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log.py
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log.py
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"""
Logarithmic functions
AUTHORS:
- Yoora Yi Tenen (2012-11-16): Add documentation for :meth:`log()` (:trac:`12113`)
- Tomas Kalvoda (2015-04-01): Add :meth:`exp_polar()` (:trac:`18085`)
"""
from sage.symbolic.function import GinacFunction, BuiltinFunction
from sage.symbolic.constants import e as const_e
from sage.symbolic.constants import pi as const_pi
from sage.libs.mpmath import utils as mpmath_utils
from sage.structure.all import parent as s_parent
from sage.symbolic.expression import Expression
from sage.rings.real_double import RDF
from sage.rings.complex_double import CDF
from sage.rings.integer import Integer
from sage.rings.integer_ring import ZZ
from sage.rings.rational_field import QQ
from sage.rings.rational import Rational
class Function_exp(GinacFunction):
def __init__(self):
r"""
The exponential function, `\exp(x) = e^x`.
EXAMPLES::
sage: exp(-1)
e^(-1)
sage: exp(2)
e^2
sage: exp(2).n(100)
7.3890560989306502272304274606
sage: exp(x^2 + log(x))
e^(x^2 + log(x))
sage: exp(x^2 + log(x)).simplify()
x*e^(x^2)
sage: exp(2.5)
12.1824939607035
sage: exp(float(2.5))
12.182493960703473
sage: exp(RDF('2.5'))
12.182493960703473
sage: exp(I*pi/12)
(1/4*I + 1/4)*sqrt(6) - (1/4*I - 1/4)*sqrt(2)
To prevent automatic evaluation, use the ``hold`` parameter::
sage: exp(I*pi,hold=True)
e^(I*pi)
sage: exp(0,hold=True)
e^0
To then evaluate again, we currently must use Maxima via
:meth:`sage.symbolic.expression.Expression.simplify`::
sage: exp(0,hold=True).simplify()
1
::
sage: exp(pi*I/2)
I
sage: exp(pi*I)
-1
sage: exp(8*pi*I)
1
sage: exp(7*pi*I/2)
-I
The precision for the result is deduced from the precision of
the input. Convert the input to a higher precision explicitly
if a result with higher precision is desired::
sage: t = exp(RealField(100)(2)); t
7.3890560989306502272304274606
sage: t.prec()
100
sage: exp(2).n(100)
7.3890560989306502272304274606
TEST::
sage: latex(exp(x))
e^{x}
sage: latex(exp(sqrt(x)))
e^{\sqrt{x}}
sage: latex(exp)
\exp
sage: latex(exp(sqrt(x))^x)
\left(e^{\sqrt{x}}\right)^{x}
sage: latex(exp(sqrt(x)^x))
e^{\left(\sqrt{x}^{x}\right)}
Test conjugates::
sage: conjugate(exp(x))
e^conjugate(x)
Test simplifications when taking powers of exp, #7264::
sage: var('a,b,c,II')
(a, b, c, II)
sage: model_exp = exp(II)**a*(b)
sage: sol1_l={b: 5.0, a: 1.1}
sage: model_exp.subs(sol1_l)
5.00000000000000*(e^II)^1.10000000000000
::
sage: exp(3)^II*exp(x)
(e^3)^II*e^x
sage: exp(x)*exp(x)
e^(2*x)
sage: exp(x)*exp(a)
e^(a + x)
sage: exp(x)*exp(a)^2
e^(2*a + x)
Another instance of the same problem, #7394::
sage: 2*sqrt(e)
2*sqrt(e)
Check that :trac:`19918` is fixed::
sage: exp(-x^2).subs(x=oo)
0
sage: exp(-x).subs(x=-oo)
+Infinity
"""
GinacFunction.__init__(self, "exp", latex_name=r"\exp",
conversions=dict(maxima='exp'))
exp = Function_exp()
class Function_log(GinacFunction):
def __init__(self):
r"""
The natural logarithm of x. See `log?` for
more information about its behavior.
