Logarithmic integral function
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In mathematics, the logarithmic integral function 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 a Cauchy 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>
Sometimes instead of li the offset logarithmic integral is used, defined as <math>{\rm Li}(x) = {\rm li}(x) - {\rm li}(2)<math>. This is often used in number theoretic applications. Neither function should be confused with the logarithmic integral whose definition is
- <math>\int_{-\infty}^\infty \frac{M(t)}{1+t^2}dt<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 finds application in many areas, in particular it is used is in estimates of prime number densities, such as the prime number theorem:
- π(x) ~ li(x) ~ Li(x)
where π(x) denotes the number of primes smaller than or equal to 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.