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 \(x_0 = x\), \(y_0 = y\) 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

\[{t\over e^t-1}=\sum_{n=0}^\infty B_n{t^n\over n!}.\]

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

\[{t\over e^t-1}=\sum_{n=0}^\infty B_n{t^n\over n!}.\]

DawsonIntegral(r): FldReElt -> FldReElt

Given a real number \(r\), compute the value of Dawson’s integral,

\[e^{-x^2}\cdot\int_0^xe^{u^2} du,\]

at \(x = r\).

ErrorFunction(r): FldReElt -> FldReElt

Erf(r): FldReElt -> FldReElt

Given a real number \(r\), calculate the value of the error function \({\operatorname{erf}}\). This is the value of

\[\sqrt{4\over \pi}\cdot \int_0^xe^{-u^2} du,\]

at \(x = r\) for \(r>0\), and for \(r<0\) it is defined by \({\operatorname{erf}}(x)=-{\operatorname{erf}}(-x)\), while \({\operatorname{erf}}(0)=0\).

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 \(y = {\rm erfc}(x) = 1 - {\operatorname{erf}}(x)\) for the error function \({\operatorname{erf}}\) 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

\[\int_{-\infty}^x{e^u\over u}du\]

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

\[\int_x^{\infty}{e^{-u}\over u}du\]

at \(x = r\).

LogIntegral(r): FldReElt -> FldReElt

Given a non-negative real number \(r\) that is not equal to 1, evaluate the logarithmic integral \(y = {\operatorname{li}}(x)\) at \(x = r\). This integral is defined to be the principal value of \(\int_0^x{1\over \log(u)}du.\)

ZetaFunction(s): FldReElt -> FldReElt

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

These functions calculate values of the Riemann \(\zeta\)-function, which is the analytic continuation of

\[\zeta(z)=\sum_{i=1}^\infty{1\over i^z}\]

(convergent for Re\((z)> 1\)). The version with one argument takes a real or complex number \(r\neq 1\) 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 \(n \neq 1\), and returns \(\zeta(n)\) in \(R\).

MPFR uses the algorithm of Jean-Luc Rémy and Sapphorain Pétermann [Pétermann and Rémy, 2006].