EXAMPLES::
sage: ln(e^2)
2
sage: ln(2)
log(2)
sage: ln(10)
log(10)
::
sage: ln(RDF(10))
2.302585092994046
sage: ln(2.718)
0.999896315728952
sage: ln(2.0)
0.693147180559945
sage: ln(float(-1))
3.141592653589793j
sage: ln(complex(-1))
3.141592653589793j
The ``hold`` parameter can be used to prevent automatic evaluation::
sage: log(-1,hold=True)
log(-1)
sage: log(-1)
I*pi
sage: I.log(hold=True)
log(I)
sage: I.log(hold=True).simplify()
1/2*I*pi
TESTS::
sage: latex(x.log())
\log\left(x\right)
sage: latex(log(1/4))
\log\left(\frac{1}{4}\right)
sage: loads(dumps(ln(x)+1))
log(x) + 1
``conjugate(log(x))==log(conjugate(x))`` unless on the branch cut which
runs along the negative real axis.::
sage: conjugate(log(x))
conjugate(log(x))
sage: var('y', domain='positive')
y
sage: conjugate(log(y))
log(y)
sage: conjugate(log(y+I))
conjugate(log(y + I))
sage: conjugate(log(-1))
-I*pi
sage: log(conjugate(-1))
I*pi
Check if float arguments are handled properly.::
sage: from sage.functions.log import function_log as log
sage: log(float(5))
1.6094379124341003
sage: log(float(0))
-inf
sage: log(float(-1))
3.141592653589793j
sage: log(x).subs(x=float(-1))
3.141592653589793j
"""
GinacFunction.__init__(self, 'log', latex_name=r'\log',
conversions=dict(maxima='log'))
def __call__(self, *args, **kwds):
"""
Return the logarithm of x to the given base.
Calls the ``log`` method of the object x when computing
the logarithm, thus allowing use of logarithm on any object
containing a ``log`` method. In other words, log works
on more than just real numbers.
EXAMPLES::
sage: log(e^2)
2
To change the base of the logarithm, add a second parameter::
sage: log(1000,10)
3
You can use
:class:`RDF<sage.rings.real_double.RealDoubleField_class>`,
:class:`~sage.rings.real_mpfr.RealField` or ``n`` to get a
numerical real approximation::
sage: log(1024, 2)
10
sage: RDF(log(1024, 2))
10.0
sage: log(10, 4)
log(10)/log(4)
sage: RDF(log(10, 4))
1.6609640474436813
sage: log(10, 2)
log(10)/log(2)
sage: n(log(10, 2))
3.32192809488736
sage: log(10, e)
log(10)
sage: n(log(10, e))
2.30258509299405
The log function works for negative numbers, complex
numbers, and symbolic numbers too, picking the branch
with angle between `-pi` and `pi`::
sage: log(-1+0*I)
I*pi
sage: log(CC(-1))
3.14159265358979*I
sage: log(-1.0)
3.14159265358979*I
For input zero, the following behavior occurs::
sage: log(0)
-Infinity
sage: log(CC(0))
-infinity
sage: log(0.0)
-infinity
The log function also works in finite fields as long as the
argument lies in the multiplicative group generated by the base::
sage: F = GF(13); g = F.multiplicative_generator(); g
2
sage: a = F(8)
sage: log(a,g); g^log(a,g)
3
8
sage: log(a,3)
Traceback (most recent call last):
...
ValueError: No discrete log of 8 found to base 3
sage: log(F(9), 3)
2
The log function also works for p-adics (see documentation for
p-adics for more information)::
sage: R = Zp(5); R
5-adic Ring with capped relative precision 20
sage: a = R(16); a
1 + 3*5 + O(5^20)
sage: log(a)
3*5 + 3*5^2 + 3*5^4 + 3*5^5 + 3*5^6 + 4*5^7 + 2*5^8 + 5^9 +
5^11 + 2*5^12 + 5^13 + 3*5^15 + 2*5^16 + 4*5^17 + 3*5^18 +
3*5^19 + O(5^20)
TESTS:
Check if :trac:`10136` is fixed::
sage: log(x).operator() is log
True
sage: log(x).operator() is ln
True
sage: log(1000, 10, base=5)
Traceback (most recent call last):
...
