Zeta, Lambert W, Wright omega, Spence and Struve functions

• sc_zeta(): the Riemann zeta function \(\zeta(x)\) when q is NULL, else the Hurwitz zeta function \(\zeta(x, q)\). The Riemann form takes complex x; the Hurwitz form, as in SciPy, only complex x on the real axis, and is NaN (a domain condition) elsewhere.

• sc_zetac(): \(\zeta(x) - 1\), accurate for large x.

• sc_lambertw(): branch k of the Lambert W function, the solution of \(w e^w = z\). As in SciPy, the result is always complex.

• sc_wrightomega(): the Wright omega function, the solution of \(\omega + \log \omega = z\), complex.

• sc_spence(): Spence's function (the dilogarithm \(\mathrm{Li}_2(1 - z)\)), complex.

• sc_struve(), sc_modstruve(): the Struve functions \(H_v(x)\) and \(L_v(x)\).

Usage

sc_zeta(x, q = NULL)

sc_zetac(x)

sc_lambertw(z, k = 0L, tol = 1e-08)

sc_wrightomega(z)

sc_spence(z)

sc_struve(v, x)

sc_modstruve(v, x)

Arguments

x

Numeric vector; for sc_zeta(), also complex.

q

NULL, or a numeric vector of Hurwitz parameters.

z

Numeric or complex vector.

k

Whole numbers within integer range, the branch.

tol

Numeric vector, the tolerance of the iteration.

v

Numeric vector of orders.

Value

A double vector, or a complex vector for complex input (always for sc_lambertw()).

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.

Branch cuts

As in SciPy: the principal branch of sc_lambertw() (k = 0) has a branch cut on the real axis from \(-\infty\) to \(-1/e\), and every other branch one from \(-\infty\) to 0; sc_spence() has a branch cut on the negative real axis.

Other packages

In gsl, for real arguments: zeta(), hzeta(s, q), zetam1() (the sc_zetac() of SciPy), lambert_W0() and lambert_Wm1() (branches 0 and -1), and dilog(x), which is sc_spence(1 - x). In lamW: lambertW0() and lambertWm1().

See also

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

Examples

sc_zeta(c(2, 4))
#> [1] 1.644934 1.082323
sc_zeta(2, 2)
#> [1] 0.6449341
sc_lambertw(1)
#> [1] 0.5671433+0i
sc_lambertw(-0.2, k = -1)
#> [1] -2.542641+0i
sc_wrightomega(1 + 2i)
#> [1] 0.7472911+1.048444i
sc_spence(0.5)
#> [1] 0.5822405
sc_struve(0.5, 2)
#> [1] 0.7989753