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r""" Piecewise-defined Functions
Sage implements a very simple class of piecewise-defined functions. Functions may be any type of symbolic expression. Infinite intervals are not supported. The endpoints of each interval must line up.
TODO:
- Implement max/min location and values,
- Need: parent object - ring of piecewise functions
- This class should derive from an element-type class, and should define ``_add_``, ``_mul_``, etc. That will automatically take care of left multiplication and proper coercion. The coercion mentioned below for scalar mult on right is bad, since it only allows ints and rationals. The right way is to use an element class and only define ``_mul_``, and have a parent, so anything gets coerced properly.
AUTHORS:
- David Joyner (2006-04): initial version
- David Joyner (2006-09): added __eq__, extend_by_zero_to, unextend, convolution, trapezoid, trapezoid_integral_approximation, riemann_sum, riemann_sum_integral_approximation, tangent_line fixed bugs in __mul__, __add__
- David Joyner (2007-03): adding Hann filter for FS, added general FS filter methods for computing and plotting, added options to plotting of FS (eg, specifying rgb values are now allowed). Fixed bug in documentation reported by Pablo De Napoli.
- David Joyner (2007-09): bug fixes due to behaviour of SymbolicArithmetic
- David Joyner (2008-04): fixed docstring bugs reported by J Morrow; added support for Laplace transform of functions with infinite support.
- David Joyner (2008-07): fixed a left multiplication bug reported by C. Boncelet (by defining __rmul__ = __mul__).
- Paul Butler (2009-01): added indefinite integration and default_variable
TESTS::
sage: R.<x> = QQ[] sage: f = Piecewise([[(0,1),1*x^0]]) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: 2*f doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. Piecewise defined function with 1 parts, [[(0, 1), 2]] """
#***************************************************************************** # Copyright (C) 2006 William Stein <wstein@gmail.com> # 2006 David Joyner <wdjoyner@gmail.com> # # Distributed under the terms of the GNU General Public License (GPL) # # This code is distributed in the hope that it will be useful, # but WITHOUT ANY WARRANTY; without even the implied warranty of # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU # General Public License for more details. # # The full text of the GPL is available at: # # http://www.gnu.org/licenses/ #*****************************************************************************
""" Deprecated spelling of :func:`sage.functions.piecewise`.
Return a piecewise function from a list of (interval, function) pairs.
``list_of_pairs`` is a list of pairs (I, fcn), where fcn is a Sage function (such as a polynomial over RR, or functions using the lambda notation), and I is an interval such as I = (1,3). Two consecutive intervals must share a common endpoint.
If the optional ``var`` is specified, then any symbolic expressions in the list will be converted to symbolic functions using ``fcn.function(var)``. (This says which variable is considered to be "piecewise".)
We assume that these definitions are consistent (ie, no checking is done).
EXAMPLES::
sage: f1(x) = -1 sage: f2(x) = 2 sage: f = Piecewise([[(0,pi/2),f1],[(pi/2,pi),f2]]) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: f(1) -1 sage: f(3) 2 sage: f = Piecewise([[(0,1),x], [(1,2),x^2]], x); f Piecewise defined function with 2 parts, [[(0, 1), x |--> x], [(1, 2), x |--> x^2]] sage: f(0.9) 0.900000000000000 sage: f(1.1) 1.21000000000000 """
""" Returns a piecewise function from a list of (interval, function) pairs.
EXAMPLES::
sage: f1(x) = -1 sage: f2(x) = 2 sage: f = Piecewise([[(0,pi/2),f1],[(pi/2,pi),f2]]) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: f(1) -1 sage: f(3) 2 """ r""" ``list_of_pairs`` is a list of pairs (I, fcn), where fcn is a Sage function (such as a polynomial over RR, or functions using the lambda notation), and I is an interval such as I = (1,3). Two consecutive intervals must share a common endpoint.
If the optional ``var`` is specified, then any symbolic expressions in the list will be converted to symbolic functions using ``fcn.function(var)``. (This says which variable is considered to be "piecewise".)
We assume that these definitions are consistent (ie, no checking is done).
EXAMPLES::
sage: f1(x) = 1 sage: f2(x) = 1 - x sage: f = Piecewise([[(0,1),f1],[(1,2),f2]]) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: f.list() [[(0, 1), x |--> 1], [(1, 2), x |--> -x + 1]] sage: f.length() 2 """ # We regenerate self._list in case self._functions was modified # above. This also protects us in case somebody mutates a list # after they use it as an argument to piecewise().
""" Returns the pieces of this function as a list of functions.
EXAMPLES::
sage: f1(x) = 1 sage: f2(x) = 1 - x sage: f = Piecewise([[(0,1),f1],[(1,2),f2]]) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: f.list() [[(0, 1), x |--> 1], [(1, 2), x |--> -x + 1]] """
""" Returns the number of pieces of this function.
EXAMPLES::
sage: f1(x) = 1 sage: f2(x) = 1 - x sage: f = Piecewise([[(0,1),f1],[(1,2),f2]]) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: f.length() 2 """
""" EXAMPLES::
sage: f1(x) = 1 sage: f2(x) = 1 - x sage: f = Piecewise([[(0,1),f1],[(1,2),f2]]); f doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. Piecewise defined function with 2 parts, [[(0, 1), x |--> 1], [(1, 2), x |--> -x + 1]] """ self.length(),self.list())
r""" EXAMPLES::
sage: f1(x) = 1 sage: f2(x) = 1 - x sage: f = Piecewise([[(0,1),f1],[(1,2),f2]]) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: latex(f) \begin{cases} x \ {\mapsto}\ 1 &\text{on $(0, 1)$}\cr x \ {\mapsto}\ -x + 1 &\text{on $(1, 2)$}\cr \end{cases}
::
sage: f(x) = sin(x*pi/2) sage: g(x) = 1-(x-1)^2 sage: h(x) = -x sage: P = Piecewise([[(0,1), f], [(1,3),g], [(3,5), h]]) sage: latex(P) \begin{cases} x \ {\mapsto}\ \sin\left(\frac{1}{2} \, \pi x\right) &\text{on $(0, 1)$}\cr x \ {\mapsto}\ -{\left(x - 1\right)}^{2} + 1 &\text{on $(1, 3)$}\cr x \ {\mapsto}\ -x &\text{on $(3, 5)$}\cr \end{cases} """
""" A piecewise non-polynomial example.