TypeError: Symbolic function log must be called as log(x),
log(x, base=b) or log(x, b)
"""
base = kwds.pop('base', None)
if base is None:
if len(args) == 1:
return GinacFunction.__call__(self, *args, **kwds)
# second argument is base
base = args[1]
args = args[:1]
if len(args) != 1:
raise TypeError("Symbolic function log must be called as "
"log(x), log(x, base=b) or log(x, b)")
try:
return args[0].log(base)
except (AttributeError, TypeError):
return GinacFunction.__call__(self, *args, **kwds) / \
GinacFunction.__call__(self, base, **kwds)
ln = log = function_log = Function_log()
class Function_polylog(GinacFunction):
def __init__(self):
r"""
The polylog function
`\text{Li}_s(z) = \sum_{k=1}^{\infty} z^k / k^s`.
This definition is valid for arbitrary complex order `s` and for
all complex arguments `z` with `|z| < 1`; it can be extended to
`|z| \ge 1` by the process of analytic continuation. So the
function may have a discontinuity at `z=1` which can cause a
`NaN` value returned for floating point arguments.
EXAMPLES::
sage: polylog(2.7, 0)
0
sage: polylog(2, 1)
1/6*pi^2
sage: polylog(2, -1)
-1/12*pi^2
sage: polylog(3, -1)
-3/4*zeta(3)
sage: polylog(2, I)
I*catalan - 1/48*pi^2
sage: polylog(4, 1/2)
polylog(4, 1/2)
sage: polylog(4, 0.5)
0.517479061673899
sage: polylog(1, x)
-log(-x + 1)
sage: polylog(2,x^2+1)
polylog(2, x^2 + 1)
sage: f = polylog(4, 1); f
1/90*pi^4
sage: f.n()
1.08232323371114
sage: polylog(4, 2).n()
2.42786280675470 - 0.174371300025453*I
sage: complex(polylog(4,2))
(2.4278628067547032-0.17437130002545306j)
sage: float(polylog(4,0.5))
0.5174790616738993
sage: z = var('z')
sage: polylog(2,z).series(z==0, 5)
1*z + 1/4*z^2 + 1/9*z^3 + 1/16*z^4 + Order(z^5)
sage: loads(dumps(polylog))
polylog
sage: latex(polylog(5, x))
{\rm Li}_{5}(x)
TESTS:
Check if :trac:`8459` is fixed::
sage: t = maxima(polylog(5,x)).sage(); t
polylog(5, x)
sage: t.operator() == polylog
True
sage: t.subs(x=.5).n()
0.50840057924226...
Check if :trac:`18386` is fixed::
sage: polylog(2.0, 1)
1.64493406684823
sage: polylog(2, 1.0)
NaN - NaN*I
sage: polylog(2.0, 1.0)
NaN - NaN*I
"""
GinacFunction.__init__(self, "polylog", nargs=2)
def _maxima_init_evaled_(self, *args):
"""
EXAMPLES:
These are indirect doctests for this function.::
sage: polylog(2, x)._maxima_()
li[2](_SAGE_VAR_x)
sage: polylog(4, x)._maxima_()
polylog(4,_SAGE_VAR_x)
"""
args_maxima = []
for a in args:
if isinstance(a, str):
args_maxima.append(a)
elif hasattr(a, '_maxima_init_'):
args_maxima.append(a._maxima_init_())
else:
args_maxima.append(str(a))
n, x = args_maxima
if int(n) in [1,2,3]:
return 'li[%s](%s)'%(n, x)
else:
return 'polylog(%s, %s)'%(n, x)
polylog = Function_polylog()
class Function_dilog(GinacFunction):
def __init__(self):
r"""
The dilogarithm function
`\text{Li}_2(z) = \sum_{k=1}^{\infty} z^k / k^2`.
