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natural_logarithm_of_complex_numbers [2016/05/05 17:06] nikolaj |
natural_logarithm_of_complex_numbers [2016/07/22 15:08] nikolaj |
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| >todo: [[Complex argument]] | >todo: [[Complex argument]] | ||
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| + | == Limits == | ||
| + | $\lim_{x\to 0}x\ln(x)=0$ | ||
| == Differentiation and integrals == | == Differentiation and integrals == | ||
| - | $\int \left(x^n\right)'\ln(x^n)\,{\mathrm d}x=x^n\left(\ln(x^n)-1\right)$ | + | $\int \ln(x^n)\,{\mathrm d}x^n=\int \left(x^n\right)'\ln(x^n)\,{\mathrm d}x=x^n\left(\ln(x^n)-1\right)$ |
| == Series == | == Series == | ||
| + | At least around $z=0$ (I think for $|z|<1$) | ||
| + | |||
| $\ln{\left(\frac{1}{1-z}\right)} = \sum_{n=1}^\infty \frac{z^n}{n}$ | $\ln{\left(\frac{1}{1-z}\right)} = \sum_{n=1}^\infty \frac{z^n}{n}$ | ||
