def _log_StirlingNegativePowers_(var, precision):
r"""
Helper function to calculate the logarithm of Stirling's approximation
formula from the negative powers of ``var`` on, i.e., it skips the
summands `n \log n - n + (\log n)/2 + \log(2\pi)/2`.
INPUT:
- ``var`` -- a string for the variable name.
- ``precision`` -- an integer specifying the number of exact summands.
If this is negative, then the result is `0`.
OUTPUT:
An asymptotic expansion.
TESTS::
sage: asymptotic_expansions._log_StirlingNegativePowers_(
....: 'm', precision=-1)
0
sage: asymptotic_expansions._log_StirlingNegativePowers_(
....: 'm', precision=0)
O(m^(-1))
sage: asymptotic_expansions._log_StirlingNegativePowers_(
....: 'm', precision=3)
1/12*m^(-1) - 1/360*m^(-3) + 1/1260*m^(-5) + O(m^(-7))
sage: _.parent()
Asymptotic Ring <m^ZZ> over Rational Field
"""
from asymptotic_ring import AsymptoticRing
from sage.rings.rational_field import QQ
A = AsymptoticRing(growth_group='{n}^ZZ'.format(n=var),
coefficient_ring=QQ)
if precision < 0:
return A.zero()
n = A.gen()
from sage.arith.all import bernoulli
from sage.arith.srange import srange
result = sum((bernoulli(k) / k / (k-1) / n**(k-1)
for k in srange(2, 2*precision + 2, 2)),
A.zero())
return result + (1 / n**(2*precision + 1)).O()
def _log_StirlingNegativePowers_(var, precision):
r"""
Helper function to calculate the logarithm of Stirling's approximation
formula from the negative powers of ``var`` on, i.e., it skips the
summands `n \log n - n + (\log n)/2 + \log(2\pi)/2`.
INPUT:
- ``var`` -- a string for the variable name.
- ``precision`` -- an integer specifying the number of exact summands.
If this is negative, then the result is `0`.
OUTPUT:
An asymptotic expansion.
TESTS::
sage: asymptotic_expansions._log_StirlingNegativePowers_(
....: 'm', precision=-1)
0
sage: asymptotic_expansions._log_StirlingNegativePowers_(
....: 'm', precision=0)
O(m^(-1))
sage: asymptotic_expansions._log_StirlingNegativePowers_(
....: 'm', precision=3)
1/12*m^(-1) - 1/360*m^(-3) + 1/1260*m^(-5) + O(m^(-7))
sage: _.parent()
Asymptotic Ring <m^ZZ> over Rational Field
"""
from asymptotic_ring import AsymptoticRing
from sage.rings.rational_field import QQ
A = AsymptoticRing(growth_group='{n}^ZZ'.format(n=var),
coefficient_ring=QQ)
if precision < 0:
return A.zero()
n = A.gen()
from sage.arith.all import bernoulli
from sage.arith.srange import srange
result = sum((bernoulli(k) / k / (k - 1) / n**(k - 1)
for k in srange(2, 2 * precision + 2, 2)), A.zero())
return result + (1 / n**(2 * precision + 1)).O()
def HarmonicNumber(var, precision=None, skip_constant_summand=False):
r"""
Return the asymptotic expansion of a harmonic number.
INPUT:
- ``var`` -- a string for the variable name.
- ``precision`` -- (default: ``None``) an integer. If ``None``, then
the default precision of the asymptotic ring is used.
- ``skip_constant_summand`` -- (default: ``False``) a
boolean. If set, then the constant summand ``euler_gamma`` is left out.
As a consequence, the coefficient ring of the output changes
from ``Symbolic Constants Subring`` (if ``False``) to
``Rational Field`` (if ``True``).
OUTPUT:
An asymptotic expansion.
