Other functions

class sage.functions.other.Function_abs

Bases: sage.symbolic.function.GinacFunction

The absolute value function.

EXAMPLES:

sage: var('x y')
(x, y)
sage: abs(x)
abs(x)
sage: abs(x^2 + y^2)
abs(x^2 + y^2)
sage: abs(-2)
2
sage: sqrt(x^2)
sqrt(x^2)
sage: abs(sqrt(x))
abs(sqrt(x))
sage: complex(abs(3*I))
(3+0j)

sage: f = sage.functions.other.Function_abs()
sage: latex(f)
\mathrm{abs}
sage: latex(abs(x))
{\left| x \right|}
sage: abs(x)._sympy_()
Abs(x)

Test pickling:

sage: loads(dumps(abs(x)))
abs(x)
class sage.functions.other.Function_arg

Bases: sage.symbolic.function.BuiltinFunction

The argument function for complex numbers.

EXAMPLES:

sage: arg(3+i)
arctan(1/3)
sage: arg(-1+i)
3/4*pi
sage: arg(2+2*i)
1/4*pi
sage: arg(2+x)
arg(x + 2)
sage: arg(2.0+i+x)
arg(x + 2.00000000000000 + 1.00000000000000*I)
sage: arg(-3)
pi
sage: arg(3)
0
sage: arg(0)
0
sage: latex(arg(x))
{\rm arg}\left(x\right)
sage: maxima(arg(x))
atan2(0,_SAGE_VAR_x)
sage: maxima(arg(2+i))
atan(1/2)
sage: maxima(arg(sqrt(2)+i))
atan(1/sqrt(2))
sage: arg(2+i)
arctan(1/2)
sage: arg(sqrt(2)+i)
arg(sqrt(2) + I)
sage: arg(sqrt(2)+i).simplify()
arctan(1/2*sqrt(2))

TESTS:

sage: arg(0.0)
0.000000000000000
sage: arg(3.0)
0.000000000000000
sage: arg(-2.5)
3.14159265358979
sage: arg(2.0+3*i)
0.982793723247329
class sage.functions.other.Function_beta

Bases: sage.symbolic.function.GinacFunction

Return the beta function. This is defined by

\[B(p,q) = \int_0^1 t^{p-1}(1-t)^{1-q} dt\]

for complex or symbolic input \(p\) and \(q\). Note that the order of inputs does not matter: \(B(p,q)=B(q,p)\).

GiNaC is used to compute \(B(p,q)\). However, complex inputs are not yet handled in general. When GiNaC raises an error on such inputs, we raise a NotImplementedError.

If either input is 1, GiNaC returns the reciprocal of the other. In other cases, GiNaC uses one of the following formulas:

\[B(p,q) = \Gamma(p)\Gamma(q)/\Gamma(p+q)\]

or

\[B(p,q) = (-1)^q B(1-p-q, q).\]

For numerical inputs, GiNaC uses the formula

\[B(p,q) = \exp[\log\Gamma(p)+\log\Gamma(q)-\log\Gamma(p+q)]\]

INPUT:

  • p - number or symbolic expression
  • q - number or symbolic expression

OUTPUT: number or symbolic expression (if input is symbolic)

EXAMPLES:

sage: beta(3,2)
1/12
sage: beta(3,1)
1/3
sage: beta(1/2,1/2)
beta(1/2, 1/2)
sage: beta(-1,1)
-1
sage: beta(-1/2,-1/2)
0
sage: beta(x/2,3)
beta(3, 1/2*x)
sage: beta(.5,.5)
3.14159265358979
sage: beta(1,2.0+I)
0.400000000000000 - 0.200000000000000*I
sage: beta(3,x+I)
beta(3, x + I)

Note that the order of arguments does not matter:

sage: beta(1/2,3*x)
beta(1/2, 3*x)

The result is symbolic if exact input is given:

sage: beta(2,1+5*I)
beta(2, 5*I + 1)
sage: beta(2, 2.)
0.166666666666667
sage: beta(I, 2.)
-0.500000000000000 - 0.500000000000000*I
sage: beta(2., 2)
0.166666666666667
sage: beta(2., I)
-0.500000000000000 - 0.500000000000000*I

