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.

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