The logarithmic integral (in the "American" convention; Abramowitz and Stegun 1972; Edwards 2001, p. 26), is defined for real as
Here, PV denotes Cauchy principal value of the integral, and the function has a singularity
at .
The logarithmic integral defined in this way is implemented in the Wolfram
Language as LogIntegral [x ].
There is a unique positive number
(3)
(OEIS A070769 ; Derbyshire 2004, p. 114) known as Soldner's constant for which , so the logarithmic integral can also be written as
(4)
for .
Special values include
(OEIS A069284 ), where is Soldner's constant
(Edwards 2001, p. 34).
The definition can also be extended to the complex plane ,
as illustrated above.
Its derivative is
(9)
and its indefinite integral is
(10)
where
is the exponential integral . It also has
the definite integral
(11)
where
(OEIS A002162 ) is the natural
logarithm of 2 .
The logarithmic integral obeys
(12)
where
is the exponential integral , as well as the
identity
(13)
(Bromwich and MacRobert 1991, p. 334; Hardy 1999, p. 25).
Nielsen showed and Ramanujan independently discovered that
(14)
where
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
(15)
where
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 :
Note that the notation is (confusingly) also used for the polylogarithm
and also for the "American" definition of (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/
Explore with Wolfram|Alpha
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 A002162 4074,
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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