EXAMPLES::
sage: asymptotic_expansions.HarmonicNumber('n', precision=5)
log(n) + euler_gamma + 1/2*n^(-1) - 1/12*n^(-2) + 1/120*n^(-4) + O(n^(-6))
TESTS::
sage: ex = asymptotic_expansions.HarmonicNumber('n', precision=5)
sage: n = ex.parent().gen()
sage: ex.compare_with_values(n, # rel tol 1e-6
....: lambda x: sum(1/k for k in srange(1, x+1)), [5, 10, 20])
[(5, 0.0038125360?), (10, 0.00392733?), (20, 0.0039579?)]
sage: asymptotic_expansions.HarmonicNumber('n')
log(n) + euler_gamma + 1/2*n^(-1) - 1/12*n^(-2) + 1/120*n^(-4)
- 1/252*n^(-6) + 1/240*n^(-8) - 1/132*n^(-10)
+ 691/32760*n^(-12) - 1/12*n^(-14) + 3617/8160*n^(-16)
- 43867/14364*n^(-18) + 174611/6600*n^(-20) - 77683/276*n^(-22)
+ 236364091/65520*n^(-24) - 657931/12*n^(-26)
+ 3392780147/3480*n^(-28) - 1723168255201/85932*n^(-30)
+ 7709321041217/16320*n^(-32)
- 151628697551/12*n^(-34) + O(n^(-36))
sage: _.parent()
Asymptotic Ring <n^ZZ * log(n)^ZZ> over Symbolic Constants Subring
::
sage: asymptotic_expansions.HarmonicNumber(
....: 'n', precision=5, skip_constant_summand=True)
log(n) + 1/2*n^(-1) - 1/12*n^(-2) + 1/120*n^(-4) + O(n^(-6))
sage: _.parent()
Asymptotic Ring <n^ZZ * log(n)^ZZ> over Rational Field
sage: for p in range(5):
....: print asymptotic_expansions.HarmonicNumber(
....: 'n', precision=p)
O(log(n))
log(n) + O(1)
log(n) + euler_gamma + O(n^(-1))
log(n) + euler_gamma + 1/2*n^(-1) + O(n^(-2))
log(n) + euler_gamma + 1/2*n^(-1) - 1/12*n^(-2) + O(n^(-4))
sage: asymptotic_expansions.HarmonicNumber('m', precision=5)
log(m) + euler_gamma + 1/2*m^(-1) - 1/12*m^(-2) + 1/120*m^(-4) + O(m^(-6))
"""
if not skip_constant_summand:
from sage.symbolic.ring import SR
coefficient_ring = SR.subring(no_variables=True)
else:
from sage.rings.rational_field import QQ
coefficient_ring = QQ
from asymptotic_ring import AsymptoticRing
A = AsymptoticRing(growth_group='{n}^ZZ * log({n})^ZZ'.format(n=var),
coefficient_ring=coefficient_ring)
n = A.gen()
if precision is None:
precision = A.default_prec
from sage.functions.log import log
result = A.zero()
if precision >= 1:
result += log(n)
if precision >= 2 and not skip_constant_summand:
from sage.symbolic.constants import euler_gamma
result += coefficient_ring(euler_gamma)
if precision >= 3:
result += 1 / (2 * n)
from sage.arith.srange import srange
from sage.arith.all import bernoulli
for k in srange(2, 2*precision - 4, 2):
result += -bernoulli(k) / k / n**k
if precision < 1:
result += (log(n)).O()
elif precision == 1:
result += A(1).O()
elif precision == 2:
result += (1 / n).O()
else:
result += (1 / n**(2*precision - 4)).O()
return result
def log_Stirling(var, precision=None, skip_constant_summand=False):
r"""
Return the logarithm of Stirling's approximation formula
for factorials.
INPUT:
- ``var`` -- a string for the variable name.
- ``precision`` -- (default: ``None``) an integer. If ``None``, then
the default precision of the asymptotic ring is used.
- ``skip_constant_summand`` -- (default: ``False``) a
boolean. If set, then the constant summand `\log(2\pi)/2` is left out.
As a consequence, the coefficient ring of the output changes
from ``Symbolic Constants Subring`` (if ``False``) to
``Rational Field`` (if ``True``).
OUTPUT:
An asymptotic expansion.
EXAMPLES::
sage: asymptotic_expansions.log_Stirling('n', precision=7)
n*log(n) - n + 1/2*log(n) + 1/2*log(2*pi) + 1/12*n^(-1)
- 1/360*n^(-3) + 1/1260*n^(-5) + O(n^(-7))
.. SEEALSO::
:meth:`Stirling`,
:meth:`~sage.rings.asymptotic.asymptotic_ring.AsymptoticExpansion.factorial`.