EXAMPLES::
sage: f1(x) = 1 sage: f2(x) = 1-x sage: f3(x) = exp(x) sage: f4(x) = sin(2*x) sage: f = Piecewise([[(0,1),f1],[(1,2),f2],[(2,3),f3],[(3,10),f4]]) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: f.intervals() [(0, 1), (1, 2), (2, 3), (3, 10)] """
""" Returns the domain of the function.
EXAMPLES::
sage: f1(x) = 1 sage: f2(x) = 1-x sage: f3(x) = exp(x) sage: f4(x) = sin(2*x) sage: f = Piecewise([[(0,1),f1],[(1,2),f2],[(2,3),f3],[(3,10),f4]]) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: f.domain() (0, 10) """
""" Returns the list of functions (the "pieces").
EXAMPLES::
sage: f1(x) = 1 sage: f2(x) = 1-x sage: f3(x) = exp(x) sage: f4(x) = sin(2*x) sage: f = Piecewise([[(0,1),f1],[(1,2),f2],[(2,3),f3],[(3,10),f4]]) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: f.functions() [x |--> 1, x |--> -x + 1, x |--> e^x, x |--> sin(2*x)] """
""" This function simply returns the piecewise defined function which is extended by 0 so it is defined on all of (xmin,xmax). This is needed to add two piecewise functions in a reasonable way.
EXAMPLES::
sage: f1(x) = 1 sage: f2(x) = 1 - x sage: f = Piecewise([[(0,1),f1],[(1,2),f2]]) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: f.extend_by_zero_to(-1, 3) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. Piecewise defined function with 4 parts, [[(-1, 0), 0], [(0, 1), x |--> 1], [(1, 2), x |--> -x + 1], [(2, 3), 0]] """
""" This removes any parts in the front or back of the function which is zero (the inverse to extend_by_zero_to).
EXAMPLES::
sage: R.<x> = QQ[] sage: f = Piecewise([[(-3,-1),1+2+x],[(-1,1),1-x^2]]) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: e = f.extend_by_zero_to(-10,10); e Piecewise defined function with 4 parts, [[(-10, -3), 0], [(-3, -1), x + 3], [(-1, 1), -x^2 + 1], [(1, 10), 0]] sage: d = e.unextend(); d doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. Piecewise defined function with 2 parts, [[(-3, -1), x + 3], [(-1, 1), -x^2 + 1]] sage: d==f True """
""" A helper function for computing Riemann sums.
INPUT:
- ``N`` - the number of subdivisions
- ``func`` - a function to apply to the endpoints of each subdivision
- ``initial`` - the starting value
EXAMPLES::
sage: f1(x) = x^2 ## example 1 sage: f2(x) = 5-x^2 sage: f = Piecewise([[(0,1),f1],[(1,2),f2]]) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: f._riemann_sum_helper(6, lambda x0, x1: (x1-x0)*f(x1)) 19/6 """
""" Returns the piecewise line function defined by the Riemann sums in numerical integration based on a subdivision into N subintervals.
Set mode="midpoint" for the height of the rectangles to be determined by the midpoint of the subinterval; set mode="right" for the height of the rectangles to be determined by the right-hand endpoint of the subinterval; the default is mode="left" (the height of the rectangles to be determined by the left-hand endpoint of the subinterval).
EXAMPLES::
sage: f1(x) = x^2 ## example 1 sage: f2(x) = 5-x^2 sage: f = Piecewise([[(0,1),f1],[(1,2),f2]]) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: f.riemann_sum_integral_approximation(6) 17/6 sage: f.riemann_sum_integral_approximation(6,mode="right") 19/6 sage: f.riemann_sum_integral_approximation(6,mode="midpoint") 3 sage: f.integral(definite=True) 3 """ else: raise ValueError("invalid mode")
""" Returns the piecewise line function defined by the Riemann sums in numerical integration based on a subdivision into N subintervals. Set mode="midpoint" for the height of the rectangles to be determined by the midpoint of the subinterval; set mode="right" for the height of the rectangles to be determined by the right-hand endpoint of the subinterval; the default is mode="left" (the height of the rectangles to be determined by the left-hand endpoint of the subinterval).
EXAMPLES::
sage: f1(x) = x^2 sage: f2(x) = 5-x^2 sage: f = Piecewise([[(0,1),f1],[(1,2),f2]]) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: f.riemann_sum(6,mode="midpoint") doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. Piecewise defined function with 6 parts, [[(0, 1/3), 1/36], [(1/3, 2/3), 1/4], [(2/3, 1), 25/36], [(1, 4/3), 131/36], [(4/3, 5/3), 11/4], [(5/3, 2), 59/36]]
::
sage: f = Piecewise([[(-1,1),(1-x^2).function(x)]]) sage: rsf = f.riemann_sum(7) sage: P = f.plot(rgbcolor=(0.7,0.1,0.5), plot_points=40) sage: Q = rsf.plot(rgbcolor=(0.7,0.6,0.6), plot_points=40) sage: L = add([line([[a,0],[a,f(x=a)]],rgbcolor=(0.7,0.6,0.6)) for (a,b),f in rsf.list()]) sage: P + Q + L Graphics object consisting of 15 graphics primitives
::
sage: f = Piecewise([[(-1,1),(1/2+x-x^3)]], x) ## example 3 sage: rsf = f.riemann_sum(8) sage: P = f.plot(rgbcolor=(0.7,0.1,0.5), plot_points=40) sage: Q = rsf.plot(rgbcolor=(0.7,0.6,0.6), plot_points=40) sage: L = add([line([[a,0],[a,f(x=a)]],rgbcolor=(0.7,0.6,0.6)) for (a,b),f in rsf.list()]) sage: P + Q + L Graphics object consisting of 17 graphics primitives """ initial=[]) rsum = self._riemann_sum_helper(N, lambda x0,x1: [[(x0,x1),SR(self(x1))]], initial=[]) initial=[]) else: raise ValueError("invalid mode")
""" Returns the piecewise line function defined by the trapezoid rule for numerical integration based on a subdivision into N subintervals.