Test pickling:

sage: loads(dumps(beta))
beta
class sage.functions.other.Function_binomial

Bases: sage.symbolic.function.GinacFunction

Return the binomial coefficient

\[\binom{x}{m} = x (x-1) \cdots (x-m+1) / m!\]

which is defined for \(m \in \ZZ\) and any \(x\). We extend this definition to include cases when \(x-m\) is an integer but \(m\) is not by

\[\binom{x}{m}= \binom{x}{x-m}\]

If \(m < 0\), return \(0\).

INPUT:

  • x, m - numbers or symbolic expressions. Either m or x-m must be an integer, else the output is symbolic.

OUTPUT: number or symbolic expression (if input is symbolic)

EXAMPLES:

sage: binomial(5,2)
10
sage: binomial(2,0)
1
sage: binomial(1/2, 0)
1
sage: binomial(3,-1)
0
sage: binomial(20,10)
184756
sage: binomial(-2, 5)
-6
sage: binomial(RealField()('2.5'), 2)
1.87500000000000
sage: n=var('n'); binomial(n,2)
1/2*(n - 1)*n
sage: n=var('n'); binomial(n,n)
1
sage: n=var('n'); binomial(n,n-1)
n
sage: binomial(2^100, 2^100)
1
sage: k, i = var('k,i')
sage: binomial(k,i)
binomial(k, i)

We can use a hold parameter to prevent automatic evaluation:

sage: SR(5).binomial(3, hold=True)
binomial(5, 3)
sage: SR(5).binomial(3, hold=True).simplify()
10

TESTS: We verify that we can convert this function to Maxima and bring it back into Sage.

sage: n,k = var('n,k')
sage: maxima(binomial(n,k))
binomial(_SAGE_VAR_n,_SAGE_VAR_k)
sage: _.sage()
binomial(n, k)
sage: binomial._maxima_init_()
'binomial'

Test pickling:

sage: loads(dumps(binomial(n,k)))
binomial(n, k)
class sage.functions.other.Function_ceil

Bases: sage.symbolic.function.BuiltinFunction

The ceiling function.

The ceiling of \(x\) is computed in the following manner.

  1. The x.ceil() method is called and returned if it is there. If it is not, then Sage checks if \(x\) is one of Python’s native numeric data types. If so, then it calls and returns Integer(int(math.ceil(x))).
  2. Sage tries to convert \(x\) into a RealIntervalField with 53 bits of precision. Next, the ceilings of the endpoints are computed. If they are the same, then that value is returned. Otherwise, the precision of the RealIntervalField is increased until they do match up or it reaches maximum_bits of precision.
  3. If none of the above work, Sage returns a Expression object.

EXAMPLES:

sage: a = ceil(2/5 + x)
sage: a
ceil(x + 2/5)
sage: a(x=4)
5
sage: a(x=4.0)
5
sage: ZZ(a(x=3))
4
sage: a = ceil(x^3 + x + 5/2); a
ceil(x^3 + x + 5/2)
sage: a.simplify()
ceil(x^3 + x + 1/2) + 2
sage: a(x=2)
13
sage: ceil(sin(8)/sin(2))
2
sage: ceil(5.4)
6
sage: type(ceil(5.4))
<type 'sage.rings.integer.Integer'>
sage: ceil(factorial(50)/exp(1))
11188719610782480504630258070757734324011354208865721592720336801
sage: ceil(SR(10^50 + 10^(-50)))
100000000000000000000000000000000000000000000000001
sage: ceil(SR(10^50 - 10^(-50)))
100000000000000000000000000000000000000000000000000

sage: ceil(sec(e))
-1

sage: latex(ceil(x))
\left \lceil x \right \rceil
sage: ceil(x)._sympy_()
ceiling(x)
sage: import numpy
sage: a = numpy.linspace(0,2,6)
sage: ceil(a)
array([ 0.,  1.,  1.,  2.,  2.,  2.])

