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Series Complex Analytic Derivative Connection Approximation Prime Counting Function Prime Number Theorem Ramify Cauchy Principle Part 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

, and for

the latter definition is interpreted as the Cauchy principal value

$\displaystyle \operatorname{li}{x} = \lim_{\varepsilon\to 0+}\left(\int_0^{1-\varepsilon}\!\frac{dt}{\ln{t}} +\int_{1+\varepsilon}^x\frac{dt}{\ln{t}}\right).$

The connection between (1) and (2) is

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

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

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

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 and the derivative $\displaystyle\frac{1}{\log{z}}$ .

1	Complex
1	Series
2	Derivative
2	Analytic
3	Connection
3	Approximation
4	Ramify
5	Prime Counting Function
5	Prime Number Theorem
11	Cauchy Principle Part Integral