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

Logarithm and friends

Dilogarithm

Log integral

Logarithm

$\ast$-integral functions

Cosine integral

Log integral

Sinh integral

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 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]]. The logarithmic integral is related to the [[exponential integral]] by the formula
+where $\log$ denotes the [[logarithm]].
-$$\mathrm{li}(x)=\mathrm{Ei}( \log(x)).$$
+<div align="center">
+<gallery>
+File:Liplot.png|Graph of $\mathrm{li}$.
+File:Complexliplot.png|[[Domain coloring]] of $\mathrm{li}$.
+</gallery>
+</div>
+=Properties=
+[[Relationship between logarithmic integral and exponential integral]]<br />
+[[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]]