## Natural logarithm of complex numbers

### Set

 definiendum $\mathrm{ln}:\mathbb C\setminus {]-\infty,0]} \to \mathbb R$ postulate $\mathrm{ln}(z):=\mathrm{ln}|z|+i\,\arg(z)$

##### Limits

$\lim_{x\to 0}x\ln(x)=0$

##### Differentiation and integrals

$\int \ln(x^n)\,{\mathrm d}x^n=\int \left(x^n\right)'\ln(x^n)\,{\mathrm d}x=x^n\left(\ln(x^n)-1\right)$

##### Series

At least around $z=0$ (I think for $|z|<1$)

$\ln{\left(\frac{1}{1-z}\right)} = \sum_{n=1}^\infty \frac{z^n}{n}$

or

$\ln{\left(1+z\right)} = -\sum_{n=1}^\infty \frac{(-z)^n}{n}$

From this series in $(-z)^n$, if we always constitutive positive and negative germs, we get

$\ln{\left(1+z\right)} = z - \sum_{n=1}^\infty \left(\dfrac{1}{2n}-\dfrac{z}{2n+1}\right) z^{2n}$

Note that this implies that $\exp$ maps the series on the right to $\frac{1}{1-z}=1+\sum_{n=1}^\infty z^n$.

The following is related to Minus twelve . Note:

$\left(\dfrac{z}{\ln(1+z)}\right)^n=1+\dfrac{n}{2}z+\dfrac{n}{2}\dfrac{3n-5}{12}z^2+\dfrac{n}{2}\dfrac{(n-2)(n-3)}{24}z^3+\dots$.

In particular, this tells us that

$\sum_{k=0}^\infty k\,z^k-\dfrac{1}{\ln(z)^2}=-\dfrac{1}{12}+{\mathcal O}\left((z-1)^1\right)$