Encyclopedia > Logarithmic integral

  Article Content

Logarithmic integral

In some 'esoteric' areas of mathematics, the logarithmic integral or integral logarithm li(x) is a non-elementary function defined for all positive real numbers x≠ 1 by the definite integral:

<math> {\rm li} (x) = \int_{0}^{x} \frac{dt}{\ln (t)} \; . </math>

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 Cauchy's principal value:

<math> {\rm li} (x) = \lim_{\varepsilon \to 0} \left( \int_{0}^{1-\varepsilon} \frac{dt}{\ln (t)} + \int_{1+\varepsilon}^{x} \frac{dt}{\ln (t)} \right) \; . </math>

The growth behavior of this function for x → ∞ is

<math> {\rm li} (x) = \Theta \left( {x\over \ln (x)} \right) \; . </math>

(see big O notation).

The logarithmic integral is mainly important because it occurs in estimates of prime number densities, especially in the prime number theorem:

π(x) ~ Li(x)

where π(x) denotes a multiplicative function - the number of primes smaller than or equal to x, and Li(x) is the offset logarithmic integral function, related to li(x) by Li(x) = li(x) - li(2).

The offset logarithmic integral gives a slightly better estimate to the π function than li(x). The function li(x) is related to the exponential integral[?] Ei(x) via the equation

li(x) = Ei (ln (x))    for all positive real x ≠ 1.

This leads to series expansions of li(x), for instance:

<math> {\rm li} (e^{u}) = \gamma + \ln \left| (u) \right| + \sum_{n=1}^{\infty} {u^{n}\over n \cdot n!} \quad {\rm for} \; u \ne 0 \; , </math>

where γ ≈ 0.57721 56649 01532 ... is the Euler-Mascheroni gamma constant. 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.



All Wikipedia text is available under the terms of the GNU Free Documentation License

 
  Search Encyclopedia

Search over one million articles, find something about almost anything!
 
 
  
  Featured Article
Brazil

... problem. Demographics Main article: Demographics of Brazil Four major groups make up the Brazilian population: the Portuguese, the original colonisers; ...

 
 
 
This page was created in 36 ms