Difference between revisions of "Logarithm"
From specialfunctionswiki
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<strong>Proposition:</strong> $\displaystyle\int \log(z) dz = z \log(z)-z$ | <strong>Proposition:</strong> $\displaystyle\int \log(z) dz = z \log(z)-z$ | ||
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| + | <strong>Proof:</strong> █ | ||
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| + | <strong>Theorem:</strong> For $|z|<1$, | ||
| + | $$\log(1+z) = -\displaystyle\sum_{k=1}^{\infty} \dfrac{(-1)^k z^k}{k}.$$ | ||
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<strong>Proof:</strong> █ | <strong>Proof:</strong> █ | ||
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</div> | </div> | ||
Revision as of 06:50, 11 February 2015
The logarithm is defined by the formula $$\log(x) = \displaystyle\int_1^x \dfrac{1}{t} dt.$$
Properties
Proposition: $\displaystyle\int \log(z) dz = z \log(z)-z$
Proof: █
Theorem: For $|z|<1$, $$\log(1+z) = -\displaystyle\sum_{k=1}^{\infty} \dfrac{(-1)^k z^k}{k}.$$
Proof: █
