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An introduction to Complex numbers

May 3, 2020 Mouctar Algebra 0

Logarithm of a complex number

The polar notation works well with the logarithm of complex numbers.
We have already seen that for a complex $z$

$z=|z|e^{i\arg(z)}$

With this notation we can find $log(z)$ by:

$log(z)=log|z|+i\arg(z)$

The books will desing the real part by $Log|z|$

Now we have the final notation

$\log(z)=Log|z|+i\arg(z)$

To find the solution to these equations we will remember to use the $i2\pi k$ added to $i\arg(z)$ .

The rest of the logarithm technics seen before will apply here:

$\log(z_1z_2)=\log(z_1)+\log(z_2)$

We should remember that there is a $modulo$ with a value of $2\pi i$

$\log(z_1/z_2)=\log(z_1)-\log(z_2)$

Example 1:

Find $\log(1+i)$

$z=1+i$
$|z|=\sqrt{2}$
$\arg(z)=\frac{\pi}{4}$

$z=\sqrt{2}e^{i\frac{\pi}{4}}$

$\log(z)=Log \sqrt{2}+i(\frac{\pi}{4}+2k\pi)$

Finally:

$\log(1+i)=Log \sqrt{2}+i(\frac{\pi}{4}+2k\pi)$

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