Logarithmic Integral Function

In Mathematics , the logarithmic integral function or '''integral logarithm''' li(''x'') is a Special Function . It occurs in problems of Physics and has Number Theoretic significance, occurring in the Prime Number Theorem as an Estimate of the number of Prime Number s less than a given value.

INTEGRAL REPRESENTATION

The logarithmic integral has an integral representation defined for all positive Real Number s

x
e 1

by the Definite Integral :

:

{
m li} (x) = \int_{0}^{x} rac{dt}{\ln (t)} \; .

Here, ln denotes the Natural Logarithm . The function 1/ln (''t'') has a Singularity at ''t'' = 1, and the integral for ''x'' > 1 has to be interpreted as a '' Cauchy Principal Value '':

:

{
m li} (x) = \lim_{arepsilon 	o 0} \left( \int_{0}^{1-arepsilon} rac{dt}{\ln (t)} + \int_{1+arepsilon}^{x} rac{dt}{\ln (t)} 
ight) \; .

OFFSET LOGARITHMIC INTEGRAL

The offset logarithmic integral or '''European logarithmic integral''' is defined as

:

{
m Li}(x) = {
m li}(x) - {
m li}(2)

or

:

{
m Li} (x) = \int_{2}^{x} rac{dt}{\ln t} \,

As such, the integral representation has the advantage of avoiding the singularity in the domain of integration.

SERIES REPRESENTATION

The function li(''x'') is related to the '' Exponential Integral '' Ei(''x'') via the equation

:

\hbox{li}(x)=\hbox{Ei}(\ln(x))

which is valid for

x > 1

. This identity provides a series representation of li(''x'') as

:

{
m li} (e^{u}) = \hbox{Ei}(u) = 
\gamma + \ln u + \sum_{n=1}^{\infty} {u^{n}\over n \cdot n!} 
\quad {
m for} \; u 
e 0 \; ,

where γ ≈ 0.57721 56649 01532 ... is the Euler-Mascheroni Gamma Constant .

SPECIAL VALUES

The function li(''x'') has a single positive zero; it occurs at ''x'' ≈ 1.45136 92348 ...; this number is known as the Ramanujan-Soldner Constant .

One has li(2) ≈ 1.04516 37801 17492 ...

ASYMPTOTIC EXPANSION

The asymptotic behavior for ''x'' → ∞ is

:

{
m li} (x) = \mathcal{O} \left( {x\over \ln (x)} 
ight) \; .

where

\mathcal{O}

refers to Big O Notation . The full Asymptotic Expansion is

:

{
m li} (x) = rac{x}{\ln x} \sum_{k=0}^{\infty} rac{k!}{(\ln x)^k}

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	CATEGORIES ABOUT LOGARITHMIC INTEGRAL FUNCTION

	special functions

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Information About

Logarithmic Integral Function