Logarithmic functions#
AUTHORS:
Yoora Yi Tenen (2012-11-16): Add documentation for
log()(Issue #12113)Tomas Kalvoda (2015-04-01): Add
exp_polar()(Issue #18085)
- class sage.functions.log.Function_dilog[source]#
Bases:
GinacFunctionThe 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: # needs sage.symbolic 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(-1.0) -0.822467033424113 sage: dilog(-1.1) -0.890838090262283 sage: dilog(1/2) 1/12*pi^2 - 1/2*log(2)^2 sage: dilog(.5) 0.582240526465012 sage: dilog(1/2).n() 0.582240526465012 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)) # needs sage.symbolic {\rm Li}_2\left(z\right)
>>> from sage.all import * >>> # needs sage.symbolic >>> dilog(Integer(1)) 1/6*pi^2 >>> dilog(Integer(1)/Integer(2)) 1/12*pi^2 - 1/2*log(2)^2 >>> dilog(x**Integer(2)+Integer(1)) dilog(x^2 + 1) >>> dilog(-Integer(1)) -1/12*pi^2 >>> dilog(-RealNumber('1.0')) -0.822467033424113 >>> dilog(-RealNumber('1.1')) -0.890838090262283 >>> dilog(Integer(1)/Integer(2)) 1/12*pi^2 - 1/2*log(2)^2 >>> dilog(RealNumber('.5')) 0.582240526465012 >>> dilog(Integer(1)/Integer(2)).n() 0.582240526465012 >>> var('z') z >>> dilog(z).diff(z, Integer(2)) log(-z + 1)/z^2 - 1/((z - 1)*z) >>> dilog(z).series(z==Integer(1)/Integer(2), Integer(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) >>> latex(dilog(z)) # needs sage.symbolic {\rm Li}_2\left(z\right)
Dilog has a branch point at \(1\). Sage’s floating point libraries may handle this differently from the symbolic package:
sage: # needs sage.symbolic sage: dilog(1) 1/6*pi^2 sage: dilog(1.) 1.64493406684823 sage: dilog(1).n() 1.64493406684823 sage: float(dilog(1)) 1.6449340668482262
>>> from sage.all import * >>> # needs sage.symbolic >>> dilog(Integer(1)) 1/6*pi^2 >>> dilog(RealNumber('1.')) 1.64493406684823 >>> dilog(Integer(1)).n() 1.64493406684823 >>> float(dilog(Integer(1))) 1.6449340668482262
- class sage.functions.log.Function_exp[source]#
Bases:
GinacFunctionThe exponential function, \(\exp(x) = e^x\).
EXAMPLES:
sage: # needs sage.symbolic 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(I*pi/12) (1/4*I + 1/4)*sqrt(6) - (1/4*I - 1/4)*sqrt(2) sage: exp(float(2.5)) 12.182493960703473 sage: exp(RDF('2.5')) # needs sage.symbolic 12.182493960703473
>>> from sage.all import * >>> # needs sage.symbolic >>> exp(-Integer(1)) e^(-1) >>> exp(Integer(2)) e^2 >>> exp(Integer(2)).n(Integer(100)) 7.3890560989306502272304274606 >>> exp(x**Integer(2) + log(x)) e^(x^2 + log(x)) >>> exp(x**Integer(2) + log(x)).simplify() x*e^(x^2) >>> exp(RealNumber('2.5')) 12.1824939607035 >>> exp(I*pi/Integer(12)) (1/4*I + 1/4)*sqrt(6) - (1/4*I - 1/4)*sqrt(2) >>> exp(float(RealNumber('2.5'))) 12.182493960703473 >>> exp(RDF('2.5')) # needs sage.symbolic 12.182493960703473
To prevent automatic evaluation, use the
holdparameter:sage: exp(I*pi, hold=True) # needs sage.symbolic e^(I*pi) sage: exp(0, hold=True) # needs sage.symbolic e^0
>>> from sage.all import * >>> exp(I*pi, hold=True) # needs sage.symbolic e^(I*pi) >>> exp(Integer(0), hold=True) # needs sage.symbolic e^0
To then evaluate again, we currently must use Maxima via
sage.symbolic.expression.Expression.simplify():sage: exp(0, hold=True).simplify() # needs sage.symbolic 1
>>> from sage.all import * >>> exp(Integer(0), hold=True).simplify() # needs sage.symbolic 1
sage: # needs sage.symbolic sage: exp(pi*I/2) I sage: exp(pi*I) -1 sage: exp(8*pi*I) 1 sage: exp(7*pi*I/2) -I
