R: Exponential and Logarithmic Integral

expint {pracma} | R Documentation

## Exponential and Logarithmic Integral

### Description

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).

### Usage

    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.

### See Also

`gsl::expint_E1,expint_Ei`, `primes`

### Examples

    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)

[Package pracma version 2.0.4 Index]