arb_series – power series over real numbers

class flint.arb_series(val=None, prec=None)

acos(s)

airy(s)

airy_ai(s)

airy_ai_prime(s)

airy_bi(s)

airy_bi_prime(s)

asin(s)

atan(s)

classmethod beta_lower(cls, a, b, z, int regularized=0)

chi(s)

ci(s)

coeffs(self)

cos(s)

cos_pi(s)

cot_pi(s)

coulomb(self, l, eta)

coulomb_f(self, l, eta)

coulomb_g(self, l, eta)

derivative(s)

ei(s)

erf(s)

erfc(s)

erfi(s)

exp(s)

static find_roots(f, a, b, maxn=100000)

Isolates the roots of a given real analytic function f on the interval [a, b]. The function f takes an arb_series as input and outputs an arb_series.

This is just a test implementation; more options including support for Newton refinement will be added in a future version.

>>> from flint import arb_series, ctx
>>> ctx.prec = 53
>>> for c in arb_series.find_roots(lambda x: x.sin(), -8, 8): print(c)
...
(-6.96875000000000, -5.93750000000000)
(-3.87500000000000, -1.81250000000000)
(-0.781250000000000, 0.250000000000000)
(2.18750000000000, 4.12500000000000)
(6.06250000000000, 7.03125000000000)
>>> for c in arb_series.find_roots(lambda x: x.riemann_siegel_z(), 0, 30): print(c)
...
(14.1210937500000, 14.1796875000000)
(20.9765625000000, 21.0351562500000)
(24.9609375000000, 25.0195312500000)

fresnel(s, bool normalized=True)

fresnel_c(s, bool normalized=True)

fresnel_s(s, bool normalized=True)

gamma(s)

classmethod gamma_lower(cls, s, z, int regularized=0)

classmethod gamma_upper(cls, s, z, int regularized=0)

integral(s)

inv(s)

lambertw(s, int branch=0)

length(self) → long

lgamma(s)

li(s, bool offset=False)

log(s)

prec

The term precision of the finitely approximated series.

>>> from flint import arb_series, ctx
>>> ctx.cap = 10
>>> s = arb_series([1,2])
>>> s
1.00000000000000 + 2.00000000000000*x + O(x^10)
>>> s.prec
10
>>> s2 = arb_series([1,2], prec=3)
>>> s2
1.00000000000000 + 2.00000000000000*x + O(x^3)
>>> s2.prec
3

repr(self, **kwargs)

reversion(s)

rgamma(s)

riemann_siegel_theta(s)

riemann_siegel_z(s)

rising(s, ulong n)

rsqrt(s)

shi(s)

si(s)

sin(s)

sin_cos(s)

sin_cos_pi(s)

sin_pi(s)

sqrt(s)

str(self, *args, **kwargs)

tan(s)

valuation(self)

zeta(s, a=1, bool deflate=0)