EXAMPLES::
sage: R.<x> = QQ[] sage: f1 = x^2 sage: f2 = 5-x^2 sage: f = Piecewise([[(0,1),f1],[(1,2),f2]]) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: f.trapezoid(4) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. Piecewise defined function with 4 parts, [[(0, 1/2), 1/2*x], [(1/2, 1), 9/2*x - 2], [(1, 3/2), 1/2*x + 2], [(3/2, 2), -7/2*x + 8]]
::
sage: R.<x> = QQ[] sage: f = Piecewise([[(-1,1),1-x^2]]) sage: tf = f.trapezoid(4) sage: P = f.plot(rgbcolor=(0.7,0.1,0.5), plot_points=40) sage: Q = tf.plot(rgbcolor=(0.7,0.6,0.6), plot_points=40) sage: L = add([line([[a,0],[a,f(a)]],rgbcolor=(0.7,0.6,0.6)) for (a,b),f in tf.list()]) sage: P+Q+L Graphics object consisting of 9 graphics primitives
::
sage: R.<x> = QQ[] sage: f = Piecewise([[(-1,1),1/2+x-x^3]]) ## example 3 sage: tf = f.trapezoid(6) sage: P = f.plot(rgbcolor=(0.7,0.1,0.5), plot_points=40) sage: Q = tf.plot(rgbcolor=(0.7,0.6,0.6), plot_points=40) sage: L = add([line([[a,0],[a,f(a)]],rgbcolor=(0.7,0.6,0.6)) for (a,b),f in tf.list()]) sage: P+Q+L Graphics object consisting of 13 graphics primitives
TESTS:
Use variables other than x (:trac:`13836`)::
sage: R.<y> = QQ[] sage: f1 = y^2 sage: f2 = 5-y^2 sage: f = Piecewise([[(0,1),f1],[(1,2),f2]]) sage: f.trapezoid(4) Piecewise defined function with 4 parts, [[(0, 1/2), 1/2*y], [(1/2, 1), 9/2*y - 2], [(1, 3/2), 1/2*y + 2], [(3/2, 2), -7/2*y + 8]]
"""
""" Returns the approximation given by the trapezoid rule for numerical integration based on a subdivision into N subintervals.
EXAMPLES::
sage: f1(x) = x^2 ## example 1 sage: f2(x) = 1-(1-x)^2 sage: f = Piecewise([[(0,1),f1],[(1,2),f2]]) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: P = f.plot(rgbcolor=(0.7,0.1,0.5), plot_points=40) sage: tf = f.trapezoid(6) sage: Q = tf.plot(rgbcolor=(0.7,0.6,0.6), plot_points=40) sage: ta = f.trapezoid_integral_approximation(6) sage: t = text('trapezoid approximation = %s'%ta, (1.5, 0.25)) sage: a = f.integral(definite=True) sage: tt = text('area under curve = %s'%a, (1.5, -0.5)) sage: P + Q + t + tt Graphics object consisting of 10 graphics primitives
::
sage: f = Piecewise([[(0,1),f1],[(1,2),f2]]) ## example 2 sage: tf = f.trapezoid(4) sage: ta = f.trapezoid_integral_approximation(4) sage: Q = tf.plot(rgbcolor=(0.7,0.6,0.6), plot_points=40) sage: t = text('trapezoid approximation = %s'%ta, (1.5, 0.25)) sage: a = f.integral(definite=True) sage: tt = text('area under curve = %s'%a, (1.5, -0.5)) sage: P+Q+t+tt Graphics object consisting of 8 graphics primitives """
""" Return the critical points of this piecewise function.
.. warning::
Uses maxima, which prints the warning to use results with caution. Only works for piecewise functions whose parts are polynomials with real critical not occurring on the interval endpoints.
EXAMPLES::
sage: R.<x> = QQ[] sage: f1 = x^0 sage: f2 = 10*x - x^2 sage: f3 = 3*x^4 - 156*x^3 + 3036*x^2 - 26208*x sage: f = Piecewise([[(0,3),f1],[(3,10),f2],[(10,20),f3]]) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: expected = [5, 12, 13, 14] sage: all(abs(e-a) < 0.001 for e,a in zip(expected, f.critical_points())) True
TESTS:
Use variables other than x (:trac:`13836`)::
sage: R.<y> = QQ[] sage: f1 = y^0 sage: f2 = 10*y - y^2 sage: f3 = 3*y^4 - 156*y^3 + 3036*y^2 - 26208*y sage: f = Piecewise([[(0,3),f1],[(3,10),f2],[(10,20),f3]]) sage: expected = [5, 12, 13, 14] sage: all(abs(e-a) < 0.001 for e,a in zip(expected, f.critical_points())) True """
""" Returns the base ring of the function pieces. This is useful when this class is extended.
EXAMPLES::
sage: f1(x) = 1 sage: f2(x) = 1-x sage: f3(x) = x^2-5 sage: f = Piecewise([[(0,1),f1],[(1,2),f2],[(2,3),f3]]) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: base_ring(f) Symbolic Ring
::
sage: R.<x> = QQ[] sage: f1 = x^0 sage: f2 = 10*x - x^2 sage: f3 = 3*x^4 - 156*x^3 + 3036*x^2 - 26208*x sage: f = Piecewise([[(0,3),f1],[(3,10),f2],[(10,20),f3]]) sage: f.base_ring() Rational Field """
""" Returns a list of all interval endpoints for this function.
