logarithmic integralLogarithmicIntegral22007-05-06 09:58:382008-06-04 15:32:17Definitioncomplex functionlogarithmic integrallogarithmus integralisEulerian logarithmic integral% this is the default PlanetMath preamble. as your knowledge
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The European or Eulerian version of {\em logarithmic integral} (in Latin {\em logarithmus integralis}) is defined as
\begin{align}
\Li{x} := \int_2^x\frac{dt}{\ln{t}},
\end{align}
and the American version is
\begin{align}
\li{x} := \int_0^x\frac{dt}{\ln{t}},
\end{align}
The integrand $\displaystyle\frac{1}{\ln{t}}$ has a singularity\, $t = 1$,\,
and for\, $x > 1$\, the latter definition is interpreted as
the Cauchy principal value
$$\li{x} =
\lim_{\varepsilon\to 0+}\left(\int_0^{1-\varepsilon}\!\frac{dt}{\ln{t}}
+\int_{1+\varepsilon}^x\frac{dt}{\ln{t}}\right).$$
The connection between (1) and (2) is
$$\Li{x} = \li{x}-\li{2}.$$
The logarithmic integral appears in some physical problems
and in a formulation of the prime number theorem ($\Li{x}$\, gives
a slightly better approximation for the prime counting function than\, $\li{x}$).
One has the asymptotic series expansion
$$\Li{x} = \frac{x}{\ln{x}}\sum_{n=0}^\infty\frac{n!}{(\ln{x})^n}.$$
The definition of the logarithmic integral may be extended to the whole
complex plane, and one gets the analytic function \, $\Li{z}$\, having
the branch point\, $z = 1$\, and the derivative \,$\displaystyle\frac{1}{\log{z}}$.