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sage.{calculus,functions,numerical,symbolic}: Docstring/doctest cosmetics, # needs #37715

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May 2, 2024
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sage.{calculus,functions,numerical,symbolic}: Docstring/doctest cosmetics, # needs #37715

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b97c954
sage.{calculus,functions,numerical,symbolic}: Docstring/doctest cosme…
mkoeppe Apr 1, 2024
41640cf
src/sage/functions/error.py: Fix # needs
mkoeppe Apr 1, 2024
e56c10b
src/sage/functions/error.py: Fix up doctest
mkoeppe Apr 28, 2024
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4 changes: 2 additions & 2 deletions src/sage/calculus/functional.py
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Original file line number Diff line number Diff line change
Expand Up @@ -215,8 +215,8 @@ def integral(f, *args, **kwds):
symbolically::

sage: f(x) = 1/(sqrt(2*pi)) * e^(-x^2/2)
sage: P = plot(f, -4, 4, hue=0.8, thickness=2)
sage: P.show(ymin=0, ymax=0.4)
sage: P = plot(f, -4, 4, hue=0.8, thickness=2) # needs sage.plot
sage: P.show(ymin=0, ymax=0.4) # needs sage.plot
sage: numerical_integral(f, -4, 4) # random output
(0.99993665751633376, 1.1101527003413533e-14)
sage: integrate(f, x)
Expand Down
3 changes: 2 additions & 1 deletion src/sage/calculus/interpolators.pyx
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Expand Up @@ -50,13 +50,14 @@ def polygon_spline(pts):
sage: ps = polygon_spline(pts)
sage: fx = lambda x: ps.value(x).real
sage: fy = lambda x: ps.value(x).imag
sage: show(parametric_plot((fx, fy), (0, 2*pi))) # needs sage.plot
sage: show(parametric_plot((fx, fy), (0, 2*pi))) # needs sage.plot sage.symbolic
sage: m = Riemann_Map([lambda x: ps.value(real(x))],
....: [lambda x: ps.derivative(real(x))], 0)
sage: show(m.plot_colored() + m.plot_spiderweb()) # needs sage.plot

Polygon approximation of a circle::

sage: # needs sage.symbolic
sage: pts = [e^(I*t / 25) for t in range(25)]
sage: ps = polygon_spline(pts)
sage: ps.derivative(2)
Expand Down
35 changes: 19 additions & 16 deletions src/sage/calculus/riemann.pyx
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Original file line number Diff line number Diff line change
Expand Up @@ -408,15 +408,15 @@ cdef class Riemann_Map:
sage: m = Riemann_Map([f], [fprime], 0)
sage: sz = m.get_szego(boundary=0)
sage: points = m.get_szego(absolute_value=True)
sage: list_plot(points)
sage: list_plot(points) # needs sage.plot
Graphics object consisting of 1 graphics primitive

Extending the points by a spline::

sage: s = spline(points)
sage: s(3*pi / 4)
0.0012158...
sage: plot(s,0,2*pi) # plot the kernel
sage: plot(s,0,2*pi) # plot the kernel # needs sage.plot
Graphics object consisting of 1 graphics primitive

The unit circle with a small hole::
Expand Down Expand Up @@ -483,7 +483,7 @@ cdef class Riemann_Map:
sage: fprime(t) = I*e^(I*t) + 0.5*I*e^(-I*t)
sage: m = Riemann_Map([f], [fprime], 0)
sage: points = m.get_theta_points()
sage: list_plot(points)
sage: list_plot(points) # needs sage.plot
Graphics object consisting of 1 graphics primitive

Extending the points by a spline::
Expand Down Expand Up @@ -740,12 +740,12 @@ cdef class Riemann_Map:

Default plot::

sage: m.plot_boundaries()
sage: m.plot_boundaries() # needs sage.plot
Graphics object consisting of 1 graphics primitive

Big blue collocation points::

sage: m.plot_boundaries(plotjoined=False, rgbcolor=[0,0,1], thickness=6)
sage: m.plot_boundaries(plotjoined=False, rgbcolor=[0,0,1], thickness=6) # needs sage.plot
Graphics object consisting of 1 graphics primitive
"""
from sage.plot.all import list_plot
Expand Down Expand Up @@ -905,17 +905,18 @@ cdef class Riemann_Map:

Default plot::

sage: m.plot_spiderweb()
sage: m.plot_spiderweb() # needs sage.plot
Graphics object consisting of 21 graphics primitives

