def stieltjes(n):
"""Computes the nth Stieltjes constant."""
n = int(n)
if n == 0:
return +euler
if n < 0:
raise ValueError("Stieltjes constants defined for n >= 0")
if n in stieltjes_cache:
prec, s = stieltjes_cache[n]
if prec >= mp.prec:
return +s
from quadrature import quadgl
def f(x):
r = exp(pi*j*x)
return (zeta(r+1) / r**n).real
orig = mp.prec
try:
p = int(log(factorial(n), 2) + 35)
mp.prec += p
u = quadgl(f, [-1, 1])
v = mpf(-1)**n * factorial(n) * u / 2
finally:
mp.prec = orig
stieltjes_cache[n] = (mp.prec, v)
return +v
def stieltjes(n):
"""Computes the nth Stieltjes constant."""
n = int(n)
if n == 0:
return +euler
if n < 0:
raise ValueError("Stieltjes constants defined for n >= 0")
if n in stieltjes_cache:
prec, s = stieltjes_cache[n]
if prec >= mp.prec:
return +s
from quadrature import quadgl
def f(x):
r = exp(pi * j * x)
return (zeta(r + 1) / r**n).real
orig = mp.prec
try:
p = int(log(factorial(n), 2) + 35)
mp.prec += p
u = quadgl(f, [-1, 1])
v = mpf(-1)**n * factorial(n) * u / 2
finally:
mp.prec = orig
stieltjes_cache[n] = (mp.prec, v)
return +v
def quadosc(f, interval, period=None, zeros=None, alt=1):
"""
Integrates f(x) over interval = [a, b] where at least one of a and
b is infinite and f is a slowly decaying oscillatory function. The
zeros of f must be provided, either by specifying a period (suitable
when f contains a pure sine or cosine factor) or providing a function
that returns the nth zero (suitable when the oscillation is not
strictly periodic).
"""
a, b = AS_POINTS(interval)
a = convert_lossless(a)
b = convert_lossless(b)
if period is None and zeros is None:
raise ValueError( \
"either the period or zeros keyword parameter must be specified")
if a == -inf and b == inf:
s1 = quadosc(f, [a, 0], zeros=zeros, period=period, alt=alt)
s2 = quadosc(f, [0, b], zeros=zeros, period=period, alt=alt)
return s1 + s2
if a == -inf:
if zeros:
return quadosc(lambda x:f(-x), [-b,-a], lambda n: zeros(-n), alt=alt)
else:
return quadosc(lambda x:f(-x), [-b,-a], period=period, alt=alt)
if b != inf:
raise ValueError("quadosc requires an infinite integration interval")
if not zeros:
zeros = lambda n: n*period/2
for n in range(1,10):
p = zeros(n)
if p > a:
break
if n >= 9:
raise ValueError("zeros do not appear to be correctly indexed")
if alt == 0:
s = quadgl(f, [a, zeros(n+1)])
s += sumrich(lambda k: quadgl(f, [zeros(2*k), zeros(2*k+2)]), [n, inf])
else:
s = quadgl(f, [a, zeros(n)])
s += sumsh(lambda k: quadgl(f, [zeros(k), zeros(k+1)]), [n, inf])
return s
