Logarithmic Integral -- From Wolfram MathWorld

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

			(1)
			(2)

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

			(5)
			(6)
			(7)
			(8)

(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 :

			(16)
			(17)

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