| Version 7 |
Version 6 |
| \PMlinkescapeword{expansion} |
\PMlinkescapeword{expansion} |
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| The European or Eulerian version of {\em logarithmic integral} (in Latin {\em logarithmus integralis}) is defined as |
The European or Eulerian version of {\em logarithmic integral} (in Latin {\em logarithmus integralis}) is defined as |
| \begin{align} |
\begin{align} |
| \Li{x} := \int_2^x\frac{dt}{\ln{t}}, |
\Li{x} := \int_2^x\frac{dt}{\ln{t}}, |
| \end{align} |
\end{align} |
| and the American version is |
and the American version is |
| \begin{align} |
\begin{align} |
| \li{x} := \int_0^x\frac{dt}{\ln{t}}, |
\li{x} := \int_0^x\frac{dt}{\ln{t}}, |
| \end{align} |
\end{align} |
| The integrand $\displaystyle\frac{1}{\ln{t}}$ has a singularity\, $t = 1$,\, |
The integrand $\displaystyle\frac{1}{\ln{t}}$ has a singularity\, $t = 1$,\, |
| and for\, $x > 1$\, the latter definition is interpreted as |
and for\, $x > 1$\, the latter definition is interpreted as |
| the Cauchy principal value |
the Cauchy principal value |
| $$\li{x} = |
$$\li{x} = |
| \lim_{\varepsilon\to 0+}\left(\int_0^{1-\varepsilon}\!\frac{dt}{\ln{t}} |
\lim_{\varepsilon\to 0+}\left(\int_0^{1-\varepsilon}\!\frac{dt}{\ln{t}} |
| +\int_{1+\varepsilon}^x\frac{dt}{\ln{t}}\right).$$ |
+\int_{1+\varepsilon}^x\frac{dt}{\ln{t}}\right).$$ |
| The connection between (1) and (2) is |
The connection between (1) and (2) is |
| $$\Li{x} = \li{x}-\li{2}.$$ |
$$\Li{x} = \li{x}-\li{2}.$$ |
| The logarithmic integral appears is some physical problems |
The logarithmic integral appears is some physical problems |
| and in a formulation of the prime number theorem ($\Li{x}$\, gives |
and in a formulation of the prime number theorem ($\Li{x}$\, gives |
| a slightly better approximation for the prime counting function than\, $\li{x}$). |
a slightly better approximation for the prime counting function than\, $\li{x}$). |
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| One has the asymptotic series expansion |
One has the asymptotic series expansion |
| $$\Li{x} = \frac{x}{\ln{x}}\sum_{n=0}^\infty\frac{n!}{(\ln{x})^n}.$$ |
$$\Li{x} = \frac{x}{\ln{x}}\sum_{n=0}^\infty\frac{n!}{(\ln{x})^n}.$$ |
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| The definition of the logarithmic integral may be extended to the whole |
The definition of the logarithmic integral may be extended to the whole |
| complex plane, and one gets the meromorphic function\, $\Li{z}$\, having |
complex plane, and one gets the meromorphic function\, $\Li{z}$\, having |
| the branch point\, $z = 1$\, and the derivative \,$\displaystyle\frac{1}{\log{z}}$. |
the branch point\, $z = 1$\, and the derivative \,$\displaystyle\frac{1}{\log{z}}$. |