EXAMPLES::
sage: f1(x) = 1 sage: f2(x) = 1-x sage: f3(x) = x^2-5 sage: f = Piecewise([[(0,1),f1],[(1,2),f2],[(2,3),f3]]) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: f.end_points() [0, 1, 2, 3] """
""" Evaluates self at x0. Returns the average value of the jump if x0 is an interior endpoint of one of the intervals of self and the usual value otherwise.
EXAMPLES::
sage: f1(x) = 1 sage: f2(x) = 1-x sage: f3(x) = exp(x) sage: f4(x) = sin(2*x) sage: f = Piecewise([[(0,1),f1],[(1,2),f2],[(2,3),f3],[(3,10),f4]]) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: f(0.5) 1 sage: f(5/2) e^(5/2) sage: f(5/2).n() 12.1824939607035 sage: f(1) 1/2 """ #x0 = QQ(x0) ## does not allow for evaluation at pi raise ValueError("Value not defined outside of domain.")
""" Returns the function piece used to evaluate self at x0.
EXAMPLES::
sage: f1(z) = z sage: f2(x) = 1-x sage: f3(y) = exp(y) sage: f4(t) = sin(2*t) sage: f = Piecewise([[(0,1),f1],[(1,2),f2],[(2,3),f3],[(3,10),f4]]) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: f.which_function(3/2) x |--> -x + 1 """ raise ValueError("Function not defined outside of domain.")
r""" Return the default variable. The default variable is defined as the first variable in the first piece that has a variable. If no pieces have a variable (each piece is a constant value), `x` is returned.
The result is cached.
AUTHOR: Paul Butler
EXAMPLES::
sage: f1(x) = 1 sage: f2(x) = 5*x sage: p = Piecewise([[(0,1),f1],[(1,4),f2]]) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: p.default_variable() x
sage: f1 = 3*var('y') sage: p = Piecewise([[(0,1),4],[(1,4),f1]]) sage: p.default_variable() y
""" except TypeError: # pass if fun is lambda function pass # default to x
r""" By default, returns the indefinite integral of the function. If definite=True is given, returns the definite integral.
AUTHOR:
- Paul Butler
EXAMPLES::
sage: f1(x) = 1-x sage: f = Piecewise([[(0,1),1],[(1,2),f1]]) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: f.integral(definite=True) 1/2
::
sage: f1(x) = -1 sage: f2(x) = 2 sage: f = Piecewise([[(0,pi/2),f1],[(pi/2,pi),f2]]) sage: f.integral(definite=True) 1/2*pi
sage: f1(x) = 2 sage: f2(x) = 3 - x sage: f = Piecewise([[(-2, 0), f1], [(0, 3), f2]]) sage: f.integral() doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. Piecewise defined function with 2 parts, [[(-2, 0), x |--> 2*x + 4], [(0, 3), x |--> -1/2*x^2 + 3*x + 4]]
sage: f1(y) = -1 sage: f2(y) = y + 3 sage: f3(y) = -y - 1 sage: f4(y) = y^2 - 1 sage: f5(y) = 3 sage: f = Piecewise([[(-4,-3),f1],[(-3,-2),f2],[(-2,0),f3],[(0,2),f4],[(2,3),f5]]) sage: F = f.integral(y) sage: F Piecewise defined function with 5 parts, [[(-4, -3), y |--> -y - 4], [(-3, -2), y |--> 1/2*y^2 + 3*y + 7/2], [(-2, 0), y |--> -1/2*y^2 - y - 1/2], [(0, 2), y |--> 1/3*y^3 - y - 1/2], [(2, 3), y |--> 3*y - 35/6]]
Ensure results are consistent with FTC::
sage: F(-3) - F(-4) -1 sage: F(-1) - F(-3) 1 sage: F(2) - F(0) 2/3 sage: f.integral(y, 0, 2) 2/3 sage: F(3) - F(-4) 19/6 sage: f.integral(y, -4, 3) 19/6 sage: f.integral(definite=True) 19/6
::
sage: f1(y) = (y+3)^2 sage: f2(y) = y+3 sage: f3(y) = 3 sage: f = Piecewise([[(-infinity, -3), f1], [(-3, 0), f2], [(0, infinity), f3]]) sage: f.integral() Piecewise defined function with 3 parts, [[(-Infinity, -3), y |--> 1/3*y^3 + 3*y^2 + 9*y + 9], [(-3, 0), y |--> 1/2*y^2 + 3*y + 9/2], [(0, +Infinity), y |--> 3*y + 9/2]]
::
sage: f1(x) = e^(-abs(x)) sage: f = Piecewise([[(-infinity, infinity), f1]]) sage: f.integral(definite=True) 2 sage: f.integral() Piecewise defined function with 1 parts, [[(-Infinity, +Infinity), x |--> -1/2*((sgn(x) - 1)*e^(2*x) - 2*e^x*sgn(x) + sgn(x) + 1)*e^(-x) - 1]]
::
sage: f = Piecewise([((0, 5), cos(x))]) sage: f.integral() Piecewise defined function with 1 parts, [[(0, 5), x |--> sin(x)]]
TESTS:
Verify that piecewise integrals of zero work (:trac:`10841`)::
sage: f0(x) = 0 sage: f = Piecewise([[(0,1),f0]]) sage: f.integral(x,0,1) 0 sage: f = Piecewise([[(0,1), 0]]) sage: f.integral(x,0,1) 0 sage: f = Piecewise([[(0,1), SR(0)]]) sage: f.integral(x,0,1) 0
"""
raise TypeError('only one endpoint given')
# The integral is computed by iterating over the pieces in order. # The definite integral for each piece is calculated and accumulated in `area`. # Thus at any time, `area` represents the definite integral of all the pieces # encountered so far. The indefinite integral of each piece is also calculated, # and the `area` before each piece is added to the piece. # # If a definite integral is requested, `area` is returned. # Otherwise, a piecewise function is constructed from the indefinite integrals # and returned. # # An exception is made if integral is called on a piecewise function # that starts at -infinity. In this case, we do not try to calculate the # definite integral of the first piece, and the value of `area` remains 0 # after the first piece.
else:
else:
""" Returns the convolution function, `f*g(t)=\int_{-\infty}^\infty f(u)g(t-u)du`, for compactly supported `f,g`.