TESTS::
sage: expansion = asymptotic_expansions.log_Stirling('n', precision=7)
sage: n = expansion.parent().gen()
sage: expansion.compare_with_values(n, lambda x: x.factorial().log(), [5, 10, 20]) # rel tol 1e-6
[(5, 0.000564287?), (10, 0.0005870?), (20, 0.0006?)]
sage: asymptotic_expansions.log_Stirling('n')
n*log(n) - n + 1/2*log(n) + 1/2*log(2*pi) + 1/12*n^(-1)
- 1/360*n^(-3) + 1/1260*n^(-5) - 1/1680*n^(-7) + 1/1188*n^(-9)
- 691/360360*n^(-11) + 1/156*n^(-13) - 3617/122400*n^(-15)
+ 43867/244188*n^(-17) - 174611/125400*n^(-19) + 77683/5796*n^(-21)
- 236364091/1506960*n^(-23) + 657931/300*n^(-25)
- 3392780147/93960*n^(-27) + 1723168255201/2492028*n^(-29)
- 7709321041217/505920*n^(-31) + O(n^(-33))
sage: _.parent()
Asymptotic Ring <n^ZZ * log(n)^ZZ> over Symbolic Constants Subring
::
sage: asymptotic_expansions.log_Stirling(
....: 'n', precision=7, skip_constant_summand=True)
n*log(n) - n + 1/2*log(n) + 1/12*n^(-1) - 1/360*n^(-3) +
1/1260*n^(-5) + O(n^(-7))
sage: _.parent()
Asymptotic Ring <n^ZZ * log(n)^ZZ> over Rational Field
sage: asymptotic_expansions.log_Stirling(
....: 'n', precision=0)
O(n*log(n))
sage: asymptotic_expansions.log_Stirling(
....: 'n', precision=1)
n*log(n) + O(n)
sage: asymptotic_expansions.log_Stirling(
....: 'n', precision=2)
n*log(n) - n + O(log(n))
sage: asymptotic_expansions.log_Stirling(
....: 'n', precision=3)
n*log(n) - n + 1/2*log(n) + O(1)
sage: asymptotic_expansions.log_Stirling(
....: 'n', precision=4)
n*log(n) - n + 1/2*log(n) + 1/2*log(2*pi) + O(n^(-1))
sage: asymptotic_expansions.log_Stirling(
....: 'n', precision=5)
n*log(n) - n + 1/2*log(n) + 1/2*log(2*pi) + 1/12*n^(-1)
+ O(n^(-3))
sage: asymptotic_expansions.log_Stirling(
....: 'm', precision=7, skip_constant_summand=True)
m*log(m) - m + 1/2*log(m) + 1/12*m^(-1) - 1/360*m^(-3) +
1/1260*m^(-5) + O(m^(-7))
"""
if not skip_constant_summand:
from sage.symbolic.ring import SR
coefficient_ring = SR.subring(no_variables=True)
else:
from sage.rings.rational_field import QQ
coefficient_ring = QQ
from asymptotic_ring import AsymptoticRing
A = AsymptoticRing(growth_group='{n}^ZZ * log({n})^ZZ'.format(n=var),
coefficient_ring=coefficient_ring)
n = A.gen()
if precision is None:
precision = AsymptoticRing.__default_prec__
from sage.functions.log import log
result = A.zero()
if precision >= 1:
result += n * log(n)
if precision >= 2:
result += -n
if precision >= 3:
result += log(n) / 2
if precision >= 4 and not skip_constant_summand:
result += log(2*coefficient_ring('pi')) / 2
result += AsymptoticExpansionGenerators._log_StirlingNegativePowers_(
var, precision - 4)
if precision < 1:
result += (n * log(n)).O()
elif precision == 1:
result += n.O()
elif precision == 2:
result += log(n).O()
elif precision == 3:
result += A(1).O()
return result
def HarmonicNumber(var, precision=None, skip_constant_summand=False):
r"""
Return the asymptotic expansion of a harmonic number.
INPUT:
- ``var`` -- a string for the variable name.
- ``precision`` -- (default: ``None``) an integer. If ``None``, then
the default precision of the asymptotic ring is used.
- ``skip_constant_summand`` -- (default: ``False``) a
boolean. If set, then the constant summand ``euler_gamma`` is left out.
As a consequence, the coefficient ring of the output changes
from ``Symbolic Constants Subring`` (if ``False``) to
``Rational Field`` (if ``True``).
