Logarithmic integral function
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- See also logarithmic integral for other senses.
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 numbers less than a given value. Image:Logarithmic integral.svg
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Integral representation
The logarithmic integral has an integral representation defined for all positive real numbers <math>x\ne 1</math> 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>
Offset logarithmic integral
The offset logarithmic integral or European logarithmic integral is defined as
- <math>{\rm Li}(x) = {\rm li}(x) - {\rm li}(2)</math>
or
- <math> {\rm Li} (x) = \int_{2}^{x} \frac{dt}{\ln t} \, </math>
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
- <math>\hbox{li}(x)=\hbox{Ei}(\ln(x))</math>
which is valid for <math>x > 1</math>. This identity provides a series representation of li(x) as
- <math> {\rm li} (e^{u}) = \hbox{Ei}(u) =
\gamma + \ln u + \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.
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
- <math> {\rm li} (x) = \mathcal{O} \left( {x\over \ln (x)} \right) \; . </math>
where <math>\mathcal{O}</math> refers to big O notation. The full asymptotic expansion is
- <math> {\rm li} (x) = \frac{x}{\ln x} \sum_{k=0}^{\infty} \frac{k!}{(\ln x)^k} </math>
or
- <math> \frac{{\rm li} (x)}{x/\ln x} = 1 + \frac{1}{\ln x} + \frac{2}{(\ln x)^2} + \cdots. </math>
Note that, as an asymptotic expansion, this series is not convergent: it is a reasonable approximation only if the series is truncated at a finite number of terms, and only large values of x are employed. This expansion follows directly from the asymptotic expansion for the exponential integral.
Number theoretic significance
The logarithmic integral is important in number theory, appearing in estimates of the number of prime numbers less than a given value. For example, the prime number theorem states that:
- <math>\pi(x)\sim\hbox{li}(x)\sim\hbox{Li}(x)</math>
where π(x) denotes the number of primes smaller than or equal to x.
See also
References
- Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. (See Chapter 5)fr:Logarithme intégral