EXAMPLES::
sage: x = PolynomialRing(QQ,'x').gen() sage: f = Piecewise([[(0,1),1*x^0]]) ## example 0 doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: g = f.convolution(f) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: h = f.convolution(g) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: P = f.plot(); Q = g.plot(rgbcolor=(1,1,0)); R = h.plot(rgbcolor=(0,1,1)); sage: # Type show(P+Q+R) to view sage: f = Piecewise([[(0,1),1*x^0],[(1,2),2*x^0],[(2,3),1*x^0]]) ## example 1 sage: g = f.convolution(f) sage: h = f.convolution(g) sage: P = f.plot(); Q = g.plot(rgbcolor=(1,1,0)); R = h.plot(rgbcolor=(0,1,1)); sage: # Type show(P+Q+R) to view sage: f = Piecewise([[(-1,1),1]]) ## example 2 sage: g = Piecewise([[(0,3),x]]) sage: f.convolution(g) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. Piecewise defined function with 3 parts, [[(-1, 1), 0], [(1, 2), -3/2*x], [(2, 4), -3/2*x]] sage: g = Piecewise([[(0,3),1*x^0],[(3,4),2*x^0]]) sage: f.convolution(g) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. Piecewise defined function with 5 parts, [[(-1, 1), x + 1], [(1, 2), 3], [(2, 3), x], [(3, 4), -x + 8], [(4, 5), -2*x + 10]] """ # this is a very, very, very ugly hack else: h = Piecewise([[(a1+b1,a1+b2),fg1],[(a1+b2,a2+b2),fg3]]) else: else:
r""" Returns the derivative (as computed by maxima) Piecewise(I,`(d/dx)(self|_I)`), as I runs over the intervals belonging to self. self must be piecewise polynomial.
EXAMPLES::
sage: f1(x) = 1 sage: f2(x) = 1-x sage: f = Piecewise([[(0,1),f1],[(1,2),f2]]) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: f.derivative() doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. Piecewise defined function with 2 parts, [[(0, 1), x |--> 0], [(1, 2), x |--> -1]] sage: f1(x) = -1 sage: f2(x) = 2 sage: f = Piecewise([[(0,pi/2),f1],[(pi/2,pi),f2]]) sage: f.derivative() Piecewise defined function with 2 parts, [[(0, 1/2*pi), x |--> 0], [(1/2*pi, pi), x |--> 0]]
::
sage: f = Piecewise([[(0,1), (x * 2)]], x) sage: f.derivative() Piecewise defined function with 1 parts, [[(0, 1), x |--> 2]] """
""" Computes the linear function defining the tangent line of the piecewise function self.
EXAMPLES::
sage: f1(x) = x^2 sage: f2(x) = 5-x^3+x sage: f = Piecewise([[(0,1),f1],[(1,2),f2]]) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: tf = f.tangent_line(0.9) ## tangent line at x=0.9 doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: P = f.plot(rgbcolor=(0.7,0.1,0.5), plot_points=40) sage: Q = tf.plot(rgbcolor=(0.7,0.2,0.2), plot_points=40) sage: P + Q Graphics object consisting of 4 graphics primitives """
""" Returns the plot of self.
Keyword arguments are passed onto the plot command for each piece of the function. E.g., the plot_points keyword affects each segment of the plot.
EXAMPLES::
sage: f1(x) = 1 sage: f2(x) = 1-x sage: f3(x) = exp(x) sage: f4(x) = sin(2*x) sage: f = Piecewise([[(0,1),f1],[(1,2),f2],[(2,3),f3],[(3,10),f4]]) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: P = f.plot(rgbcolor=(0.7,0.1,0), plot_points=40) sage: P Graphics object consisting of 4 graphics primitives
Remember: to view this, type show(P) or P.save("path/myplot.png") and then open it in a graphics viewer such as GIMP.
TESTS:
We should not add each piece to the legend individually, since this creates duplicates (:trac:`12651`). This tests that only one of the graphics objects in the plot has a non-``None`` ``legend_label``::
sage: f1 = sin(x) sage: f2 = cos(x) sage: f = piecewise([[(-1,0), f1],[(0,1), f2]]) sage: p = f.plot(legend_label='$f(x)$') sage: lines = [ ....: line ....: for line in p._objects ....: if line.options()['legend_label'] is not None ] sage: len(lines) 1 """
# If it's the first piece, pass all arguments. Otherwise, # filter out 'legend_label' so that we don't add each # piece to the legend separately (trac #12651). del kwds['legend_label']
r""" Returns the n-th Fourier series coefficient of `\cos(n\pi x/L)`, `a_n`.
INPUT:
- ``self`` - the function f(x), defined over -L x L
- ``n`` - an integer n=0
- ``L`` - (the period)/2
OUTPUT: `a_n = \frac{1}{L}\int_{-L}^L f(x)\cos(n\pi x/L)dx`
EXAMPLES::
sage: f(x) = x^2 sage: f = Piecewise([[(-1,1),f]]) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: f.fourier_series_cosine_coefficient(2,1) pi^(-2) sage: f(x) = x^2 sage: f = Piecewise([[(-pi,pi),f]]) sage: f.fourier_series_cosine_coefficient(2,pi) 1 sage: f1(x) = -1 sage: f2(x) = 2 sage: f = Piecewise([[(-pi,pi/2),f1],[(pi/2,pi),f2]]) sage: f.fourier_series_cosine_coefficient(5,pi) -3/5/pi """ for (a,b), f in self.list()]) return result
r""" Returns the n-th Fourier series coefficient of `\sin(n\pi x/L)`, `b_n`.
INPUT:
- ``self`` - the function f(x), defined over -L x L
- ``n`` - an integer n0
- ``L`` - (the period)/2
OUTPUT: `b_n = \frac{1}{L}\int_{-L}^L f(x)\sin(n\pi x/L)dx`
EXAMPLES::
sage: f(x) = x^2 sage: f = Piecewise([[(-1,1),f]]) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: f.fourier_series_sine_coefficient(2,1) # L=1, n=2 0 """ for (a,b), f in self.list()]) return result
r""" A helper function for the construction of Fourier series. The argument scale_function is a function which takes in n, representing the `n^{th}` coefficient, and return an expression to scale the sine and cosine coefficients by.