Simplified plot with many discrete points::

sage: m.plot_spiderweb(spokes=4, circles=1, pts=400, linescale=0.95, plotjoined=False)
sage: m.plot_spiderweb(spokes=4, circles=1, pts=400, # needs sage.plot
....: linescale=0.95, plotjoined=False)
Graphics object consisting of 6 graphics primitives

Plot with thick, red lines::

sage: m.plot_spiderweb(rgbcolor=[1,0,0], thickness=3)
sage: m.plot_spiderweb(rgbcolor=[1,0,0], thickness=3) # needs sage.plot
Graphics object consisting of 21 graphics primitives

To generate the unit circle map, it's helpful to see what the
Expand All @@ -924,7 +925,7 @@ cdef class Riemann_Map:
sage: f(t) = e^(I*t)
sage: fprime(t) = I*e^(I*t)
sage: m = Riemann_Map([f], [fprime], 0, 1000)
sage: m.plot_spiderweb()
sage: m.plot_spiderweb() # needs sage.plot
Graphics object consisting of 21 graphics primitives

A multiply connected region with corners. We set ``min_mag`` higher
Expand All @@ -934,8 +935,10 @@ cdef class Riemann_Map:
sage: z1 = lambda t: ps.value(t); z1p = lambda t: ps.derivative(t)
sage: z2(t) = -2+exp(-I*t); z2p(t) = -I*exp(-I*t)
sage: z3(t) = 2+exp(-I*t); z3p(t) = -I*exp(-I*t)
sage: m = Riemann_Map([z1,z2,z3],[z1p,z2p,z3p],0,ncorners=4) # long time
sage: p = m.plot_spiderweb(withcolor=True,plot_points=500, thickness = 2.0, min_mag=0.1) # long time
sage: m = Riemann_Map([z1,z2,z3], [z1p,z2p,z3p], 0, # long time
....: ncorners=4)
sage: p = m.plot_spiderweb(withcolor=True, plot_points=500, # long time, needs sage.plot
....: thickness=2.0, min_mag=0.1)
"""
from sage.plot.complex_plot import ComplexPlot
from sage.plot.all import list_plot, Graphics
Expand Down Expand Up @@ -1028,17 +1031,17 @@ cdef class Riemann_Map:
sage: f(t) = e^(I*t) - 0.5*e^(-I*t)
sage: fprime(t) = I*e^(I*t) + 0.5*I*e^(-I*t)
sage: m = Riemann_Map([f], [fprime], 0)
sage: m.plot_colored()
sage: m.plot_colored() # needs sage.plot
Graphics object consisting of 1 graphics primitive

Plot zoomed in on a specific spot::

sage: m.plot_colored(plot_range=[0,1,.25,.75])
sage: m.plot_colored(plot_range=[0,1,.25,.75]) # needs sage.plot
Graphics object consisting of 1 graphics primitive

High resolution plot::

sage: m.plot_colored(plot_points=1000) # long time (29s on sage.math, 2012)
sage: m.plot_colored(plot_points=1000) # long time (29s on sage.math, 2012), needs sage.plot
Graphics object consisting of 1 graphics primitive

To generate the unit circle map, it's helpful to see what the
Expand All @@ -1047,7 +1050,7 @@ cdef class Riemann_Map:
sage: f(t) = e^(I*t)
sage: fprime(t) = I*e^(I*t)
sage: m = Riemann_Map([f], [fprime], 0, 1000)
sage: m.plot_colored()
sage: m.plot_colored() # needs sage.plot
Graphics object consisting of 1 graphics primitive
"""
from sage.plot.complex_plot import ComplexPlot
Expand Down Expand Up @@ -1081,7 +1084,7 @@ cdef comp_pt(clist, loop=True):
sage: f(t) = e^(I*t) - 0.5*e^(-I*t)
sage: fprime(t) = I*e^(I*t) + 0.5*I*e^(-I*t)
sage: m = Riemann_Map([f], [fprime], 0)
sage: m.plot_spiderweb()
sage: m.plot_spiderweb() # needs sage.plot
Graphics object consisting of 21 graphics primitives
"""
list2 = [(c.real, c.imag) for c in clist]
Expand Down
6 changes: 5 additions & 1 deletion src/sage/calculus/transforms/dft.py
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Expand Up @@ -294,13 +294,15 @@ def dft(self, chi=None):
Indexed sequence: [6, 0, 0, 0, 0, 0]
indexed by [0, 1, 2, 3, 4, 5]