This is simply an alias for polylog(2, z).
EXAMPLES::
sage: dilog(1)
1/6*pi^2
sage: dilog(1/2)
1/12*pi^2 - 1/2*log(2)^2
sage: dilog(x^2+1)
dilog(x^2 + 1)
sage: dilog(-1)
-1/12*pi^2
sage: dilog(-1.1)
-0.890838090262283
sage: float(dilog(1))
1.6449340668482262
sage: var('z')
z
sage: dilog(z).diff(z, 2)
log(-z + 1)/z^2 - 1/((z - 1)*z)
sage: dilog(z).series(z==1/2, 3)
(1/12*pi^2 - 1/2*log(2)^2) + (-2*log(1/2))*(z - 1/2) + (2*log(1/2) + 2)*(z - 1/2)^2 + Order(1/8*(2*z - 1)^3)
sage: latex(dilog(z))
{\rm Li}_2\left(z\right)
TESTS:
``conjugate(dilog(x))==dilog(conjugate(x))`` unless on the branch cuts
which run along the positive real axis beginning at 1.::
sage: conjugate(dilog(x))
conjugate(dilog(x))
sage: var('y',domain='positive')
y
sage: conjugate(dilog(y))
conjugate(dilog(y))
sage: conjugate(dilog(1/19))
dilog(1/19)
sage: conjugate(dilog(1/2*I))
dilog(-1/2*I)
sage: dilog(conjugate(1/2*I))
dilog(-1/2*I)
sage: conjugate(dilog(2))
conjugate(dilog(2))
"""
GinacFunction.__init__(self, 'dilog',
conversions=dict(maxima='li[2]'))
dilog = Function_dilog()
class Function_lambert_w(BuiltinFunction):
r"""
The integral branches of the Lambert W function `W_n(z)`.
This function satisfies the equation
.. math::
z = W_n(z) e^{W_n(z)}
INPUT:
- ``n`` - an integer. `n=0` corresponds to the principal branch.
- ``z`` - a complex number
If called with a single argument, that argument is ``z`` and the branch ``n`` is
assumed to be 0 (the principal branch).
ALGORITHM:
Numerical evaluation is handled using the mpmath and SciPy libraries.
REFERENCES:
- :wikipedia:`Lambert_W_function`
EXAMPLES:
Evaluation of the principal branch::
sage: lambert_w(1.0)
0.567143290409784
sage: lambert_w(-1).n()
-0.318131505204764 + 1.33723570143069*I
sage: lambert_w(-1.5 + 5*I)
1.17418016254171 + 1.10651494102011*I
Evaluation of other branches::
sage: lambert_w(2, 1.0)
-2.40158510486800 + 10.7762995161151*I
Solutions to certain exponential equations are returned in terms of lambert_w::
sage: S = solve(e^(5*x)+x==0, x, to_poly_solve=True)
sage: z = S[0].rhs(); z
-1/5*lambert_w(5)
sage: N(z)
-0.265344933048440
Check the defining equation numerically at `z=5`::
sage: N(lambert_w(5)*exp(lambert_w(5)) - 5)
0.000000000000000
There are several special values of the principal branch which
are automatically simplified::
sage: lambert_w(0)
0
sage: lambert_w(e)
1
sage: lambert_w(-1/e)
-1
Integration (of the principal branch) is evaluated using Maxima::
sage: integrate(lambert_w(x), x)
(lambert_w(x)^2 - lambert_w(x) + 1)*x/lambert_w(x)
sage: integrate(lambert_w(x), x, 0, 1)
(lambert_w(1)^2 - lambert_w(1) + 1)/lambert_w(1) - 1
sage: integrate(lambert_w(x), x, 0, 1.0)
0.3303661247616807
Warning: The integral of a non-principal branch is not implemented,
neither is numerical integration using GSL. The :meth:`numerical_integral`
function does work if you pass a lambda function::
sage: numerical_integral(lambda x: lambert_w(x), 0, 1)
(0.33036612476168054, 3.667800782666048e-15)
"""
def __init__(self):
r"""
See the docstring for :meth:`Function_lambert_w`.
