Exponential integrals

• sc_exp1(): \(E_1(z) = \int_1^\infty e^{-zt} / t \, dt\), complex, with a branch cut on the negative real axis.

• sc_expi(): \(\mathrm{Ei}(x)\), complex.

• sc_expn(): the generalised \(E_n(x)\) for whole n.

Usage

sc_exp1(z)

sc_expi(x)

sc_expn(n, x)

Arguments

z

Numeric or complex vector.

x

Numeric vector; in sc_expi(), also complex.

n

Whole numbers within integer range. NaN gives NaN, as in SciPy. A non-whole n is an error: SciPy truncates it with a warning, and zuspecial refuses it so that a fractional order cannot slip through.

Value

A double vector, or a complex vector for complex input.

Details

Arguments are recycled: each must have length one or the common length. NA gives NA; NaN is evaluated as SciPy does. The result takes its dimensions, or else its names, from the arguments as base R arithmetic does, and so does each element of a list result. Conditions the kernels raise, such as a pole, are handled as sc_seterr() says.

Windows

R on Windows is built with Rtools, whose C library (mingw-w64 v11) evaluates sin() and cos() inaccurately for arguments beyond about 1e5, as base R's own sin() shows there. Complex arguments with a part that large, and real ones where the kernel takes a sine of such a value, are correspondingly inaccurate on Windows only.

Other packages

In gsl, for real arguments: expint_E1(), expint_Ei() and expint_En(n, x). In expint: expint_E1(), expint_Ei() and expint_En().

See also

https://docs.scipy.org/doc/scipy/reference/special.html#other-special-functions

Examples

sc_exp1(c(0.5, 1, 2))
#> [1] 0.55977359 0.21938393 0.04890051
sc_expi(-1 + 0i)
#> [1] -0.2193839+0i
sc_expn(2, 1)
#> [1] 0.1484955