EXAMPLES::
sage: f(x) = x^2 sage: f = Piecewise([[(-1,1),f]]) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: f._fourier_series_helper(3, 1, lambda n: 1) cos(2*pi*x)/pi^2 - 4*cos(pi*x)/pi^2 + 1/3 """ self.fourier_series_sine_coefficient(n,L)*sin(n*pi*x/L))* scale_function(n) for n in srange(1,N)])
r""" Returns the partial sum
.. MATH::
f(x) \sim \frac{a_0}{2} + \sum_{n=1}^N [a_n\cos(\frac{n\pi x}{L}) + b_n\sin(\frac{n\pi x}{L})],
as a string.
EXAMPLES::
sage: f(x) = x^2 sage: f = Piecewise([[(-1,1),f]]) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: f.fourier_series_partial_sum(3,1) cos(2*pi*x)/pi^2 - 4*cos(pi*x)/pi^2 + 1/3 sage: f1(x) = -1 sage: f2(x) = 2 sage: f = Piecewise([[(-pi,pi/2),f1],[(pi/2,pi),f2]]) sage: f.fourier_series_partial_sum(3,pi) -3*cos(x)/pi - 3*sin(2*x)/pi + 3*sin(x)/pi - 1/4 """
r""" Returns the Cesaro partial sum
.. MATH::
f(x) \sim \frac{a_0}{2} + \sum_{n=1}^N (1-n/N)*[a_n\cos(\frac{n\pi x}{L}) + b_n\sin(\frac{n\pi x}{L})],
as a string. This is a "smoother" partial sum - the Gibbs phenomenon is mollified.
EXAMPLES::
sage: f(x) = x^2 sage: f = Piecewise([[(-1,1),f]]) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: f.fourier_series_partial_sum_cesaro(3,1) 1/3*cos(2*pi*x)/pi^2 - 8/3*cos(pi*x)/pi^2 + 1/3 sage: f1(x) = -1 sage: f2(x) = 2 sage: f = Piecewise([[(-pi,pi/2),f1],[(pi/2,pi),f2]]) sage: f.fourier_series_partial_sum_cesaro(3,pi) -2*cos(x)/pi - sin(2*x)/pi + 2*sin(x)/pi - 1/4 """
r""" Returns the Hann-filtered partial sum (named after von Hann, not Hamming)
.. MATH::
f(x) \sim \frac{a_0}{2} + \sum_{n=1}^N H_N(n)*[a_n\cos(\frac{n\pi x}{L}) + b_n\sin(\frac{n\pi x}{L})],
as a string, where `H_N(x) = (1+\cos(\pi x/N))/2`. This is a "smoother" partial sum - the Gibbs phenomenon is mollified.
EXAMPLES::
sage: f(x) = x^2 sage: f = Piecewise([[(-1,1),f]]) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: f.fourier_series_partial_sum_hann(3,1) 1/4*cos(2*pi*x)/pi^2 - 3*cos(pi*x)/pi^2 + 1/3 sage: f1(x) = -1 sage: f2(x) = 2 sage: f = Piecewise([[(-pi,pi/2),f1],[(pi/2,pi),f2]]) sage: f.fourier_series_partial_sum_hann(3,pi) -9/4*cos(x)/pi - 3/4*sin(2*x)/pi + 9/4*sin(x)/pi - 1/4 """
r""" Returns the "filtered" partial sum
.. MATH::
f(x) \sim \frac{a_0}{2} + \sum_{n=1}^N F_n*[a_n\cos(\frac{n\pi x}{L}) + b_n\sin(\frac{n\pi x}{L})],
as a string, where `F = [F_1,F_2, ..., F_{N}]` is a list of length `N` consisting of real numbers. This can be used to plot FS solutions to the heat and wave PDEs.
EXAMPLES::
sage: f(x) = x^2 sage: f = Piecewise([[(-1,1),f]]) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: f.fourier_series_partial_sum_filtered(3,1,[1,1,1]) cos(2*pi*x)/pi^2 - 4*cos(pi*x)/pi^2 + 1/3 sage: f1(x) = -1 sage: f2(x) = 2 sage: f = Piecewise([[(-pi,pi/2),f1],[(pi/2,pi),f2]]) sage: f.fourier_series_partial_sum_filtered(3,pi,[1,1,1]) -3*cos(x)/pi - 3*sin(2*x)/pi + 3*sin(x)/pi - 1/4 """
r""" Plots the partial sum
.. MATH::
f(x) \sim \frac{a_0}{2} + sum_{n=1}^N [a_n\cos(\frac{n\pi x}{L}) + b_n\sin(\frac{n\pi x}{L})],
over xmin x xmin.
EXAMPLES::
sage: f1(x) = -2 sage: f2(x) = 1 sage: f3(x) = -1 sage: f4(x) = 2 sage: f = Piecewise([[(-pi,-pi/2),f1],[(-pi/2,0),f2],[(0,pi/2),f3],[(pi/2,pi),f4]]) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: P = f.plot_fourier_series_partial_sum(3,pi,-5,5) # long time sage: f1(x) = -1 sage: f2(x) = 2 sage: f = Piecewise([[(-pi,pi/2),f1],[(pi/2,pi),f2]]) sage: P = f.plot_fourier_series_partial_sum(15,pi,-5,5) # long time
Remember, to view this type show(P) or P.save("path/myplot.png") and then open it in a graphics viewer such as GIMP. """ from sage.plot.all import plot return plot(self.fourier_series_partial_sum(N,L), xmin, xmax, **kwds)
r""" Plots the partial sum
.. MATH::
f(x) \sim \frac{a_0}{2} + \sum_{n=1}^N (1-n/N)*[a_n\cos(\frac{n\pi x}{L}) + b_n\sin(\frac{n\pi x}{L})],
over xmin x xmin. This is a "smoother" partial sum - the Gibbs phenomenon is mollified.