EXAMPLES::
sage: lambert_w(0, 1.0)
0.567143290409784
"""
BuiltinFunction.__init__(self, "lambert_w", nargs=2,
conversions={'mathematica': 'ProductLog',
'maple': 'LambertW',
'matlab': 'lambertw',
'maxima': 'generalized_lambert_w'})
def __call__(self, *args, **kwds):
r"""
Custom call method allows the user to pass one argument or two. If
one argument is passed, we assume it is ``z`` and that ``n=0``.
EXAMPLES::
sage: lambert_w(1)
lambert_w(1)
sage: lambert_w(1, 2)
lambert_w(1, 2)
"""
if len(args) == 2:
return BuiltinFunction.__call__(self, *args, **kwds)
elif len(args) == 1:
return BuiltinFunction.__call__(self, 0, args[0], **kwds)
else:
raise TypeError("lambert_w takes either one or two arguments.")
def _eval_(self, n, z):
"""
EXAMPLES::
sage: lambert_w(6.0)
1.43240477589830
sage: lambert_w(1)
lambert_w(1)
sage: lambert_w(x+1)
lambert_w(x + 1)
There are three special values which are automatically simplified::
sage: lambert_w(0)
0
sage: lambert_w(e)
1
sage: lambert_w(-1/e)
-1
sage: lambert_w(SR(0))
0
The special values only hold on the principal branch::
sage: lambert_w(1,e)
lambert_w(1, e)
sage: lambert_w(1, e.n())
-0.532092121986380 + 4.59715801330257*I
TESTS:
When automatic simplication occurs, the parent of the output value should be
either the same as the parent of the input, or a Sage type::
sage: parent(lambert_w(int(0)))
<type 'int'>
sage: parent(lambert_w(Integer(0)))
Integer Ring
sage: parent(lambert_w(e))
Symbolic Ring
"""
if not isinstance(z, Expression):
if n == 0 and z == 0:
return s_parent(z)(0)
elif n == 0:
if z.is_trivial_zero():
return s_parent(z)(Integer(0))
elif (z-const_e).is_trivial_zero():
return s_parent(z)(Integer(1))
elif (z+1/const_e).is_trivial_zero():
return s_parent(z)(Integer(-1))
def _evalf_(self, n, z, parent=None, algorithm=None):
"""
EXAMPLES::
sage: N(lambert_w(1))
0.567143290409784
sage: lambert_w(RealField(100)(1))
0.56714329040978387299996866221
SciPy is used to evaluate for float, RDF, and CDF inputs::
sage: lambert_w(RDF(1))
0.5671432904097838
sage: lambert_w(float(1))
0.5671432904097838
sage: lambert_w(CDF(1))
0.5671432904097838
sage: lambert_w(complex(1))
(0.5671432904097838+0j)
sage: lambert_w(RDF(-1)) # abs tol 2e-16
-0.31813150520476413 + 1.3372357014306895*I
sage: lambert_w(float(-1)) # abs tol 2e-16
(-0.31813150520476413+1.3372357014306895j)
"""
R = parent or s_parent(z)
if R is float or R is RDF:
from scipy.special import lambertw
res = lambertw(z, n)
# SciPy always returns a complex value, make it real if possible
if not res.imag:
return R(res.real)
elif R is float:
return complex(res)
else:
return CDF(res)
elif R is complex or R is CDF:
from scipy.special import lambertw
return R(lambertw(z, n))
else:
import mpmath
return mpmath_utils.call(mpmath.lambertw, z, n, parent=R)
def _derivative_(self, n, z, diff_param=None):
"""
The derivative of `W_n(x)` is `W_n(x)/(x \cdot W_n(x) + x)`.