Test pickling:

sage: loads(dumps(ceil))
ceil
class sage.functions.other.Function_conjugate

Bases: sage.symbolic.function.GinacFunction

Returns the complex conjugate of the input.

It is possible to prevent automatic evaluation using the hold parameter:

sage: conjugate(I,hold=True)
conjugate(I)

To then evaluate again, we currently must use Maxima via sage.symbolic.expression.Expression.simplify():

sage: conjugate(I,hold=True).simplify()
-I

TESTS:

sage: x,y = var('x,y')
sage: x.conjugate()
conjugate(x)
sage: latex(conjugate(x))
\overline{x}
sage: f = function('f')
sage: latex(f(x).conjugate())
\overline{f\left(x\right)}
sage: f = function('psi',x,y)
sage: latex(f.conjugate())
\overline{\psi\left(x, y\right)}
sage: x.conjugate().conjugate()
x
sage: x.conjugate().operator()
conjugate
sage: x.conjugate().operator() == conjugate
True

Check if #8755 is fixed:

sage: conjugate(sqrt(-3))
conjugate(sqrt(-3))
sage: conjugate(sqrt(3))
sqrt(3)
sage: conjugate(sqrt(x))
conjugate(sqrt(x))
sage: conjugate(x^2)
conjugate(x)^2
sage: var('y',domain='positive')
y
sage: conjugate(sqrt(y))
sqrt(y)

Check if #10964 is fixed:

sage: z= I*sqrt(-3); z
I*sqrt(-3)
sage: conjugate(z)
-I*conjugate(sqrt(-3))
sage: var('a')
a
sage: conjugate(a*sqrt(-2)*sqrt(-3))
conjugate(sqrt(-2))*conjugate(sqrt(-3))*conjugate(a)

Test pickling:

sage: loads(dumps(conjugate))
conjugate
class sage.functions.other.Function_erf

Bases: sage.symbolic.function.BuiltinFunction

The error function, defined for real values as

\(\text{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} dt\).

This function is also defined for complex values, via analytic continuation.

EXAMPLES:

We can evaluate numerically:

sage: erf(2)
erf(2)
sage: erf(2).n()
0.995322265018953
sage: erf(2).n(100)
0.99532226501895273416206925637
sage: erf(ComplexField(100)(2+3j))
-20.829461427614568389103088452 + 8.6873182714701631444280787545*I

Basic symbolic properties are handled by Sage and Maxima:

sage: x = var("x")
sage: diff(erf(x),x)
2*e^(-x^2)/sqrt(pi)
sage: integrate(erf(x),x)
x*erf(x) + e^(-x^2)/sqrt(pi)

ALGORITHM:

Sage implements numerical evaluation of the error function via the erf() function from mpmath. Symbolics are handled by Sage and Maxima.

REFERENCES:

TESTS:

Check limits:

   sage: limit(erf(x),x=0)
   0
   sage: limit(erf(x),x=infinity)
   1

Check that it's odd::

    sage: erf(1.0)
    0.842700792949715
    sage: erf(-1.0)
    -0.842700792949715

Check against other implementations and against the definition:

sage: erf(3).n()
0.999977909503001
sage: maxima.erf(3).n()
0.999977909503001
sage: (1-pari(3).erfc())
0.999977909503001
sage: RR(3).erf()
0.999977909503001
sage: (integrate(exp(-x**2),(x,0,3))*2/sqrt(pi)).n()
0.999977909503001

trac ticket #9044:

sage: N(erf(sqrt(2)),200)
0.95449973610364158559943472566693312505644755259664313203267

trac ticket #11626:

sage: n(erf(2),100)
0.99532226501895273416206925637
sage: erf(2).n(100)
0.99532226501895273416206925637

Test (indirectly) trac ticket #11885:

sage: erf(float(0.5))
0.5204998778130465
sage: erf(complex(0.5))
(0.5204998778130465+0j)

