Difference between revisions of "Logarithmic integral"
From specialfunctionswiki
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The logarithmic integral is | The logarithmic integral is | ||
$$\mathrm{li}(x) = \displaystyle\int_0^x \dfrac{dt}{\log(t)},$$ | $$\mathrm{li}(x) = \displaystyle\int_0^x \dfrac{dt}{\log(t)},$$ | ||
| − | where $\log$ denotes the [[logarithm]]. | + | where $\log$ denotes the [[logarithm]]. |
| − | |||
<div align="center"> | <div align="center"> | ||
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=Properties= | =Properties= | ||
| − | + | {{:Relationship between logarithmic integral and exponential integral}} | |
{{:Prime number theorem, logarithmic integral}} | {{:Prime number theorem, logarithmic integral}} | ||
Revision as of 06:44, 5 April 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 $[0,6]$.
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.
