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Homelogarithmic integral

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# logarithmic integral

The European or Eulerian version of logarithmic integral (in Latin logarithmus integralis) is defined as

$\displaystyle\operatorname{Li}{x}:=\int_{2}^{x}\frac{dt}{\ln{t}},$ | (1) |

and the American version is

$\displaystyle\operatorname{li}{x}:=\int_{0}^{x}\frac{dt}{\ln{t}},$ | (2) |

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

$\operatorname{li}{x}=\lim_{{\varepsilon\to 0+}}\left(\int_{0}^{{1-\varepsilon}% }\!\frac{dt}{\ln{t}}+\int_{{1+\varepsilon}}^{x}\frac{dt}{\ln{t}}\right).$ |

The connection between (1) and (2) is

$\operatorname{Li}{x}=\operatorname{li}{x}-\operatorname{li}{2}.$ |

The logarithmic integral appears in some physical problems and in a formulation of the prime number theorem ($\operatorname{Li}{x}$ gives a slightly better approximation for the prime counting function than $\operatorname{li}{x}$).

One has the asymptotic series expansion

$\operatorname{Li}{x}=\frac{x}{\ln{x}}\sum_{{n=0}}^{\infty}\frac{n!}{(\ln{x})^{% n}}.$ |

The definition of the logarithmic integral may be extended to the whole complex plane, and one gets the analytic function $\operatorname{Li}{z}$ having the branch point $z=1$ and the derivative $\displaystyle\frac{1}{\log{z}}$.

## Mathematics Subject Classification

30E20*no label found*33E20

*no label found*26A36

*no label found*

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## Comments

## Pole or essential singularity?

Hi, I am unsure whether z = 1 is a pole or an essential singularity of the logarithmic integral. It is clear that z = 1 is a pole of order 1 of 1/ln{z}.

Jussi

## Re: Pole or essential singularity?

Neither --- it is a branch point. When one integrates a simple pole,

one genrates a logarithm. you can think of it this way. Write

1/ln{z} as 1/(z-1) + f(z) where f is analytic at z = 1 (in other

words, separate out the pole). Integrating, we find that li{z} =

log (z-1) + F(z), where F is the antiderivative of f, which makes it

analytic at z = 1.

## Re: Pole or essential singularity?

Thank you, Ray! Thus I must remove the attribute "meromorphic" from the entry.

Jussi