Ensure conversion from maxima elements works:

sage: merf = maxima(erf(x)).sage().operator()
sage: merf == erf
True

Make sure we can dump and load it:

sage: loads(dumps(erf(2)))
erf(2)

Special-case 0 for immediate evaluation:

sage: erf(0)
0
sage: solve(erf(x)==0,x)
[x == 0]

Make sure that we can hold:

sage: erf(0,hold=True)
erf(0)
sage: simplify(erf(0,hold=True))
0

Check that high-precision ComplexField inputs work:

sage: CC(erf(ComplexField(1000)(2+3j)))
-20.8294614276146 + 8.68731827147016*I
class sage.functions.other.Function_factorial

Bases: sage.symbolic.function.GinacFunction

Returns the factorial of \(n\).

INPUT:

  • n - any complex argument (except negative integers) or any symbolic expression

OUTPUT: an integer or symbolic expression

EXAMPLES:

sage: x = var('x')
sage: factorial(0)
1
sage: factorial(4)
24
sage: factorial(10)
3628800
sage: factorial(6) == 6*5*4*3*2
True
sage: f = factorial(x + factorial(x)); f
factorial(x + factorial(x))
sage: f(x=3)
362880
sage: factorial(x)^2
factorial(x)^2

To prevent automatic evaluation use the hold argument:

sage: factorial(5,hold=True)
factorial(5)

To then evaluate again, we currently must use Maxima via sage.symbolic.expression.Expression.simplify():

sage: factorial(5,hold=True).simplify()
120

We can also give input other than nonnegative integers. For other nonnegative numbers, the gamma() function is used:

sage: factorial(1/2)
1/2*sqrt(pi)
sage: factorial(3/4)
gamma(7/4)
sage: factorial(2.3)
2.68343738195577

But negative input always fails:

sage: factorial(-32)
Traceback (most recent call last):
...
ValueError: factorial -- self = (-32) must be nonnegative

TESTS:

We verify that we can convert this function to Maxima and bring it back into Sage.:

sage: z = var('z')
sage: factorial._maxima_init_()
'factorial'
sage: maxima(factorial(z))
factorial(_SAGE_VAR_z)
sage: _.sage()
factorial(z)
sage: k = var('k')
sage: factorial(k)
factorial(k)

sage: factorial(3.14)
7.173269190187...

Test latex typesetting:

sage: latex(factorial(x))
x!
sage: latex(factorial(2*x))
\left(2 \, x\right)!
sage: latex(factorial(sin(x)))
\sin\left(x\right)!
sage: latex(factorial(sqrt(x+1)))
\left(\sqrt{x + 1}\right)!
sage: latex(factorial(sqrt(x)))
\sqrt{x}!
sage: latex(factorial(x^(2/3)))
\left(x^{\frac{2}{3}}\right)!

sage: latex(factorial)
{\rm factorial}

Check that #11539 is fixed:

sage: (factorial(x) == 0).simplify()
factorial(x) == 0
sage: maxima(factorial(x) == 0).sage()
factorial(x) == 0
sage: y = var('y')
sage: (factorial(x) == y).solve(x)
[factorial(x) == y]

Test pickling:

sage: loads(dumps(factorial))
factorial
class sage.functions.other.Function_floor

Bases: sage.symbolic.function.BuiltinFunction

The floor function.

The floor of \(x\) is computed in the following manner.

  1. The x.floor() method is called and returned if it is there. If it is not, then Sage checks if \(x\) is one of Python’s native numeric data types. If so, then it calls and returns Integer(int(math.floor(x))).
  2. Sage tries to convert \(x\) into a RealIntervalField with 53 bits of precision. Next, the floors of the endpoints are computed. If they are the same, then that value is returned. Otherwise, the precision of the RealIntervalField is increased until they do match up or it reaches maximum_bits of precision.
  3. If none of the above work, Sage returns a symbolic Expression object.

