Difference between revisions of "Logarithmic integral"

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The logarithmic integral is
 
The logarithmic integral is
$$\mathrm{li}(x) = \displaystyle\int_0^x \dfrac{dt}{\log(t)},$$
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$$\mathrm{li}(x) = \displaystyle\int_0^x \dfrac{1}{\log(t)} \mathrm{d}t,$$
where $\log$ denotes the [[logarithm]]. The logarithmic integral is related to the [[exponential integral]] by the formula
+
where $\log$ denotes the [[logarithm]].  
$$\mathrm{li}(x)=\mathrm{Ei}( \log(x)).$$
 
  
[[File:Logarithmicintegral.png|500px]]
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<div align="center">
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<gallery>
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File:Liplot.png|Graph of $\mathrm{li}$.
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File:Complexliplot.png|[[Domain coloring]] of $\mathrm{li}$.
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</gallery>
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</div>
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=Properties=
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[[Relationship between logarithmic integral and exponential integral]]<br />
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[[Prime number theorem, logarithmic integral]]<br />
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=See Also=
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[[Prime counting function]] <br />
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=References=
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* {{PaperReference|On certain definite integrals involving the exponential-integral|1881|James Whitbread Lee Glaisher|prev=Exponential integral Ei series|next=Relationship between logarithmic integral and exponential integral}}
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* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Exponential integral Ei|next=Exponential integral E}}: $5.1.3$
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{{:Logarithm and friends footer}}
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{{:*-integral functions footer}}
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[[Category:SpecialFunction]]

Latest revision as of 03:33, 17 March 2018

The logarithmic integral is $$\mathrm{li}(x) = \displaystyle\int_0^x \dfrac{1}{\log(t)} \mathrm{d}t,$$ where $\log$ denotes the logarithm.

Properties

Relationship between logarithmic integral and exponential integral
Prime number theorem, logarithmic integral

See Also

Prime counting function

References

Logarithm and friends
$\ast$-integral functions