>>> from sage.all import * >>> # needs sage.symbolic >>> exp(pi*I/Integer(2)) I >>> exp(pi*I) -1 >>> exp(Integer(8)*pi*I) 1 >>> exp(Integer(7)*pi*I/Integer(2)) -I
For the sake of simplification, the argument is reduced modulo the period of the complex exponential function, \(2\pi i\):
sage: k = var('k', domain='integer') # needs sage.symbolic sage: exp(2*k*pi*I) # needs sage.symbolic 1 sage: exp(log(2) + 2*k*pi*I) # needs sage.symbolic 2
>>> from sage.all import * >>> k = var('k', domain='integer') # needs sage.symbolic >>> exp(Integer(2)*k*pi*I) # needs sage.symbolic 1 >>> exp(log(Integer(2)) + Integer(2)*k*pi*I) # needs sage.symbolic 2
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 # needs sage.rings.real_mpfr 7.3890560989306502272304274606 sage: t.prec() # needs sage.rings.real_mpfr 100 sage: exp(2).n(100) # needs sage.symbolic 7.3890560989306502272304274606
>>> from sage.all import * >>> t = exp(RealField(Integer(100))(Integer(2))); t # needs sage.rings.real_mpfr 7.3890560989306502272304274606 >>> t.prec() # needs sage.rings.real_mpfr 100 >>> exp(Integer(2)).n(Integer(100)) # needs sage.symbolic 7.3890560989306502272304274606
- class sage.functions.log.Function_exp_polar[source]#
Bases:
BuiltinFunctionRepresentation 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) # needs sage.symbolic I sage: x = var('x', domain='real') # needs sage.symbolic sage: exp_polar(-1/2*I*pi + x) # needs sage.symbolic e^(-1/2*I*pi + x)
>>> from sage.all import * >>> exp_polar(pi*I/Integer(2)) # needs sage.symbolic I >>> x = var('x', domain='real') # needs sage.symbolic >>> exp_polar(-Integer(1)/Integer(2)*I*pi + x) # needs sage.symbolic 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) # needs sage.symbolic exp_polar(2*I*pi) sage: exp_polar(-4*pi*I) # needs sage.symbolic exp_polar(-4*I*pi)
>>> from sage.all import * >>> exp_polar(Integer(2)*pi*I) # needs sage.symbolic exp_polar(2*I*pi) >>> exp_polar(-Integer(4)*pi*I) # needs sage.symbolic exp_polar(-4*I*pi)
This fixes Issue #18085:
sage: integrate(1/sqrt(1+x^3), x, algorithm='sympy') # needs sage.symbolic 1/3*x*gamma(1/3)*hypergeometric((1/3, 1/2), (4/3,), -x^3)/gamma(4/3)
>>> from sage.all import * >>> integrate(Integer(1)/sqrt(Integer(1)+x**Integer(3)), x, algorithm='sympy') # needs sage.symbolic 1/3*x*gamma(1/3)*hypergeometric((1/3, 1/2), (4/3,), -x^3)/gamma(4/3)
REFERENCES:
- class sage.functions.log.Function_harmonic_number[source]#
Bases:
BuiltinFunctionHarmonic number function, defined by:
\[ \begin{align}\begin{aligned}H_{n}=H_{n,1}=\sum_{k=1}^n\frac1k\\H_{s}=\int_0^1\frac{1-x^s}{1-x}\end{aligned}\end{align} \]See the docstring for
Function_harmonic_number_generalized().This class exists as callback for
harmonic_numberreturned by Maxima.
- class sage.functions.log.Function_harmonic_number_generalized[source]#
Bases:
BuiltinFunctionHarmonic and generalized harmonic number functions, defined by:
\[ \begin{align}\begin{aligned}H_{n}=H_{n,1}=\sum_{k=1}^n\frac{1}{k}\\H_{n,m}=\sum_{k=1}^n\frac{1}{k^m}\end{aligned}\end{align} \]They are also well-defined for complex argument, through:
\[ \begin{align}\begin{aligned}H_{s}=\int_0^1\frac{1-x^s}{1-x}\\H_{s,m}=\zeta(m)-\zeta(m,s-1)\end{aligned}\end{align} \]If called with a single argument, that argument is
sandmis assumed to be 1 (the normal harmonic numbers \(H_s\)).ALGORITHM:
Numerical evaluation is handled using the mpmath and FLINT libraries.
