The Exponential Integral and Variants

Computes the exponential integral \(Ei(x)\) for real values, as well as \(\exp(-x) \times Ei(x)\) and \(E_1(x)\) and their derivatives (up to the 3rd derivative).

expint(x, deriv = 0)
expexpint(x, deriv = 0)
expint.E1(x, deriv = 0)

Arguments

x

Numeric. Ideally a vector of positive reals.

deriv

Integer. Either 0, 1, 2 or 3.

Details

The exponential integral \(Ei(x)\) function is the integral of \(\exp(t) / t\) from 0 to \(x\), for positive real \(x\). The function \(E_1(x)\) is the integral of \(\exp(-t) / t\) from \(x\) to infinity, for positive real \(x\).

Value

Function expint(x, deriv = n) returns the \(n\)th derivative of \(Ei(x)\) (up to the 3rd), function expexpint(x, deriv = n) returns the \(n\)th derivative of \(\exp(-x) \times Ei(x)\) (up to the 3rd), function expint.E1(x, deriv = n) returns the \(n\)th derivative of \(E_1(x)\) (up to the 3rd).

References

https://netlib.org/specfun/ei.

Author

T. W. Yee has simply written a small wrapper function to call the NETLIB FORTRAN code. Xiangjie Xue modified the functions to calculate derivatives. Higher derivatives can actually be calculated—please let me know if you need it.

Warning

These functions have not been tested thoroughly.

See also

log, exp. There is also a package called expint.

Examples

 if (FALSE) { # \dontrun{
par(mfrow = c(2, 2))
curve(expint, 0.01, 2, xlim = c(0, 2), ylim = c(-3, 5),
      las = 1, col = "orange")
abline(v = (-3):5, h = (-4):5, lwd = 2, lty = "dotted", col = "gray")
abline(h = 0, v = 0, lty = "dashed", col = "blue")

curve(expexpint, 0.01, 2, xlim = c(0, 2), ylim = c(-3, 2),
      las = 1, col = "orange")
abline(v = (-3):2, h = (-4):5, lwd = 2, lty = "dotted", col = "gray")
abline(h = 0, v = 0, lty = "dashed", col = "blue")

curve(expint.E1, 0.01, 2, xlim = c(0, 2), ylim = c(0, 5),
      las = 1, col = "orange")
abline(v = (-3):2, h = (-4):5, lwd = 2, lty = "dotted", col = "gray")
abline(h = 0, v = 0, lty = "dashed", col = "blue")
} # }