EXAMPLES:

sage: floor(5.4)
5
sage: type(floor(5.4))
<type 'sage.rings.integer.Integer'>
sage: var('x')
x
sage: a = floor(5.4 + x); a
floor(x + 5.40000000000000)
sage: a.simplify()
floor(x + 0.4000000000000004) + 5
sage: a(x=2)
7
sage: floor(cos(8)/cos(2))
0
sage: floor(factorial(50)/exp(1))
11188719610782480504630258070757734324011354208865721592720336800
sage: floor(SR(10^50 + 10^(-50)))
100000000000000000000000000000000000000000000000000
sage: floor(SR(10^50 - 10^(-50)))
99999999999999999999999999999999999999999999999999
sage: floor(int(10^50))
100000000000000000000000000000000000000000000000000
sage: import numpy
sage: a = numpy.linspace(0,2,6)
sage: floor(a)
array([ 0.,  0.,  0.,  1.,  1.,  2.])

Test pickling:

sage: loads(dumps(floor))
floor
class sage.functions.other.Function_gamma

Bases: sage.symbolic.function.GinacFunction

The Gamma function. This is defined by

\[\Gamma(z) = \int_0^\infty t^{z-1}e^{-t} dt\]

for complex input \(z\) with real part greater than zero, and by analytic continuation on the rest of the complex plane (except for negative integers, which are poles).

It is computed by various libraries within Sage, depending on the input type.

EXAMPLES:

sage: from sage.functions.other import gamma1
sage: gamma1(CDF(0.5,14))
-4.0537030780372815e-10 - 5.773299834553605e-10*I
sage: gamma1(CDF(I))
-0.15494982830181067 - 0.49801566811835607*I

Recall that \(\Gamma(n)\) is \(n-1\) factorial:

sage: gamma1(11) == factorial(10)
True
sage: gamma1(6)
120
sage: gamma1(1/2)
sqrt(pi)
sage: gamma1(-1)
Infinity
sage: gamma1(I)
gamma(I)
sage: gamma1(x/2)(x=5)
3/4*sqrt(pi)

sage: gamma1(float(6))  # For ARM: rel tol 3e-16
120.0
sage: gamma(6.)
120.000000000000
sage: gamma1(x)
gamma(x)
sage: gamma1(pi)
gamma(pi)
sage: gamma1(i)
gamma(I)
sage: gamma1(i).n()
-0.154949828301811 - 0.498015668118356*I
sage: gamma1(int(5))
24
sage: conjugate(gamma(x))
gamma(conjugate(x))
sage: plot(gamma1(x),(x,1,5))
Graphics object consisting of 1 graphics primitive

To prevent automatic evaluation use the hold argument:

sage: gamma1(1/2,hold=True)
gamma(1/2)

To then evaluate again, we currently must use Maxima via sage.symbolic.expression.Expression.simplify():

sage: gamma1(1/2,hold=True).simplify()
sqrt(pi)

TESTS:

We verify that we can convert this function to Maxima and convert back to Sage:

sage: z = var('z')
sage: maxima(gamma1(z)).sage()
gamma(z)
sage: latex(gamma1(z))
\Gamma\left(z\right)

Test that Trac ticket 5556 is fixed:

sage: gamma1(3/4)
gamma(3/4)

sage: gamma1(3/4).n(100)
1.2254167024651776451290983034

Check that negative integer input works:

sage: (-1).gamma()
Infinity
sage: (-1.).gamma()
NaN
sage: CC(-1).gamma()
Infinity
sage: RDF(-1).gamma()
NaN
sage: CDF(-1).gamma()
Infinity

Check if #8297 is fixed:

sage: latex(gamma(1/4))
\Gamma\left(\frac{1}{4}\right)

Test pickling:

sage: loads(dumps(gamma(x)))
gamma(x)
class sage.functions.other.Function_gamma_inc

Bases: sage.symbolic.function.BuiltinFunction

The incomplete gamma function.