OUTPUT:
An asymptotic expansion.
EXAMPLES::
sage: asymptotic_expansions.HarmonicNumber('n', precision=5)
log(n) + euler_gamma + 1/2*n^(-1) - 1/12*n^(-2) + 1/120*n^(-4) + O(n^(-6))
TESTS::
sage: ex = asymptotic_expansions.HarmonicNumber('n', precision=5)
sage: n = ex.parent().gen()
sage: ex.compare_with_values(n, # rel tol 1e-6
....: lambda x: sum(1/k for k in srange(1, x+1)), [5, 10, 20])
[(5, 0.0038125360?), (10, 0.00392733?), (20, 0.0039579?)]
sage: asymptotic_expansions.HarmonicNumber('n')
log(n) + euler_gamma + 1/2*n^(-1) - 1/12*n^(-2) + 1/120*n^(-4)
- 1/252*n^(-6) + 1/240*n^(-8) - 1/132*n^(-10)
+ 691/32760*n^(-12) - 1/12*n^(-14) + 3617/8160*n^(-16)
- 43867/14364*n^(-18) + 174611/6600*n^(-20) - 77683/276*n^(-22)
+ 236364091/65520*n^(-24) - 657931/12*n^(-26)
+ 3392780147/3480*n^(-28) - 1723168255201/85932*n^(-30)
+ 7709321041217/16320*n^(-32)
- 151628697551/12*n^(-34) + O(n^(-36))
sage: _.parent()
Asymptotic Ring <n^ZZ * log(n)^ZZ> over Symbolic Constants Subring
::
sage: asymptotic_expansions.HarmonicNumber(
....: 'n', precision=5, skip_constant_summand=True)
log(n) + 1/2*n^(-1) - 1/12*n^(-2) + 1/120*n^(-4) + O(n^(-6))
sage: _.parent()
Asymptotic Ring <n^ZZ * log(n)^ZZ> over Rational Field
sage: for p in range(5):
....: print asymptotic_expansions.HarmonicNumber(
....: 'n', precision=p)
O(log(n))
log(n) + O(1)
log(n) + euler_gamma + O(n^(-1))
log(n) + euler_gamma + 1/2*n^(-1) + O(n^(-2))
log(n) + euler_gamma + 1/2*n^(-1) - 1/12*n^(-2) + O(n^(-4))
sage: asymptotic_expansions.HarmonicNumber('m', precision=5)
log(m) + euler_gamma + 1/2*m^(-1) - 1/12*m^(-2) + 1/120*m^(-4) + O(m^(-6))
"""
if not skip_constant_summand:
from sage.symbolic.ring import SR
coefficient_ring = SR.subring(no_variables=True)
else:
from sage.rings.rational_field import QQ
coefficient_ring = QQ
from asymptotic_ring import AsymptoticRing
A = AsymptoticRing(growth_group='{n}^ZZ * log({n})^ZZ'.format(n=var),
coefficient_ring=coefficient_ring)
n = A.gen()
if precision is None:
precision = A.default_prec
from sage.functions.log import log
result = A.zero()
if precision >= 1:
result += log(n)
if precision >= 2 and not skip_constant_summand:
from sage.symbolic.constants import euler_gamma
result += coefficient_ring(euler_gamma)
if precision >= 3:
result += 1 / (2 * n)
from sage.arith.srange import srange
from sage.arith.all import bernoulli
for k in srange(2, 2 * precision - 4, 2):
result += -bernoulli(k) / k / n**k
if precision < 1:
result += (log(n)).O()
elif precision == 1:
result += A(1).O()
elif precision == 2:
result += (1 / n).O()
else:
result += (1 / n**(2 * precision - 4)).O()
return result
def log_Stirling(var, precision=None, skip_constant_summand=False):
r"""
Return the logarithm of Stirling's approximation formula
for factorials.
INPUT:
- ``var`` -- a string for the variable name.
- ``precision`` -- (default: ``None``) an integer. If ``None``, then
the default precision of the asymptotic ring is used.
- ``skip_constant_summand`` -- (default: ``False``) a
boolean. If set, then the constant summand `\log(2\pi)/2` is left out.
As a consequence, the coefficient ring of the output changes
from ``Symbolic Constants Subring`` (if ``False``) to
``Rational Field`` (if ``True``).
OUTPUT:
An asymptotic expansion.
