Difference between revisions of "Logarithmic integral"

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The logarithmic integral is
 
The logarithmic integral is
$$\mathrm{li}(x) = \displaystyle\int_0^x \dfrac{dt}{\log(t)},$$
+
$$\mathrm{li}(x) = \displaystyle\int_0^x \dfrac{1}{\log(t)} \mathrm{d}t,$$
 
where $\log$ denotes the [[logarithm]].  
 
where $\log$ denotes the [[logarithm]].  
  
 
<div align="center">
 
<div align="center">
 
<gallery>
 
<gallery>
File:Logarithmicintegral.png|Graph of $\mathrm{li}$ on $[0,6]$.
+
File:Liplot.png|Graph of $\mathrm{li}$.
File:Domain coloring of log integral.png|[[Domain coloring]] of [[analytic continuation]] of $\mathrm{li}$.
+
File:Complexliplot.png|[[Domain coloring]] of $\mathrm{li}$.
 
</gallery>
 
</gallery>
 
</div>
 
</div>
  
 
=Properties=
 
=Properties=
{{:Relationship between logarithmic integral and exponential integral}}
+
[[Relationship between logarithmic integral and exponential integral]]<br />
{{:Prime number theorem, logarithmic integral}}
+
[[Prime number theorem, logarithmic integral]]<br />
 +
 
 +
=See Also=
 +
[[Prime counting function]] <br />
 +
 
 +
=References=
 +
* {{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}}
 +
* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Exponential integral Ei|next=Exponential integral E}}: $5.1.3$
 +
 
 +
{{:Logarithm and friends footer}}
 +
{{:*-integral functions footer}}
 +
 
 +
 
 +
[[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