Definition:Logarithmic Integral

Definition

The logarithmic integral is defined as:

$\ds \map \li x = \PV_0^x \frac {\d t} {\ln t}$

where:

$\ln$ denotes the natural logarithm function

$\PV$ denotes the Cauchy principal value.

That is, as $\dfrac 1 {\ln t}$ has discontinuities at $t = 0$ and $t = 1$:

$\map \li x = \begin {cases} \ds \lim_{\varepsilon \mathop \to 0^+} \paren {\int_\varepsilon^x \frac {\rd t} {\ln t} } & 0 < x < 1 \\ \ds \lim_{\varepsilon \mathop \to 0^+} \paren {\int_\varepsilon^{1 - \varepsilon} \frac {\rd t} {\ln t} + \int_{1 + \varepsilon}^x \frac {\rd t} {\ln t} } & x > 1 \end {cases}$

Eulerian Logarithmic Interval

The Eulerian logarithmic integral is defined as:

$\ds \map \Li x = \int_2^x \frac {\d t} {\ln t}$

Also defined as

By defining the integrand of the logarithmic integral to be $0$ at $t = 0$, the lower limit can be taken in the first integral to be $0$.

Hence:

$\map \li x = \begin {cases} \ds \int_0^x \frac {\rd t} {\ln t} & : 0 \le x < 1 \\ \ds \lim_{\varepsilon \mathop \to 0^+} \paren {\int_0^{1 - \varepsilon} \frac {\rd t} {\ln t} + \int_{1 + \varepsilon}^x \frac {\rd t} {\ln t} } & : x > 1 \end {cases}$

Also known as

The logarithmic integral is also seen referred to as the integral logarithm.

Warning

The logarithmic integral and the Eulerian logarithmic integral are not consistently denoted in the literature (some sources use $\map \li x$ to indicate the Eulerian version, for example).

It is therefore important to take care which is being referred to at any point.

Also see

• Primitive of Reciprocal of Logarithm of x

• Results about the logarithmic integral can be found here.

Sources

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions: $5$. Exponential Integral and Related Functions: $5.1$ Exponential Integral: Definitions: $5.1.3$