REFERENCES:
EXAMPLES:
Evaluation of integer, rational, or complex argument:
sage: harmonic_number(5) # needs mpmath 137/60 sage: # needs sage.symbolic 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
>>> from sage.all import * >>> harmonic_number(Integer(5)) # needs mpmath 137/60 >>> # needs sage.symbolic >>> harmonic_number(Integer(3), Integer(3)) 251/216 >>> harmonic_number(Integer(5)/Integer(2)) -2*log(2) + 46/15 >>> harmonic_number(RealNumber('3.'), Integer(3)) zeta(3) - 0.0400198661225573 >>> harmonic_number(RealNumber('3.'), RealNumber('3.')) 1.16203703703704 >>> harmonic_number(Integer(3), Integer(3)).n(Integer(200)) 1.16203703703703703703703... >>> harmonic_number(Integer(1) + I, Integer(5)) harmonic_number(I + 1, 5) >>> harmonic_number(Integer(5), RealNumber('1.') + I) 1.57436810798989 - 1.06194728851357*I
Solutions to certain sums are returned in terms of harmonic numbers:
sage: k = var('k') # needs sage.symbolic sage: sum(1/k^7,k,1,x) # needs sage.symbolic harmonic_number(x, 7)
>>> from sage.all import * >>> k = var('k') # needs sage.symbolic >>> sum(Integer(1)/k**Integer(7),k,Integer(1),x) # needs sage.symbolic 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)) # needs sage.symbolic True
>>> from sage.all import * >>> n = randint(Integer(10),Integer(100)) >>> bool(SR(integrate((Integer(1)-x**n)/(Integer(1)-x),x,Integer(0),Integer(1))) == harmonic_number(n)) # needs sage.symbolic True
There are several special values which are automatically simplified:
sage: harmonic_number(0) # needs mpmath 0 sage: harmonic_number(1) # needs mpmath 1 sage: harmonic_number(x, 1) # needs sage.symbolic harmonic_number(x)
>>> from sage.all import * >>> harmonic_number(Integer(0)) # needs mpmath 0 >>> harmonic_number(Integer(1)) # needs mpmath 1 >>> harmonic_number(x, Integer(1)) # needs sage.symbolic harmonic_number(x)
- class sage.functions.log.Function_lambert_w[source]#
Bases:
BuiltinFunctionThe integral branches of the Lambert W function \(W_n(z)\).
This function satisfies the equation
\[z = W_n(z) e^{W_n(z)}\]INPUT:
n– an integer. \(n=0\) corresponds to the principal branch.z– a complex number
If called with a single argument, that argument is
zand the branchnis assumed to be 0 (the principal branch).ALGORITHM:
Numerical evaluation is handled using the mpmath and SciPy libraries.
REFERENCES:
EXAMPLES:
Evaluation of the principal branch:
sage: lambert_w(1.0) # needs scipy 0.567143290409784 sage: lambert_w(-1).n() # needs mpmath -0.318131505204764 + 1.33723570143069*I sage: lambert_w(-1.5 + 5*I) # needs mpmath sage.symbolic 1.17418016254171 + 1.10651494102011*I
>>> from sage.all import * >>> lambert_w(RealNumber('1.0')) # needs scipy 0.567143290409784 >>> lambert_w(-Integer(1)).n() # needs mpmath -0.318131505204764 + 1.33723570143069*I >>> lambert_w(-RealNumber('1.5') + Integer(5)*I) # needs mpmath sage.symbolic 1.17418016254171 + 1.10651494102011*I
Evaluation of other branches:
sage: lambert_w(2, 1.0) # needs scipy -2.40158510486800 + 10.7762995161151*I
>>> from sage.all import * >>> lambert_w(Integer(2), RealNumber('1.0')) # needs scipy -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) # needs sage.symbolic sage: z = S[0].rhs(); z # needs sage.symbolic -1/5*lambert_w(5) sage: N(z) # needs sage.symbolic -0.265344933048440
>>> from sage.all import * >>> S = solve(e**(Integer(5)*x)+x==Integer(0), x, to_poly_solve=True) # needs sage.symbolic >>> z = S[Integer(0)].rhs(); z # needs sage.symbolic -1/5*lambert_w(5) >>> N(z) # needs sage.symbolic -0.265344933048440
Check the defining equation numerically at \(z=5\):
sage: N(lambert_w(5)*exp(lambert_w(5)) - 5) # needs mpmath 0.000000000000000
>>> from sage.all import * >>> N(lambert_w(Integer(5))*exp(lambert_w(Integer(5))) - Integer(5)) # needs mpmath 0.000000000000000
There are several special values of the principal branch which are automatically simplified:
sage: lambert_w(0) # needs mpmath 0 sage: lambert_w(e) # needs sage.symbolic 1 sage: lambert_w(-1/e) # needs sage.symbolic -1
>>> from sage.all import * >>> lambert_w(Integer(0)) # needs mpmath 0 >>> lambert_w(e) # needs sage.symbolic 1 >>> lambert_w(-Integer(1)/e) # needs sage.symbolic -1