EXAMPLES:

sage: gamma_inc(CDF(0,1), 3)
0.003208574993369116 + 0.012406185811871568*I
sage: gamma_inc(RDF(1), 3)
0.049787068367863944
sage: gamma_inc(3,2)
gamma(3, 2)
sage: gamma_inc(x,0)
gamma(x)
sage: latex(gamma_inc(3,2))
\Gamma\left(3, 2\right)
sage: loads(dumps((gamma_inc(3,2))))
gamma(3, 2)
sage: i = ComplexField(30).0; gamma_inc(2, 1 + i)
0.70709210 - 0.42035364*I
sage: gamma_inc(2., 5)
0.0404276819945128
class sage.functions.other.Function_imag_part

Bases: sage.symbolic.function.GinacFunction

Returns the imaginary part of the (possibly complex) input.

It is possible to prevent automatic evaluation using the hold parameter:

sage: imag_part(I,hold=True)
imag_part(I)

To then evaluate again, we currently must use Maxima via sage.symbolic.expression.Expression.simplify():

sage: imag_part(I,hold=True).simplify()
1

TESTS:

sage: z = 1+2*I
sage: imaginary(z)
2
sage: imag(z)
2
sage: imag(complex(3, 4))
4.0
sage: loads(dumps(imag_part))
imag_part
sage: imag_part(x)._sympy_()
im(x)

Check if #6401 is fixed:

sage: latex(x.imag())
\Im \left( x \right)

sage: f(x) = function('f',x)
sage: latex( f(x).imag())
\Im \left( f\left(x\right) \right)
class sage.functions.other.Function_log_gamma

Bases: sage.symbolic.function.GinacFunction

The principal branch of the logarithm of Gamma function. Gamma is defined for complex input \(z\) with real part greater than zero, and by analytic continuation on the rest of the complex plane (except for negative integers, which are poles).

It is computed by the \(log_gamma\) function for the number type, or by \(lgamma\) in Ginac, failing that.

EXAMPLES:

Numerical evaluation happens when appropriate, to the appropriate accuracy (see #10072):

sage: log_gamma(6)
log(120)
sage: log_gamma(6.)
4.78749174278205
sage: log_gamma(6).n()
4.78749174278205
sage: log_gamma(RealField(100)(6))
4.7874917427820459942477009345
sage: log_gamma(2.4+i)
-0.0308566579348816 + 0.693427705955790*I
sage: log_gamma(-3.1)
0.400311696703985

Symbolic input works (see #10075):

sage: log_gamma(3*x)
log_gamma(3*x)
sage: log_gamma(3+i)
log_gamma(I + 3)
sage: log_gamma(3+i+x)
log_gamma(x + I + 3)

To get evaluation of input for which gamma is negative and the ceiling is even, we must explicitly make the input complex. This is a known issue, see #12521:

sage: log_gamma(-2.1)
NaN
sage: log_gamma(CC(-2.1))
1.53171380819509 + 3.14159265358979*I

In order to prevent evaluation, use the \(hold\) argument; to evaluate a held expression, use the \(n()\) numerical evaluation method:

sage: log_gamma(SR(5),hold=True)
log_gamma(5)
sage: log_gamma(SR(5),hold=True).n()
3.17805383034795

TESTS:

sage: log_gamma(-2.1+i)
-1.90373724496982 - 0.901638463592247*I
sage: log_gamma(pari(6))
4.78749174278205
sage: log_gamma(CC(6))
4.78749174278205
sage: log_gamma(CC(-2.5))
-0.0562437164976740 + 3.14159265358979*I
sage: log_gamma(x)._sympy_()
loggamma(x)

conjugate(log_gamma(x))==log_gamma(conjugate(x)) unless on the branch cut, which runs along the negative real axis.:

sage: conjugate(log_gamma(x))
conjugate(log_gamma(x))
sage: var('y', domain='positive')
y
sage: conjugate(log_gamma(y))
log_gamma(y)
sage: conjugate(log_gamma(y+I))
conjugate(log_gamma(y + I))
sage: log_gamma(-2)
+Infinity
sage: conjugate(log_gamma(-2))
+Infinity
class sage.functions.other.Function_psi1

Bases: sage.symbolic.function.GinacFunction

The digamma function, \(\psi(x)\), is the logarithmic derivative of the gamma function.

