Logarithmic Integral?

Question

li (x) = ∫ dt / ln t [x = 0, x]
Is there a formula of li (x) which is more readily to be input into a calculator?

Answer 1

You can calculate the logarithmic integral function from its series representation:

Li(x)
= γ + ln(|ln(x)|) + ln(x) + [ln(x)]² / (2 · 2!) + ... + [ln(x)]^n / (n · n!) + ....
= γ + ln(|ln(x)|) + Σ|n=1 → ∞| [ln(x)]^n / (n · n!)

or

Li(x) =
= γ + ln(|ln(x)|) + Σ|n=1 → ∞|{ (-1)^(n-1) · [ln(x)]^n / (n! · 2^(n-1)) · Σ|k=1 → (n-1)/2| 1/(2k+1) }

The later one converges more rapidly.

γ is the Euler Mascheroni constant
γ = 0.57721...


An alternative for large values of x is the asymptotic expansion:
Li(x) =
= x/ln(x) + Σ|n=0 → ∞|{ n! / [ln(x)]^n }·

Answer 2

Unfortunately, no.
li(x) is nonelementary and the only way
to get a grip on it is to use infinite series.
BTW, li(x) is important in the study of prime numbers.
It gives an estimate for the number of primes
less than or equal to x.

Answer 3

do your own homework