EXAMPLES::
sage: x = var('x')
sage: derivative(lambert_w(x), x)
lambert_w(x)/(x*lambert_w(x) + x)
sage: derivative(lambert_w(2, exp(x)), x)
e^x*lambert_w(2, e^x)/(e^x*lambert_w(2, e^x) + e^x)
TESTS:
Differentiation in the first parameter raises an error :trac:`14788`::
sage: n = var('n')
sage: lambert_w(n, x).diff(n)
Traceback (most recent call last):
...
ValueError: cannot differentiate lambert_w in the first parameter
"""
if diff_param == 0:
raise ValueError("cannot differentiate lambert_w in the first parameter")
return lambert_w(n, z)/(z*lambert_w(n, z)+z)
def _maxima_init_evaled_(self, n, z):
"""
EXAMPLES:
These are indirect doctests for this function.::
sage: lambert_w(0, x)._maxima_()
lambert_w(_SAGE_VAR_x)
sage: lambert_w(1, x)._maxima_()
generalized_lambert_w(1,_SAGE_VAR_x)
TESTS::
sage: lambert_w(x)._maxima_()._sage_()
lambert_w(x)
sage: lambert_w(2, x)._maxima_()._sage_()
lambert_w(2, x)
"""
if isinstance(z, str):
maxima_z = z
elif hasattr(z, '_maxima_init_'):
maxima_z = z._maxima_init_()
else:
maxima_z = str(z)
if n == 0:
return "lambert_w(%s)" % maxima_z
else:
return "generalized_lambert_w(%s,%s)" % (n, maxima_z)
def _print_(self, n, z):
"""
Custom _print_ method to avoid printing the branch number if
it is zero.
EXAMPLES::
sage: lambert_w(1)
lambert_w(1)
sage: lambert_w(0,x)
lambert_w(x)
"""
if n == 0:
return "lambert_w(%s)" % z
else:
return "lambert_w(%s, %s)" % (n, z)
def _print_latex_(self, n, z):
"""
Custom _print_latex_ method to avoid printing the branch
number if it is zero.
EXAMPLES::
sage: latex(lambert_w(1))
\operatorname{W}({1})
sage: latex(lambert_w(0,x))
\operatorname{W}({x})
sage: latex(lambert_w(1,x))
\operatorname{W_{1}}({x})
sage: latex(lambert_w(1,x+exp(x)))
\operatorname{W_{1}}({x + e^{x}})
"""
if n == 0:
return r"\operatorname{W}({%s})" % z._latex_()
else:
return r"\operatorname{W_{%s}}({%s})" % (n, z._latex_())
lambert_w = Function_lambert_w()
class Function_exp_polar(BuiltinFunction):
def __init__(self):
r"""
Representation of a complex number in a polar form.
INPUT:
- ``z`` - a complex number `z = a + ib`.
OUTPUT:
A complex number with modulus `\exp(a)` and argument `b`.
If `-\pi < b \leq \pi` then `\operatorname{exp\_polar}(z)=\exp(z)`.
For other values of `b` the function is left unevaluated.
