mpfr-special-functions {Rmpfr}

Special Mathematical Functions (MPFR)

Description

Special Mathematical Functions, supported by the MPFR Library.

Note that additionally, all the Math and Math2 group member functions are “mpfr-ified”, too; ditto, for many more standard R functions. See see the methods listed in mpfr (aka ?`mpfr-class`).

Usage

zeta(x)
Ei(x)
Li2(x)

erf(x)
erfc(x)

Arguments

x

a numeric or mpfr vector.

Details

zeta(x) computes Riemann's Zeta function \zeta(x) important in analytical number theory and related fields. The traditional definition is

\zeta(x) = \sum_{n=1}^\infty \frac{1}{n^x}.

Ei(x) computes the exponential integral,

\int_{-\infty}^{x} \frac{e^t}{t} \; dt.

Li2(x) computes the dilogarithm,

\int_{0}^{x} \frac{-log(1-t)}{t} \; dt.

erf(x) and erfc(x) are the error, respectively complementary error function which are both reparametrizations of pnorm, erf(x) = 2*pnorm(sqrt(2)*x) and erfc(x) = 2* pnorm(sqrt(2)*x, lower=FALSE), and hence Rmpfr provides its own version of pnorm.

Value

A vector of the same length as x, of class mpfr.

See Also

pnorm in standard package stats; the class description mpfr mentioning the generic arithmetic and mathematical functions (sin, log, ..., etc) for which "mpfr" methods are available.

Note the (integer order, non modified) Bessel functions j_0(), y_n(), etc, named j0, yn etc, and Airy function Ai() in Bessel_mpfr.

Examples

curve(Ei,  0, 5, n=2001)

## As we now require (mpfrVersion() >= "2.4.0"):
curve(Li2,  0,    5, n=2001)
curve(Li2, -2,   13, n=2000); abline(h=0,v=0, lty=3)
curve(Li2, -200,400, n=2000); abline(h=0,v=0, lty=3)

curve(erf, -3,3, col = "red", ylim = c(-1,2))
curve(erfc, add = TRUE, col = "blue")
abline(h=0, v=0, lty=3)
legend(-3,1, c("erf(x)", "erfc(x)"), col = c("red","blue"), lty=1)