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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.
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
xi, yi where x0 = x, y0 = y and
xi + 1 = (xi + yi)/2, yi + 1 = Sqrt(xi yi).
The function calculates both sequences, and when the numbers are
within the desired precision of each other, it returns one of them.
For a non-negative integer n, return the value of the n-th Bernoulli
number Bn, defined by
(t/et - 1)=∑n=0^∞Bn(tn/n!).
For a non-negative integer n, return an approximation in the field of real
numbers to
the value of the n-th Bernoulli
number Bn, defined by
(t/et - 1)=∑n=0^∞Bn(tn/n!).
Given a real number r, compute the value of
Dawson's integral, e - x2 times the integral from 0 to x
of eu2 with respect to u,
at x = r.
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.
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.
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 eu / u with respect to u
at x = r.
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.
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(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].
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