Logarithmic integral

The logarithmic integral is $$\mathrm{li}(x) = \displaystyle\int_0^x \dfrac{1}{\log(t)} \mathrm{d}t,$$ where $\log$ denotes the logarithm.

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  Graph of $\mathrm{li}$.

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  Domain coloring of $\mathrm{li}$.

Properties

Relationship between logarithmic integral and exponential integral
Prime number theorem, logarithmic integral

See Also

Prime counting function

References

• James Whitbread Lee Glaisher: On certain definite integrals involving the exponential-integral (1881)... (previous)... (next)
• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $5.1.3$

Logarithm and friends

Dilogarithm

Log integral

Logarithm

Polylogarithm

$\ast$-integral functions

Cosine integral

Cosh integral

Exp integral $E_n$

Exp integral $\mathrm{Ei}$

Fresnel $C$

Fresnel $S$

Gudermannian

Inverse $\mathrm{gd}$

Sine integral

Log integral

Sinh integral