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Logarithmic Integral

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LogIntegral

The logarithmic integral (in the "American" convention; Abramowitz and Stegun 1972; Edwards 2001, p. 26), is defined for real x as

li(x)={int_0^x(dt)/(lnt) for 0<x<1; PVint_0^x(dt)/(lnt) for x>1
(1)
={int_0^x(dt)/(lnt) for 0<x<1; lim_(epsilon->0^+)[int_0^(1-epsilon)(dt)/(lnt)+int_(1+epsilon)^x(dt)/(lnt)] for x>1
(2)

Here, PV denotes Cauchy principal value of the integral, and the function has a singularity at x=1.

The logarithmic integral defined in this way is implemented in the Wolfram Language as LogIntegral[x].

There is a unique positive number

mu=1.4513692348...
(3)

(OEIS A070769; Derbyshire 2004, p. 114) known as Soldner's constant for which li(x)=0, so the logarithmic integral can also be written as

li(x)=int_mu^x(dt)/(lnt)
(4)

for x>mu.

Special values include

li(0)=0
(5)
li(1)=-infty
(6)
li(mu)=0
(7)
li(2)=1.0451637801174...,
(8)

(OEIS A069284), where mu is Soldner's constant (Edwards 2001, p. 34).

LogIntegralReImAbs
Min Max
Re
Im Powered by webMathematica

The definition can also be extended to the complex plane, as illustrated above.

Its derivative is

(dli(z))/(dz)=1/(lnz),
(9)

and its indefinite integral is

intli(z)dz=zli(z)-Ei(2lnz),
(10)

where Ei(z) is the exponential integral. It also has the definite integral

int_0^1li(z)dz=-ln2,
(11)

where ln2=0.69314... (OEIS A002162) is the natural logarithm of 2.

The logarithmic integral obeys

li(z)=Ei(lnz),
(12)

where Ei(z) is the exponential integral, as well as the identity

li(z^(1/m))=gamma+lnlnz-lnm+sum_(n=1)^infty((lnz)^n)/(n·n!m^n)
(13)

(Bromwich and MacRobert 1991, p. 334; Hardy 1999, p. 25).

Nielsen showed and Ramanujan independently discovered that

li(x)=gamma+lnlnx+sum_(k=1)^infty((lnx)^k)/(k!k),
(14)

where gamma is the Euler-Mascheroni constant (Nielsen 1965, pp. 3 and 11; Berndt 1994; Finch 2003; Havil 2003, p. 106). Another formula due to Ramanujan which converges more rapidly is

li(x)=gamma+lnlnx+sqrt(x)sum_(n=1)^infty((-1)^(n-1)(lnx)^n)/(n!2^(n-1))sum_(k=0)^(|_(n-1)/2_|)1/(2k+1),
(15)

where |_x_| is the floor function (Berndt 1994).

The form of this function appearing in the prime number theorem (used for example by Landau as well as Havil 2003, pp. 105 and 175) and sometimes referred to as the "European" definition (Derbyshire 2004, p. 373) is defined so that Li(2)=0:

Li(x)=int_2^x(du)/(lnu)
(16)
=li(x)-li(2).
(17)

Note that the notation Li_n(z) is (confusingly) also used for the polylogarithm and also for the "American" definition of li(x) (Edwards 2001, p. 26).

See also

Polylogarithm, Prime Constellation, Prime Counting Function, Prime Number Theorem, Skewes Number

Related Wolfram sites

http://functions.wolfram.com/GammaBetaErf/LogIntegral/

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References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 879, 1972.Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 126-131, 1994.Bromwich, T. J. I'A. and MacRobert, T. M. An Introduction to the Theory of Infinite Series, 3rd ed. New York: Chelsea, p. 334, 1991.de Morgan, A. The Differential and Integral Calculus, Containing Differentiation, Integration, Development, Series, Differential Equations, Differences, Summation, Equations of Differences, Calculus of Variations, Definite Integrals--With Applications to Algebra, Plane Geometry, Solid Geometry, and Mechanics. London: Robert Baldwin, p. 662, 1839.Derbyshire, J. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. New York: Penguin, pp. 114-117 and 373, 2004.Edwards, H. M. Riemann's Zeta Function. New York: Dover, 2001.Finch, S. R. "Euler-Gompertz Constant." §6.2 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 423-428, 2003.Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999.Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 105-106 and 175-176, 2003.Koosis, P. The Logarithmic Integral I. Cambridge, England: Cambridge University Press, 1998.Nielsen, N. "Theorie des Integrallograrithmus und Verwandter Transzendenten." Part II in Die Gammafunktion. New York: Chelsea, 1965.Vardi, I. Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, p. 151, 1991.Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, p. 45, 1999.Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 39, 1983.Sloane, N. J. A. Sequences A0021624074, A069284 and A070769 in "The On-Line Encyclopedia of Integer Sequences."Soldner. Abhandlungen 2, 333, 1812.

Referenced on Wolfram|Alpha

Logarithmic Integral

Cite this as:

Weisstein, Eric W. "Logarithmic Integral." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LogarithmicIntegral.html

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