===== Natural logarithm of complex numbers =====
==== Set ====
| @#FFBB00: definiendum | @#FFBB00: $\mathrm{ln}:\mathbb C\setminus {]-\infty,0]} \to \mathbb R$ |
| @#55EE55: postulate   | @#55EE55: $\mathrm{ln}(z):=\mathrm{ln}|z|+i\,\arg(z)$ |

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>todo: [[Complex argument]]

== 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

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

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=== Requirements ===
[[Natural logarithm of real numbers]]