Documentation

Other Special Functions

ArithmeticGeometricMean(x, y) : RngSerElt, RngSerElt -> RngSerElt

AGM(f, g) : RngSerElt, RngSerElt -> RngSerElt

Return the hyperbolic arithmetic-geometric mean of the series f and g defined over a field. The valuations of f and g must be equal.

ArithmeticGeometricMean(x, y) : FldReElt, FldReElt -> FldReElt

AGM(x, y) : FldReElt, FldReElt -> FldReElt

Returns the arithmetic-geometric mean of the real or complex numbers x and y, defined as the limit of either of the sequences x_i, y_i where x₀ = x, y₀ = y and x_{i + 1} = (x_i + y_i)/2, y_{i + 1} = Sqrt(x_i y_i). The function calculates both sequences, and when the numbers are within the desired precision of each other, it returns one of them.

BernoulliNumber(n) : RngIntElt -> FldRatElt

For a non-negative integer n, return the value of the n-th Bernoulli number B_n, defined by (t/e^t - 1)=∑_n=0^∞B_n(tⁿ/n!).

BernoulliApproximation(n) : RngIntElt -> FldReElt

For a non-negative integer n, return an approximation in the field of real numbers to the value of the n-th Bernoulli number B_n, defined by (t/e^t - 1)=∑_n=0^∞B_n(tⁿ/n!).

DawsonIntegral(r) : FldReElt -> FldReElt

Given a real number r, compute the value of Dawson's integral, e^{- x²} times the integral from 0 to x of e^u² with respect to u, at x = r.

ErrorFunction(r) : FldReElt -> FldReElt

Erf(r) : FldReElt -> FldReElt

Given a real number r, calculate the value of the error function erf. This is the value of the square root of (4/pi) times the integral from 0 to x of e^{- u²} with respect to u, at x = r for r>0, and for r<0 it is defined by erf(x)= - erf( - x), while erf(0)=0.

ComplementaryErrorFunction(r) : FldReElt -> FldReElt

Erfc(r) : FldReElt -> FldReElt

Given a real number r, calculate the value of the complementary error function. This is the value of y = (erfc)(x) = 1 - erf(x) for the error function erf as defined above.

ExponentialIntegral(r) : FldReElt -> FldReElt

Given a real number r, calculate the value of the exponential integral, that is, the principal value of the integral from minus infinity to x of e^u / u with respect to u at x = r.

ExponentialIntegralE1(r) : FldReElt -> FldReElt

Given a real number r, calculate the value of the exponential integral E1, that is, the principal value of the integral from x to infinity of e^{- u} / u with respect to u at x = r.

LogIntegral(r) : FldReElt -> FldReElt

Given a non-negative real number r that is not equal to 1, evaluate the logarithmic integral y = li(x) at x = r. This integral is defined to be the principal value of the integral from 0 to x of 1 / log(u) with respect to u.

ZetaFunction(s) : FldReElt -> FldReElt

ZetaFunction(R, n) : FldRe, RngIntElt -> FldReElt

These functions calculate values of the Riemann ζ-function, which is the analytic continuation of ζ(z) = the sum from i equals 1 to infinity of 1 / iⁿ (convergent for Re(z)> 1). The version with one argument takes a real or complex number r != 1 and returns a real or complex number.
The version with two arguments is much more restricted; it takes a real field R and an integer n != 1, and returns ζ(n) in R.

MPFR uses the algorithm of Jean-Luc Rémy and Sapphorain Pétermann [PR06].

V2.28, 13 July 2023