Logarithmic Integral Calculator

Logarithmic Integral Calculator
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The Logarithmic Integral function li(x) is defined by the integral

li(x) = x0  dt/ln(t)

The antiderivative of the function f(t) = 1/ln(t) cannot be written explicitly in terms of elementary functions, thus the logarithmic integral function cannot be evaluated in terms of other elementary functions such as ln(x) or ex. Because the function 1/ln(t) has a singularity at t = 0, an alternative form of the logarithmic integral is sometimes used:

Li(x) = x2  dt/ln(t)  =  li(x) - li(2)

The logarithmic integral is important in number theory since the number of primes less than or equal to n is roughly Li(n).

Other Properties of li(x)

The logarithmic integral can be expressed as an infinite series:

li(x) = γ + ln(ln(x)) + Σ n=1   ln(x)n / (n⋅n!)

where γ is the Euler-Mascheroni constant 0.57721566490153286...

For large values of x, li(x) can be evaluated with the asymptotic formula

li(x) ∼ x/ln(x) + x/ln(x)² + 2x/ln(x)³ + 6x/ln(x)⁴ + 24x/ln(x)⁵

At x = 1.45136923488338105... the function li(x) is equal to 0.

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