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.
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[edit] Integral representation
The logarithmic integral has an integral representation defined for all positive real numbers by the definite integral:
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:
[edit] Offset logarithmic integral
The offset logarithmic integral or Eulerean logarithmic integral is defined as
- Li(x) = li(x) − li(2)
or
As such, the integral representation has the advantage of avoiding the singularity in the domain of integration.
[edit] Series representation
The function li(x) is related to the exponential integral Ei(x) via the equation
- li(x) = Ei(ln(x))
which is valid for x > 1. This identity provides a series representation of li(x) as
where γ ≈ 0.57721 56649 01532 ... is the Euler-Mascheroni gamma constant.
[edit] 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 ...
[edit] Asymptotic expansion
The asymptotic behavior for x → ∞ is
where refers to big O notation. The full asymptotic expansion is
or
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.
[edit] 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:
where π(x) denotes the number of primes smaller than or equal to x.
[edit] See also
[edit] 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)