Special Mathematical Functions (MPFR)

Description

Special Mathematical Functions, supported by the MPFR Library.

Usage

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

erf(x)
erfc(x)

Arguments

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..Inf; 1/(n^x)].

Ei(x) computes the exponential integral,

Integral(-Inf,x; e^t/t dt).

Li2(x) computes the dilogarithm,

Integral(0,x; -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.

Examples

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

if(mpfrVersion() >= "2.4.0") { ## Li2() is not available in older MPFR versions
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)