- #1

bigplanet401

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## Homework Statement

Show that, for any two nonzero complex numbers z_1 and z_2,

[tex]

\text{Log } (z_1 z_2) = \text{Log } z_1 + \text{Log } z_2 + 2 N \pi i \, ,

[/tex]

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

## Homework Equations

The logarithm on the principal branch is:

[tex]

\begin{align*}

&\text{Log } z = \ln r + i \Theta \, ,\\

\intertext{with}

&r > 0 \text{ and } -\pi < \Theta < \pi \, .

\end{align*}

[/tex]

## 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

[tex]

\log z = \ln |z| + i \arg z

[/tex]

in general.

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