\[\psi(x) = \frac{d}{dx} \log(\Gamma(x)) = \frac{\Gamma'(x)}{\Gamma(x)}\]

EXAMPLES:

sage: from sage.functions.other import psi1
sage: psi1(x)
psi(x)
sage: psi1(x).derivative(x)
psi(1, x)
sage: psi1(3)
-euler_gamma + 3/2
sage: psi(.5)
-1.96351002602142
sage: psi(RealField(100)(.5))
-1.9635100260214234794409763330

TESTS:

sage: latex(psi1(x))
\psi\left(x\right)
sage: loads(dumps(psi1(x)+1))
psi(x) + 1

sage: t = psi1(x); t
psi(x)
sage: t.subs(x=.2)
-5.28903989659219
sage: psi(x)._sympy_()
polygamma(0, x)
class sage.functions.other.Function_psi2

Bases: sage.symbolic.function.GinacFunction

Derivatives of the digamma function \(\psi(x)\). T

EXAMPLES:

sage: from sage.functions.other import psi2
sage: psi2(2, x)
psi(2, x)
sage: psi2(2, x).derivative(x)
psi(3, x)
sage: n = var('n')
sage: psi2(n, x).derivative(x)
psi(n + 1, x)
sage: psi2(0, x)
psi(x)
sage: psi2(-1, x)
log(gamma(x))
sage: psi2(3, 1)
1/15*pi^4
sage: psi2(2, .5).n()
-16.8287966442343
sage: psi2(2, .5).n(100)
-16.828796644234319995596334261

TESTS:

sage: psi2(n, x).derivative(n)
Traceback (most recent call last):
...
RuntimeError: cannot diff psi(n,x) with respect to n

sage: latex(psi2(2,x))
\psi\left(2, x\right)
sage: loads(dumps(psi2(2,x)+1))
psi(2, x) + 1
sage: psi(2, x)._sympy_()
polygamma(2, x)
class sage.functions.other.Function_real_part

Bases: sage.symbolic.function.GinacFunction

Returns the real part of the (possibly complex) input.

It is possible to prevent automatic evaluation using the hold parameter:

sage: real_part(I,hold=True)
real_part(I)

To then evaluate again, we currently must use Maxima via sage.symbolic.expression.Expression.simplify():

sage: real_part(I,hold=True).simplify()
0

EXAMPLES:

sage: z = 1+2*I
sage: real(z)
1
sage: real(5/3)
5/3
sage: a = 2.5
sage: real(a)
2.50000000000000
sage: type(real(a))
<type 'sage.rings.real_mpfr.RealLiteral'>
sage: real(1.0r)
1.0
sage: real(complex(3, 4))
3.0

TESTS:

sage: loads(dumps(real_part))
real_part
sage: real_part(x)._sympy_()
re(x)

Check if #6401 is fixed:

sage: latex(x.real())
\Re \left( x \right)

sage: f(x) = function('f',x)
sage: latex( f(x).real())
\Re \left( f\left(x\right) \right)
class sage.functions.other.Function_sqrt

Bases: object

x.__init__(...) initializes x; see help(type(x)) for signature

sage.functions.other.gamma(a, *args, **kwds)

Gamma and incomplete gamma functions. This is defined by the integral

\[\Gamma(a, z) = \int_z^\infty t^{a-1}e^{-t} dt\]

EXAMPLES:

Recall that `\Gamma(n)` is `n-1` factorial::

    sage: gamma(11) == factorial(10)
    True
    sage: gamma(6)
    120
    sage: gamma(1/2)
    sqrt(pi)
    sage: gamma(-4/3)
    gamma(-4/3)
    sage: gamma(-1)
    Infinity
    sage: gamma(0)
    Infinity

::

    sage: gamma_inc(3,2)
    gamma(3, 2)
    sage: gamma_inc(x,0)
    gamma(x)

::

    sage: gamma(5, hold=True)
    gamma(5)
    sage: gamma(x, 0, hold=True)
    gamma(x, 0)

