Logarithmic integral function
The logarithmic integral is an analytical function of the real numbers x ≥ 0, x ≠ 1 (or x > 1) in the real numbers.
Definition
Two definitions common that differ by a constant. For one of the most important applications - as asymptotic comparison variable for the prime number function in the prime number theorem - makes the difference between the two definitions do not matter.
A definition in the field is
It must be because of the singularity at be defined for a threshold value ( Cauchy principal value):
A different definition for
When there is not a pole but a logarithmic singularity.
Properties
Some values :
It shall be the Integralexponentialfunktion, it obtains the series representation
Where ( sequence A001620 in OEIS ) is the Euler - Mascheroni constant.
From the definition of is obtained by linear substitution
Must be being used for because of the singularity at the Cauchy principal value. Furthermore, we have for
Moreover, for
For we obtain In the limit is
Another formula is
The Golomb - Dickman constant ( sequence A084945 in OEIS ) occurs in the theory of random permutations in estimating the length of the longest Zykels a permutation and in number theory in estimating the size of the largest prime factor of a number.