EXAMPLES::
sage: f1(x) = -2 sage: f2(x) = 1 sage: f3(x) = -1 sage: f4(x) = 2 sage: f = Piecewise([[(-pi,-pi/2),f1],[(-pi/2,0),f2],[(0,pi/2),f3],[(pi/2,pi),f4]]) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: P = f.plot_fourier_series_partial_sum_cesaro(3,pi,-5,5) # long time sage: f1(x) = -1 sage: f2(x) = 2 sage: f = Piecewise([[(-pi,pi/2),f1],[(pi/2,pi),f2]]) sage: P = f.plot_fourier_series_partial_sum_cesaro(15,pi,-5,5) # long time
Remember, to view this type show(P) or P.save("path/myplot.png") and then open it in a graphics viewer such as GIMP. """ from sage.plot.all import plot return plot(self.fourier_series_partial_sum_cesaro(N,L), xmin, xmax, **kwds)
r""" Plots the partial sum
.. MATH::
f(x) \sim \frac{a_0}{2} + \sum_{n=1}^N H_N(n)*[a_n\cos(\frac{n\pi x}{L}) + b_n\sin(\frac{n\pi x}{L})],
over xmin x xmin, where H_N(x) = (0.5)+(0.5)\*cos(x\*pi/N) is the N-th Hann filter.
EXAMPLES::
sage: f1(x) = -2 sage: f2(x) = 1 sage: f3(x) = -1 sage: f4(x) = 2 sage: f = Piecewise([[(-pi,-pi/2),f1],[(-pi/2,0),f2],[(0,pi/2),f3],[(pi/2,pi),f4]]) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: P = f.plot_fourier_series_partial_sum_hann(3,pi,-5,5) # long time sage: f1(x) = -1 sage: f2(x) = 2 sage: f = Piecewise([[(-pi,pi/2),f1],[(pi/2,pi),f2]]) sage: P = f.plot_fourier_series_partial_sum_hann(15,pi,-5,5) # long time
Remember, to view this type show(P) or P.save("path/myplot.png") and then open it in a graphics viewer such as GIMP. """ from sage.plot.all import plot return plot(self.fourier_series_partial_sum_hann(N,L), xmin, xmax, **kwds)
r""" Plots the partial sum
.. MATH::
f(x) \sim \frac{a_0}{2} + \sum_{n=1}^N F_n*[a_n\cos(\frac{n\pi x}{L}) + b_n\sin(\frac{n\pi x}{L})],
over xmin x xmin, where `F = [F_1,F_2, ..., F_{N}]` is a list of length `N` consisting of real numbers. This can be used to plot FS solutions to the heat and wave PDEs.
EXAMPLES::
sage: f1(x) = -2 sage: f2(x) = 1 sage: f3(x) = -1 sage: f4(x) = 2 sage: f = Piecewise([[(-pi,-pi/2),f1],[(-pi/2,0),f2],[(0,pi/2),f3],[(pi/2,pi),f4]]) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: P = f.plot_fourier_series_partial_sum_filtered(3,pi,[1]*3,-5,5) # long time sage: f1(x) = -1 sage: f2(x) = 2 sage: f = Piecewise([[(-pi,-pi/2),f1],[(-pi/2,0),f2],[(0,pi/2),f1],[(pi/2,pi),f2]]) sage: P = f.plot_fourier_series_partial_sum_filtered(15,pi,[1]*15,-5,5) # long time
Remember, to view this type show(P) or P.save("path/myplot.png") and then open it in a graphics viewer such as GIMP. """ from sage.plot.all import plot return plot(self.fourier_series_partial_sum_filtered(N,L,F), xmin, xmax, **kwds)
r""" Returns the value of the Fourier series coefficient of self at `x`,
.. MATH::
f(x) \sim \frac{a_0}{2} + \sum_{n=1}^\infty [a_n\cos(\frac{n\pi x}{L}) + b_n\sin(\frac{n\pi x}{L})], \ \ \ -L<x<L.
This method applies to piecewise non-polynomial functions as well.
INPUT:
- ``self`` - the function f(x), defined over -L x L
- ``x`` - a real number
- ``L`` - (the period)/2
OUTPUT: `(f^*(x+)+f^*(x-)/2`, where `f^*` denotes the function `f` extended to `\RR` with period `2L` (Dirichlet's Theorem for Fourier series).
EXAMPLES::
sage: f1(x) = 1 sage: f2(x) = 1-x sage: f3(x) = exp(x) sage: f4(x) = sin(2*x) sage: f = Piecewise([[(-10,1),f1],[(1,2),f2],[(2,3),f3],[(3,10),f4]]) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: f.fourier_series_value(101,10) 1/2 sage: f.fourier_series_value(100,10) 1 sage: f.fourier_series_value(10,10) 1/2*sin(20) + 1/2 sage: f.fourier_series_value(20,10) 1 sage: f.fourier_series_value(30,10) 1/2*sin(20) + 1/2 sage: f1(x) = -1 sage: f2(x) = 2 sage: f = Piecewise([[(-pi,0),lambda x:0],[(0,pi/2),f1],[(pi/2,pi),f2]]) sage: f.fourier_series_value(-1,pi) 0 sage: f.fourier_series_value(20,pi) -1 sage: f.fourier_series_value(pi/2,pi) 1/2 """ else:
r""" Returns the n-th cosine series coefficient of `\cos(n\pi x/L)`, `a_n`.
INPUT:
- ``self`` - the function f(x), defined over 0 x L (no checking is done to insure this)
- ``n`` - an integer n=0
- ``L`` - (the period)/2
OUTPUT: `a_n = \frac{2}{L}\int_{-L}^L f(x)\cos(n\pi x/L)dx` such that
.. MATH::
f(x) \sim \frac{a_0}{2} + \sum_{n=1}^\infty a_n\cos(\frac{n\pi x}{L}),\ \ 0<x<L.