::

    sage: gamma(CDF(0.5,14))
    -4.0537030780372815e-10 - 5.773299834553605e-10*I
    sage: gamma(CDF(I))
    -0.15494982830181067 - 0.49801566811835607*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 = gamma(RealField(100)(2.5)); t
    1.3293403881791370204736256125
    sage: t.prec()
    100

    sage: gamma(6)
    120

    sage: gamma(pi).n(100)
    2.2880377953400324179595889091

    sage: gamma(3/4).n(100)
    1.2254167024651776451290983034
    
The gamma function only works with input that can be coerced to the
Symbolic Ring::

    sage: Q.<i> = NumberField(x^2+1)
    sage: gamma(i)
    Traceback (most recent call last):
    ...
    TypeError: cannot coerce arguments: no canonical coercion...

We make an exception for elements of AA or QQbar, which cannot be
coerced into symbolic expressions to allow this usage::

    sage: t = QQbar(sqrt(2)) + sqrt(3); t
    3.146264369941973?
    sage: t.parent()
    Algebraic Field

Symbolic functions convert the arguments to symbolic expressions if they
are in QQbar or AA::

    sage: gamma(QQbar(I))
    -0.154949828301811 - 0.498015668118356*I
sage.functions.other.psi(x, *args, **kwds)

The digamma function, \(\psi(x)\), is the logarithmic derivative of the gamma function.

\[\psi(x) = \frac{d}{dx} \log(\Gamma(x)) = \frac{\Gamma'(x)}{\Gamma(x)}\]

We represent the \(n\)-th derivative of the digamma function with \(\psi(n, x)\) or \(psi(n, x)\).

EXAMPLES:

sage: psi(x)
psi(x)
sage: psi(.5)
-1.96351002602142
sage: psi(3)
-euler_gamma + 3/2
sage: psi(1, 5)
1/6*pi^2 - 205/144
sage: psi(1, x)
psi(1, x)
sage: psi(1, x).derivative(x)
psi(2, x)
sage: psi(3, hold=True)
psi(3)
sage: psi(1, 5, hold=True)
psi(1, 5)

TESTS:

sage: psi(2, x, 3)
Traceback (most recent call last):
...
TypeError: Symbolic function psi takes at most 2 arguments (3 given)
sage.functions.other.sqrt(x, *args, **kwds)

INPUT:

  • x - a number
  • prec - integer (default: None): if None, returns an exact square root; otherwise returns a numerical square root if necessary, to the given bits of precision.
  • extend - bool (default: True); this is a place holder, and is always ignored or passed to the sqrt function for x, since in the symbolic ring everything has a square root.
  • all - bool (default: False); if True, return all square roots of self, instead of just one.

EXAMPLES:

sage: sqrt(-1)
I
sage: sqrt(2)
sqrt(2)
sage: sqrt(2)^2
2
sage: sqrt(4)
2
sage: sqrt(4,all=True)
[2, -2]
sage: sqrt(x^2)
sqrt(x^2)
sage: sqrt(2).n()
1.41421356237310

To prevent automatic evaluation, one can use the hold parameter after coercing to the symbolic ring:

sage: sqrt(SR(4),hold=True)
sqrt(4)
sage: sqrt(4,hold=True)
Traceback (most recent call last):
...
TypeError: _do_sqrt() got an unexpected keyword argument 'hold'

This illustrates that the bug reported in #6171 has been fixed:

sage: a = 1.1
sage: a.sqrt(prec=100)  # this is supposed to fail
Traceback (most recent call last):
...
TypeError: sqrt() got an unexpected keyword argument 'prec'
sage: sqrt(a, prec=100)
1.0488088481701515469914535137
sage: sqrt(4.00, prec=250)
2.0000000000000000000000000000000000000000000000000000000000000000000000000

One can use numpy input as well:

sage: import numpy
sage: a = numpy.arange(2,5)
sage: sqrt(a)
array([ 1.41421356,  1.73205081,  2.        ])

Previous topic

Orthogonal Polynomials

Next topic

Special Functions

This Page