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^∞ \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_{-∞}^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 π
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 π(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)
x |
vector of real or complex numbers. |
For x in [-38, 2] we use a series expansion,
otherwise a continued fraction, see the references below, chapter 5.
Returns a vector of real or complex numbers, the vectorized exponential integral, resp. the logarithmic integral.
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”.
Abramowitz, M., and I.A. Stegun (1965). Handbook of Mathematical Functions. Dover Publications, New York.
gsl::expint_E1,expint_Ei, primes
expint_E1(1:10) # 0.2193839 0.0489005 0.0130484 0.0037794 0.0011483 # 0.0003601 0.0001155 0.0000377 0.0000124 0.0000042 expint_Ei(1:10) ## Not run: estimPi <- function(n) round(Re(li(n) - li(2))) # estimated number of primes primesPi <- function(n) length(primes(n)) # true number of primes <= n N <- 1e6 (estimPi(N) - primesPi(N)) / estimPi(N) # deviation is 0.16 percent! ## End(Not run)
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