EXAMPLES::
sage: f(x) = x sage: f = Piecewise([[(0,1),f]]) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: f.cosine_series_coefficient(2,1) 0 sage: f.cosine_series_coefficient(3,1) -4/9/pi^2 sage: f1(x) = -1 sage: f2(x) = 2 sage: f = Piecewise([[(0,pi/2),f1],[(pi/2,pi),f2]]) sage: f.cosine_series_coefficient(2,pi) 0 sage: f.cosine_series_coefficient(3,pi) 2/pi sage: f.cosine_series_coefficient(111,pi) 2/37/pi sage: f1 = lambda x: x*(pi-x) sage: f = Piecewise([[(0,pi),f1]]) sage: f.cosine_series_coefficient(0,pi) 1/3*pi^2
""" for (a,b), f in self.list()]) return result
r""" Returns the n-th sine series coefficient of `\sin(n\pi x/L)`, `b_n`.
INPUT:
- ``self`` - the function f(x), defined over 0 x L (no checking is done to insure this)
- ``n`` - an integer n0
- ``L`` - (the period)/2
OUTPUT:
`b_n = \frac{2}{L}\int_{-L}^L f(x)\sin(n\pi x/L)dx` such that
.. MATH::
f(x) \sim \sum_{n=1}^\infty b_n\sin(\frac{n\pi x}{L}),\ \ 0<x<L.
EXAMPLES::
sage: f(x) = 1 sage: f = Piecewise([[(0,1),f]]) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: f.sine_series_coefficient(2,1) 0 sage: f.sine_series_coefficient(3,1) 4/3/pi """ for (a,b), f in self.list()]) return result
r""" Returns the Laplace transform of self with respect to the variable var.
INPUT:
- ``x`` - variable of self
- ``s`` - variable of Laplace transform.
We assume that a piecewise function is 0 outside of its domain and that the left-most endpoint of the domain is 0.
EXAMPLES::
sage: x, s, w = var('x, s, w') sage: f = Piecewise([[(0,1),1],[(1,2), 1-x]]) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: f.laplace(x, s) -e^(-s)/s + (s + 1)*e^(-2*s)/s^2 + 1/s - e^(-s)/s^2 sage: f.laplace(x, w) -e^(-w)/w + (w + 1)*e^(-2*w)/w^2 + 1/w - e^(-w)/w^2
::
sage: y, t = var('y, t') sage: f = Piecewise([[(1,2), 1-y]]) sage: f.laplace(y, t) (t + 1)*e^(-2*t)/t^2 - e^(-t)/t^2
::
sage: s = var('s') sage: t = var('t') sage: f1(t) = -t sage: f2(t) = 2 sage: f = Piecewise([[(0,1),f1],[(1,infinity),f2]]) sage: f.laplace(t,s) (s + 1)*e^(-s)/s^2 + 2*e^(-s)/s - 1/s^2 """ for (a,b),f in self.list()])
""" Returns self and other extended to be defined on the same domain as well as a refinement of their intervals. This is used for adding and multiplying piecewise functions.
EXAMPLES::
sage: R.<x> = QQ[] sage: f1 = Piecewise([[(0, 2), x]]) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: f2 = Piecewise([[(1, 3), x^2]]) sage: f1._make_compatible(f2) (Piecewise defined function with 2 parts, [[(0, 2), x], [(2, 3), 0]], Piecewise defined function with 2 parts, [[(0, 1), 0], [(1, 3), x^2]], [(0, 1), (1, 2), (2, 3)]) """
""" Returns the piecewise defined function which is the sum of self and other. Does not require both domains be the same.
EXAMPLES::
sage: x = PolynomialRing(QQ,'x').gen() sage: f1 = x^0 sage: f2 = 1-x sage: f3 = 2*x sage: f4 = 10-x sage: f = Piecewise([[(0,1),f1],[(1,2),f2],[(2,3),f3],[(3,10),f4]]) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: g1 = x-2 sage: g2 = x-5 sage: g = Piecewise([[(0,5),g1],[(5,10),g2]]) sage: h = f+g sage: h Piecewise defined function with 5 parts, [[(0, 1), x - 1], [(1, 2), -1], [(2, 3), 3*x - 2], [(3, 5), 8], [(5, 10), 5]]
Note that in this case the functions must be defined using polynomial expressions *not* using the lambda notation. """
r""" Returns the piecewise defined function which is the product of one piecewise function (self) with another one (other).
EXAMPLES::
sage: x = PolynomialRing(QQ,'x').gen() sage: f1 = x^0 sage: f2 = 1-x sage: f3 = 2*x sage: f4 = 10-x sage: f = Piecewise([[(0,1),f1],[(1,2),f2],[(2,3),f3],[(3,10),f4]]) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: g1 = x-2 sage: g2 = x-5 sage: g = Piecewise([[(0,5),g1],[(5,10),g2]]) sage: h = f*g doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: h Piecewise defined function with 5 parts, [[(0, 1), x - 2], [(1, 2), -x^2 + 3*x - 2], [(2, 3), 2*x^2 - 4*x], [(3, 5), -x^2 + 12*x - 20], [(5, 10), -x^2 + 15*x - 50]] sage: g*(11/2) Piecewise defined function with 2 parts, [[(0, 5), 11/2*x - 11], [(5, 10), 11/2*x - 55/2]]
Note that in this method the functions must be defined using polynomial expressions *not* using the lambda notation. """ ## needed for scalar multiplication else:
""" Implements Boolean == operator.
EXAMPLES::
sage: f1 = x^0 sage: f2 = 1-x sage: f3 = 2*x sage: f4 = 10-x sage: f = Piecewise([[(0,1),f1],[(1,2),f2],[(2,3),f3],[(3,10),f4]]) doctest:...: DeprecationWarning: use lower-case piecewise instead See http://trac.sagemath.org/14801 for details. sage: g = Piecewise([[(0,1),1],[(1,2),f2],[(2,3),f3],[(3,10),f4]]) sage: f==g True """ |