EXAMPLES:
The following expressions are evaluated using the exponential
function::
sage: exp_polar(pi*I/2)
I
sage: x = var('x', domain='real')
sage: exp_polar(-1/2*I*pi + x)
e^(-1/2*I*pi + x)
The function is left unevaluated when the imaginary part of the
input `z` does not satisfy `-\pi < \Im(z) \leq \pi`::
sage: exp_polar(2*pi*I)
exp_polar(2*I*pi)
sage: exp_polar(-4*pi*I)
exp_polar(-4*I*pi)
This fixes :trac:`18085`::
sage: integrate(1/sqrt(1+x^3),x,algorithm='sympy')
1/3*x*hypergeometric((1/3, 1/2), (4/3,), -x^3)*gamma(1/3)/gamma(4/3)
SEEALSO:
`Examples in Sympy documentation <http://docs.sympy.org/latest/modules/functions/special.html?highlight=exp_polar>`_,
`Sympy source code of exp_polar <http://docs.sympy.org/0.7.4/_modules/sympy/functions/elementary/exponential.html>`_
REFERENCES:
:wikipedia:`Complex_number#Polar_form`
"""
BuiltinFunction.__init__(self, "exp_polar",
latex_name=r"\operatorname{exp\_polar}",
conversions=dict(sympy='exp_polar'))
def _evalf_(self, z, parent=None, algorithm=None):
r"""
EXAMPLES:
If the imaginary part of `z` obeys `-\pi < z \leq \pi`, then
`\operatorname{exp\_polar}(z)` is evaluated as `\exp(z)`::
sage: exp_polar(1.0 + 2.0*I)
-1.13120438375681 + 2.47172667200482*I
If the imaginary part of `z` is outside of that interval the
expression is left unevaluated::
sage: exp_polar(-5.0 + 8.0*I)
exp_polar(-5.00000000000000 + 8.00000000000000*I)
An attempt to numerically evaluate such an expression raises an error::
sage: exp_polar(-5.0 + 8.0*I).n()
Traceback (most recent call last):
...
ValueError: invalid attempt to numerically evaluate exp_polar()
"""
from sage.functions.other import imag
if (not isinstance(z, Expression)
and bool(-const_pi < imag(z) <= const_pi)):
return exp(z)
else:
raise ValueError("invalid attempt to numerically evaluate exp_polar()")
def _eval_(self, z):
"""
EXAMPLES::
sage: exp_polar(3*I*pi)
exp_polar(3*I*pi)
sage: x = var('x', domain='real')
sage: exp_polar(4*I*pi + x)
exp_polar(4*I*pi + x)
"""
if (isinstance(z, Expression)
and bool(-const_pi < z.imag_part() <= const_pi)):
return exp(z)
else:
return None
exp_polar = Function_exp_polar()
class Function_harmonic_number_generalized(BuiltinFunction):
r"""
Harmonic and generalized harmonic number functions,
defined by:
.. math::
H_{n}=H_{n,1}=\sum_{k=1}^n\frac1k
H_{n,m}=\sum_{k=1}^n\frac1{k^m}
They are also well-defined for complex argument, through:
.. math::
H_{s}=\int_0^1\frac{1-x^s}{1-x}
H_{s,m}=\zeta(m)-\zeta(m,s-1)
If called with a single argument, that argument is ``s`` and ``m`` is
assumed to be 1 (the normal harmonic numbers ``H_s``).
ALGORITHM:
Numerical evaluation is handled using the mpmath and FLINT libraries.
REFERENCES:
- :wikipedia:`/Harmonic_number`
EXAMPLES:
Evaluation of integer, rational, or complex argument::
sage: harmonic_number(5)
137/60
sage: harmonic_number(3,3)
251/216
sage: harmonic_number(5/2)
-2*log(2) + 46/15
sage: harmonic_number(3.,3)
zeta(3) - 0.0400198661225573
sage: harmonic_number(3.,3.)
1.16203703703704
sage: harmonic_number(3,3).n(200)
1.16203703703703703703703...
sage: harmonic_number(1+I,5)
harmonic_number(I + 1, 5)
sage: harmonic_number(5,1.+I)
1.57436810798989 - 1.06194728851357*I
Solutions to certain sums are returned in terms of harmonic numbers::
sage: k=var('k')
sage: sum(1/k^7,k,1,x)
harmonic_number(x, 7)
Check the defining integral at a random integer::
sage: n=randint(10,100)
sage: bool(SR(integrate((1-x^n)/(1-x),x,0,1)) == harmonic_number(n))
True
There are several special values which are automatically simplified::
sage: harmonic_number(0)
0
sage: harmonic_number(1)
1
sage: harmonic_number(x,1)
harmonic_number(x)
Arguments are swapped with respect to the same functions in
Maxima::
sage: maxima(harmonic_number(x,2)) # maxima expect interface