Exponential and Logarithmic Integral

The exponential integral functions E1 and Ei and the logarithmic integral Li.

The exponential integral is defined for $x > 0$ as $$\int_x^\infty \frac{e^{-t}}{t} dt$$ and by analytic continuation in the complex plane. It can also be defined as the Cauchy principal value of the integral $$\int_{-\infty}^x \frac{e^t}{t} dt$$ This is denoted as $Ei(x)$ and the relationship between Ei and expint(x) for x real, x > 0 is as follows: $$Ei(x) = - E1(-x) -i \pi$$

The logarithmic integral $li(x)$ for real $x, x > 0$, is defined as $$li(x) = \int_0^x \frac{dt}{log(t)}$$ and the Eulerian logarithmic integral as $Li(x) = li(x) - li(2)$.

The integral $Li$ approximates the prime number function $\pi(n)$, i.e., the number of primes below or equal to n (see the examples).

expint(x)
expint_E1(x)

expint_Ei(x)
li(x)

Arguments

x: vector of real or complex numbers.

Details

For x in [-38, 2] we use a series expansion, otherwise a continued fraction, see the references below, chapter 5.

Value

Returns a vector of real or complex numbers, the vectorized exponential integral, resp. the logarithmic integral.

Note

The logarithmic integral li(10^i)-li(2) is an approximation of the number of primes below 10^i, i.e., Pi(10^i), see “?primes”.

References

Abramowitz, M., and I.A. Stegun (1965). Handbook of Mathematical Functions. Dover Publications, New York.