the values -1, 0, 1.

Homework Equations

The logarithm on the principal branch is:
<br /> \begin{align*}<br /> &amp;\text{Log } z = \ln r + i \Theta \, ,\\<br /> \intertext{with}<br /> &amp;r &gt; 0 \text{ and } -\pi &lt; \Theta &lt; \pi \, .<br /> \end{align*}<br />

The Attempt at a Solution

I tried writing z_1 z_2 as exp(log(z_1) + log(z_2)) and taking the log that way, and I ended up getting the result above, but with N being allowed to take on any integer value. Note that
<br /> \log z = \ln |z| + i \arg z<br />

in general.

 

[AKG]

Use the equation you have for log under "attempt at a solution," because that's really the relevant equation. And don't express z₁z₂ as exp(log(z₁) + log(z₂)), express it in the form re^iq.

 

[StatusX]

The point with the N=1,0,-1 business is that when you add two numbers in (-pi,pi), you might not get a number in (-pi,pi), although you will get one in (-2pi,2pi). So, since adding an integer multiple of 2pi to the arg doesn't affect the result, you can either add or subtract 2pi and get back in the necessary (-pi,pi) range.