EXAMPLES::
sage: asymptotic_expansions.log_Stirling('n', precision=7)
n*log(n) - n + 1/2*log(n) + 1/2*log(2*pi) + 1/12*n^(-1)
- 1/360*n^(-3) + 1/1260*n^(-5) + O(n^(-7))
.. SEEALSO::
:meth:`Stirling`,
:meth:`~sage.rings.asymptotic.asymptotic_ring.AsymptoticExpansion.factorial`.
TESTS::
sage: expansion = asymptotic_expansions.log_Stirling('n', precision=7)
sage: n = expansion.parent().gen()
sage: expansion.compare_with_values(n, lambda x: x.factorial().log(), [5, 10, 20]) # rel tol 1e-6
[(5, 0.000564287?), (10, 0.0005870?), (20, 0.0006?)]
sage: asymptotic_expansions.log_Stirling('n')
n*log(n) - n + 1/2*log(n) + 1/2*log(2*pi) + 1/12*n^(-1)
- 1/360*n^(-3) + 1/1260*n^(-5) - 1/1680*n^(-7) + 1/1188*n^(-9)
- 691/360360*n^(-11) + 1/156*n^(-13) - 3617/122400*n^(-15)
+ 43867/244188*n^(-17) - 174611/125400*n^(-19) + 77683/5796*n^(-21)
- 236364091/1506960*n^(-23) + 657931/300*n^(-25)
- 3392780147/93960*n^(-27) + 1723168255201/2492028*n^(-29)
- 7709321041217/505920*n^(-31) + O(n^(-33))
sage: _.parent()
Asymptotic Ring <n^ZZ * log(n)^ZZ> over Symbolic Constants Subring
::
sage: asymptotic_expansions.log_Stirling(
....: 'n', precision=7, skip_constant_summand=True)
n*log(n) - n + 1/2*log(n) + 1/12*n^(-1) - 1/360*n^(-3) +
1/1260*n^(-5) + O(n^(-7))
sage: _.parent()
Asymptotic Ring <n^ZZ * log(n)^ZZ> over Rational Field
sage: asymptotic_expansions.log_Stirling(
....: 'n', precision=0)
O(n*log(n))
sage: asymptotic_expansions.log_Stirling(
....: 'n', precision=1)
n*log(n) + O(n)
sage: asymptotic_expansions.log_Stirling(
....: 'n', precision=2)
n*log(n) - n + O(log(n))
sage: asymptotic_expansions.log_Stirling(
....: 'n', precision=3)
n*log(n) - n + 1/2*log(n) + O(1)
sage: asymptotic_expansions.log_Stirling(
....: 'n', precision=4)
n*log(n) - n + 1/2*log(n) + 1/2*log(2*pi) + O(n^(-1))
sage: asymptotic_expansions.log_Stirling(
....: 'n', precision=5)
n*log(n) - n + 1/2*log(n) + 1/2*log(2*pi) + 1/12*n^(-1)
+ O(n^(-3))
sage: asymptotic_expansions.log_Stirling(
....: 'm', precision=7, skip_constant_summand=True)
m*log(m) - m + 1/2*log(m) + 1/12*m^(-1) - 1/360*m^(-3) +
1/1260*m^(-5) + O(m^(-7))
"""
if not skip_constant_summand:
from sage.symbolic.ring import SR
coefficient_ring = SR.subring(no_variables=True)
else:
from sage.rings.rational_field import QQ
coefficient_ring = QQ
from asymptotic_ring import AsymptoticRing
A = AsymptoticRing(growth_group='{n}^ZZ * log({n})^ZZ'.format(n=var),
coefficient_ring=coefficient_ring)
n = A.gen()
if precision is None:
precision = AsymptoticRing.__default_prec__
from sage.functions.log import log
result = A.zero()
if precision >= 1:
result += n * log(n)
if precision >= 2:
result += -n
if precision >= 3:
result += log(n) / 2
if precision >= 4 and not skip_constant_summand:
result += log(2 * coefficient_ring('pi')) / 2
result += AsymptoticExpansionGenerators._log_StirlingNegativePowers_(
var, precision - 4)
if precision < 1:
result += (n * log(n)).O()
elif precision == 1:
result += n.O()
elif precision == 2:
result += log(n).O()
elif precision == 3:
result += A(1).O()
return result