Difference between revisions of "Logarithmic integral"
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File:Logarithmicintegral.png|Graph of $\mathrm{li}$ on $[0,6]$. | File:Logarithmicintegral.png|Graph of $\mathrm{li}$ on $[0,6]$. | ||
| + | File:Domain coloring of log integral.png|[[Domain coloring]] of [[analytic continuation]] of $\mathrm{li}$. | ||
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Revision as of 19:08, 25 July 2015
The logarithmic integral is $$\mathrm{li}(x) = \displaystyle\int_0^x \dfrac{dt}{\log(t)},$$ where $\log$ denotes the logarithm.
- Logarithmicintegral.png
Graph of $\mathrm{li}$ on $[0,6]$.
Domain coloring of analytic continuation of $\mathrm{li}$.
Properties
Theorem
The following formula holds: $$\mathrm{li}(x)=\mathrm{Ei}( \log(x)),$$ where $\mathrm{li}$ denotes the logarithmic integral, $\mathrm{Ei}$ denotes the exponential integral Ei, and $\log$ denotes the logarithm.
Proof
References
- James Whitbread Lee Glaisher: On certain definite integrals involving the exponential-integral (1881)... (previous)... (next) (note: expresses this relationship as $\mathrm{Ei}(x)=\mathrm{li}(e^x)$)
Theorem
The following formula holds: $$\lim_{x \rightarrow \infty} \dfrac{\pi(x)}{\mathrm{li}(x)}=1,$$ where $\pi$ denotes the prime counting function and $\mathrm{li}$ denotes the logarithmic integral.
