Logarithm

From specialfunctionswiki
Revision as of 06:36, 5 April 2015 by Tom (talk | contribs)
Jump to: navigation, search

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:

Theorem

The following formula holds for $x>0$: $$\mathrm{Ei}(x) = \gamma + \log x + \displaystyle\sum_{k=1}^{\infty} \dfrac{x^k}{kk!},$$ where $\mathrm{Ei}$ denotes the exponential integral Ei, $\log$ denotes the logarithm, and $\gamma$ denotes the Euler-Mascheroni constant.

Proof

References

Theorem

The function $\pi(x)$ obeys the formula $$\lim_{x \rightarrow \infty} \dfrac{\pi(x)}{\frac{x}{\log(x)}}=1,$$ where $\pi$ denotes the prime counting function and $\log$ denotes the logarithm.

Proof

References