LogIntegral

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Values at fixed points

Values at infinities

General characteristics

Domain and analyticity

li(z) is an analytical function of z which is defined over the whole complex z-plane.

Symmetries and periodicities

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

The function li(z) does not have poles and essential singularities.

Branch points

Branch cuts

Series representations

Generalized power series

Expansions at generic point z==(Subscript[z, 0])

Expansions at z==1

Eric Weisstein

Expansions on branch cuts

Pavlyk 0. (2006)

Expansions at Log[z]==0

Asymptotic series expansions

Residue representations

Other series representations

Integral representations

On the real axis

Of the direct function

Contour integral representations

Differential equations

Ordinary nonlinear differential equations

Complex characteristics

Real part

Imaginary part

Absolute value

Argument

Conjugate value

Signum value

Differentiation

Low-order differentiation

Symbolic differentiation

Fractional integro-differentiation

Integration

Indefinite integration

Involving only one direct function

Involving one direct function and elementary functions

Definite integration

Involving the direct function

Representations through more general functions

Through hypergeometric functions

Involving HypergeometricPFQ

Through Meijer G

Classical cases for the direct function itself

Through other functions

Representations through equivalent functions

With related functions

Theorems

The prime number theorem

({{π(x)(Underscript[∼,  x->∞])li(x)+(∑_(k=2)^n)((μ(k))/k)li((x^(1/k))).      , }})

History

– L. Euler (1768)
– L. Mascheroni (1790)
– T. Caluso (1805)
– J. von Soldner (1809) introduced the notation li
–  C. A. Bretschneider (1837)
– A.de Morgan (1842)
–  H. Amstein (1895) introduced branch cut for complex argument

Copyright