Integration (of the principal branch) is evaluated using Maxima:
sage: integrate(lambert_w(x), x) # needs sage.symbolic (lambert_w(x)^2 - lambert_w(x) + 1)*x/lambert_w(x) sage: integrate(lambert_w(x), x, 0, 1) # needs sage.symbolic (lambert_w(1)^2 - lambert_w(1) + 1)/lambert_w(1) - 1 sage: integrate(lambert_w(x), x, 0, 1.0) # needs sage.symbolic 0.3303661247616807
>>> from sage.all import * >>> integrate(lambert_w(x), x) # needs sage.symbolic (lambert_w(x)^2 - lambert_w(x) + 1)*x/lambert_w(x) >>> integrate(lambert_w(x), x, Integer(0), Integer(1)) # needs sage.symbolic (lambert_w(1)^2 - lambert_w(1) + 1)/lambert_w(1) - 1 >>> integrate(lambert_w(x), x, Integer(0), RealNumber('1.0')) # needs sage.symbolic 0.3303661247616807
Warning: The integral of a non-principal branch is not implemented, neither is numerical integration using GSL. The
numerical_integral()function does work if you pass a lambda function:sage: numerical_integral(lambda x: lambert_w(x), 0, 1) # needs sage.modules (0.33036612476168054, 3.667800782666048e-15)
>>> from sage.all import * >>> numerical_integral(lambda x: lambert_w(x), Integer(0), Integer(1)) # needs sage.modules (0.33036612476168054, 3.667800782666048e-15)
- class sage.functions.log.Function_log1[source]#
Bases:
GinacFunctionThe natural logarithm of
x.See
log()for extensive documentation.EXAMPLES:
sage: ln(e^2) # needs sage.symbolic 2 sage: ln(2) # needs sage.symbolic log(2) sage: ln(10) # needs sage.symbolic log(10)
>>> from sage.all import * >>> ln(e**Integer(2)) # needs sage.symbolic 2 >>> ln(Integer(2)) # needs sage.symbolic log(2) >>> ln(Integer(10)) # needs sage.symbolic log(10)
- class sage.functions.log.Function_log2[source]#
Bases:
GinacFunctionReturn the logarithm of x to the given base.
See
log()for extensive documentation.EXAMPLES:
sage: from sage.functions.log import logb sage: logb(1000, 10) # needs sage.symbolic 3
>>> from sage.all import * >>> from sage.functions.log import logb >>> logb(Integer(1000), Integer(10)) # needs sage.symbolic 3
- class sage.functions.log.Function_polylog[source]#
Bases:
GinacFunctionThe polylog function \(\text{Li}_s(z) = \sum_{k=1}^{\infty} z^k / k^s\).
The first argument is \(s\) (usually an integer called the weight) and the second argument is \(z\):
polylog(s, z).This definition is valid for arbitrary complex numbers \(s\) and \(z\) with \(|z| < 1\). It can be extended to \(|z| \ge 1\) by the process of analytic continuation, with a branch cut along the positive real axis from \(1\) to \(+\infty\). A
NaNvalue may be returned for floating point arguments that are on the branch cut.EXAMPLES:
sage: # needs sage.symbolic sage: polylog(2.7, 0) 0.000000000000000 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: # needs sage.symbolic sage: polylog(1, x) -log(-x + 1) sage: polylog(2, x^2 + 1) dilog(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)) # needs sage.symbolic {\rm Li}_{5}(x) sage: polylog(x, x)._sympy_() # needs sympy sage.symbolic polylog(x, x)
>>> from sage.all import * >>> # needs sage.symbolic >>> polylog(RealNumber('2.7'), Integer(0)) 0.000000000000000 >>> polylog(Integer(2), Integer(1)) 1/6*pi^2 >>> polylog(Integer(2), -Integer(1)) -1/12*pi^2 >>> polylog(Integer(3), -Integer(1)) -3/4*zeta(3) >>> polylog(Integer(2), I) I*catalan - 1/48*pi^2 >>> polylog(Integer(4), Integer(1)/Integer(2)) polylog(4, 1/2) >>> polylog(Integer(4), RealNumber('0.5')) 0.517479061673899 >>> # needs sage.symbolic >>> polylog(Integer(1), x) -log(-x + 1) >>> polylog(Integer(2), x**Integer(2) + Integer(1)) dilog(x^2 + 1) >>> f = polylog(Integer(4), Integer(1)); f 1/90*pi^4 >>> f.n() 1.08232323371114 >>> polylog(Integer(4), Integer(2)).n() 2.42786280675470 - 0.174371300025453*I >>> complex(polylog(Integer(4), Integer(2))) (2.4278628067547032-0.17437130002545306j) >>> float(polylog(Integer(4), RealNumber('0.5'))) 0.5174790616738993 >>> z = var('z') >>> polylog(Integer(2), z).series(z==Integer(0), Integer(5)) 1*z + 1/4*z^2 + 1/9*z^3 + 1/16*z^4 + Order(z^5) >>> loads(dumps(polylog)) polylog >>> latex(polylog(Integer(5), x)) # needs sage.symbolic {\rm Li}_{5}(x) >>> polylog(x, x)._sympy_() # needs sympy sage.symbolic polylog(x, x)