sage: # needs sage.groups
sage: # needs sage.combinat sage.groups
sage: G = SymmetricGroup(3)
sage: J = G.conjugacy_classes_representatives()
sage: s = IndexedSequence([1,2,3], J) # 1,2,3 are the values of a class fcn on G
sage: s.dft() # the "scalar-valued Fourier transform" of this class fcn
Indexed sequence: [8, 2, 2]
indexed by [(), (1,2), (1,2,3)]

sage: # needs sage.rings.number_field
sage: J = AbelianGroup(2, [2,3], names='ab')
sage: s = IndexedSequence([1,2,3,4,5,6], J)
sage: s.dft() # the precision of output is somewhat random and architecture dependent.
Expand All @@ -311,6 +313,8 @@ def dft(self, chi=None):
-0.00000000000000976996261670137 - 0.0000000000000159872115546022*I,
-0.00000000000000621724893790087 - 0.0000000000000106581410364015*I]
indexed by Multiplicative Abelian group isomorphic to C2 x C3

sage: # needs sage.groups sage.rings.number_field
sage: J = CyclicPermutationGroup(6)
sage: s = IndexedSequence([1,2,3,4,5,6], J)
sage: s.dft() # the precision of output is somewhat random and architecture dependent.
Expand Down
17 changes: 9 additions & 8 deletions src/sage/calculus/transforms/fft.pyx
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Expand Up @@ -243,7 +243,7 @@ cdef class FastFourierTransform_complex(FastFourierTransform_base):
EXAMPLES::

sage: a = FastFourierTransform(4)
sage: a._plot_polar(0,2) # needs sage.plot
sage: a._plot_polar(0,2) # needs sage.plot sage.symbolic
Graphics object consisting of 2 graphics primitives

"""
Expand Down Expand Up @@ -304,9 +304,9 @@ cdef class FastFourierTransform_complex(FastFourierTransform_base):

- ``style`` -- Style of the plot, options are ``"rect"`` or ``"polar"``

- ``rect`` -- height represents real part, color represents
- ``"rect"`` -- height represents real part, color represents
imaginary part.
- ``polar`` -- height represents absolute value, color
- ``"polar"`` -- height represents absolute value, color
represents argument.

- ``xmin`` -- The lower bound of the slice to plot. 0 by default.
Expand All @@ -319,21 +319,22 @@ cdef class FastFourierTransform_complex(FastFourierTransform_base):

EXAMPLES::

sage: # needs sage.plot
sage: a = FastFourierTransform(16)
sage: for i in range(16): a[i] = (random(),random())
sage: A = plot(a) # needs sage.plot
sage: B = plot(a, style='polar') # needs sage.plot
sage: type(A) # needs sage.plot
sage: A = plot(a)
sage: B = plot(a, style='polar') # needs sage.symbolic
sage: type(A)
<class 'sage.plot.graphics.Graphics'>
sage: type(B) # needs sage.plot
sage: type(B) # needs sage.symbolic
<class 'sage.plot.graphics.Graphics'>
sage: a = FastFourierTransform(125)
sage: b = FastFourierTransform(125)
sage: for i in range(1, 60): a[i]=1
sage: for i in range(1, 60): b[i]=1
sage: a.forward_transform()
sage: a.inverse_transform()
sage: a.plot() + b.plot() # needs sage.plot
sage: a.plot() + b.plot()
Graphics object consisting of 250 graphics primitives

"""
Expand Down
2 changes: 1 addition & 1 deletion src/sage/functions/error.py
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Expand Up @@ -265,7 +265,7 @@ def _evalf_(self, x, parent=None, algorithm=None):

sage: gp.set_real_precision(59) # random # needs sage.libs.pari
38
sage: print(gp.eval("1 - erfc(1)")); print(erf(1).n(200)) # needs sage.libs.pari
sage: print(gp.eval("1 - erfc(1)")); print(erf(1).n(200)) # needs mpmath sage.libs.pari
0.84270079294971486934122063508260925929606699796630290845994
0.84270079294971486934122063508260925929606699796630290845994

Expand Down
4 changes: 2 additions & 2 deletions src/sage/functions/exp_integral.py
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Expand Up @@ -1470,7 +1470,7 @@ def exponential_integral_1(x, n=0):

EXAMPLES::

sage: # needs sage.libs.pari
sage: # needs sage.libs.pari sage.rings.real_mpfr
sage: exponential_integral_1(2)
0.0489005107080611
sage: exponential_integral_1(2, 4) # abs tol 1e-18
Expand Down Expand Up @@ -1519,7 +1519,7 @@ def exponential_integral_1(x, n=0):
....: if e >= c:
....: print("exponential_integral_1(%s, %s)[%s] with precision %s has error of %s >= %s"%(a, n, i, prec, e, c))

ALGORITHM: use the PARI C-library function ``eint1``.
ALGORITHM: use the PARI C-library function :pari:`eint1`.

REFERENCE:

Expand Down
29 changes: 15 additions & 14 deletions src/sage/functions/gamma.py
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Expand Up @@ -40,9 +40,9 @@ def __init__(self):
EXAMPLES::

sage: from sage.functions.gamma import gamma1
sage: gamma1(CDF(0.5, 14)) # needs sage.libs.pari
sage: gamma1(CDF(0.5, 14)) # needs sage.libs.pari sage.rings.complex_double
-4.0537030780372815e-10 - 5.773299834553605e-10*I
sage: gamma1(CDF(I)) # needs sage.libs.pari sage.symbolic
sage: gamma1(CDF(I)) # needs sage.libs.pari sage.rings.complex_double sage.symbolic
-0.15494982830181067 - 0.49801566811835607*I

Recall that `\Gamma(n)` is `n-1` factorial::
Expand Down Expand Up @@ -99,7 +99,7 @@ def __init__(self):
1*x^(-2) + (-2*euler_gamma)*x^(-1)
+ (2*euler_gamma^2 + 1/6*pi^2) + Order(x)

To prevent automatic evaluation use the ``hold`` argument::
To prevent automatic evaluation, use the ``hold`` argument::

sage: gamma1(1/2, hold=True) # needs sage.symbolic
gamma(1/2)
Expand Down Expand Up @@ -138,9 +138,9 @@ def __init__(self):
Infinity
sage: (-1.).gamma() # needs sage.rings.real_mpfr
NaN
sage: CC(-1).gamma() # needs sage.libs.pari
sage: CC(-1).gamma() # needs sage.libs.pari sage.rings.real_mpfr
Infinity
sage: RDF(-1).gamma()
sage: RDF(-1).gamma() # needs sage.rings.real_mpfr
NaN
sage: CDF(-1).gamma() # needs sage.libs.pari sage.rings.complex_double
Infinity
Expand Down Expand Up @@ -691,9 +691,9 @@ def gamma(a, *args, **kwds):

::

sage: gamma(CDF(I)) # needs sage.libs.pari sage.symbolic
sage: gamma(CDF(I)) # needs sage.libs.pari sage.rings.complex_double sage.symbolic
-0.15494982830181067 - 0.49801566811835607*I
sage: gamma(CDF(0.5, 14)) # needs sage.libs.pari
sage: gamma(CDF(0.5, 14)) # needs sage.libs.pari sage.rings.complex_double
-4.0537030780372815e-10 - 5.773299834553605e-10*I

Use ``numerical_approx`` to get higher precision from
Expand Down Expand Up @@ -721,7 +721,8 @@ def gamma(a, *args, **kwds):
sage: gamma(i) # needs sage.rings.number_field sage.symbolic
Traceback (most recent call last):
...
TypeError: cannot coerce arguments: no canonical coercion from Number Field in i with defining polynomial x^2 + 1 to Symbolic Ring
TypeError: cannot coerce arguments: no canonical coercion
from Number Field in i with defining polynomial x^2 + 1 to Symbolic Ring

.. SEEALSO::

Expand Down Expand Up @@ -1004,9 +1005,9 @@ def __init__(self):

INPUT:

- ``p`` - number or symbolic expression
- ``p`` -- number or symbolic expression

- ``q`` - number or symbolic expression
- ``q`` -- number or symbolic expression


OUTPUT: number or symbolic expression (if input is symbolic)
Expand All @@ -1016,18 +1017,18 @@ def __init__(self):
sage: # needs sage.symbolic
sage: beta(3, 2)
1/12
sage: beta(3,1)
sage: beta(3, 1)
1/3
sage: beta(1/2, 1/2)
beta(1/2, 1/2)
sage: beta(-1,1)
sage: beta(-1, 1)
-1
sage: beta(-1/2,-1/2)
sage: beta(-1/2, -1/2)
0
sage: ex = beta(x/2, 3)
sage: set(ex.operands()) == set([1/2*x, 3])
True
sage: beta(.5,.5)
sage: beta(.5, .5)
3.14159265358979
sage: beta(1, 2.0+I)
0.400000000000000 - 0.200000000000000*I
Expand Down
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