functional.py

"""
Functional notation support for common calculus methods

EXAMPLES: We illustrate each of the calculus functional functions.

::

    sage: simplify(x - x)
    0
    sage: a = var('a')
    sage: derivative(x^a + sin(x), x)
    a*x^(a - 1) + cos(x)
    sage: diff(x^a + sin(x), x)
    a*x^(a - 1) + cos(x)
    sage: derivative(x^a + sin(x), x)
    a*x^(a - 1) + cos(x)
    sage: integral(a*x*sin(x), x)
    -(x*cos(x) - sin(x))*a
    sage: integrate(a*x*sin(x), x)
    -(x*cos(x) - sin(x))*a
    sage: limit(a*sin(x)/x, x=0)
    a
    sage: taylor(a*sin(x)/x, x, 0, 4)
    1/120*a*x^4 - 1/6*a*x^2 + a
    sage: expand( (x-a)^3 )
    -a^3 + 3*a^2*x - 3*a*x^2 + x^3
    sage: laplace( e^(x+a), x, a)
    e^a/(a - 1)
    sage: inverse_laplace( e^a/(a-1), x, a)
    ilt(e^a/(a - 1), x, a)
"""

from calculus import SR
from sage.symbolic.expression import Expression

def simplify(f):
    r"""
    Simplify the expression `f`.
    
    EXAMPLES: We simplify the expression `i + x - x`.
    
    ::
    
        sage: f = I + x - x; simplify(f)
        I
    
    In fact, printing `f` yields the same thing - i.e., the
    simplified form.
    """
    try:
        return f.simplify()
    except AttributeError:
        return f

def derivative(f, *args, **kwds):
    """
    The derivative of `f`.
    
    Repeated differentiation is supported by the syntax given in the
    examples below.
    
    ALIAS: diff
    
    EXAMPLES: We differentiate a callable symbolic function::
    
        sage: f(x,y) = x*y + sin(x^2) + e^(-x)
        sage: f 
        (x, y) |--> x*y + e^(-x) + sin(x^2)
        sage: derivative(f, x)
        (x, y) |--> 2*x*cos(x^2) + y - e^(-x)
        sage: derivative(f, y)
        (x, y) |--> x
    
    We differentiate a polynomial::
    
        sage: t = polygen(QQ, 't')
        sage: f = (1-t)^5; f
        -t^5 + 5*t^4 - 10*t^3 + 10*t^2 - 5*t + 1
        sage: derivative(f)
        -5*t^4 + 20*t^3 - 30*t^2 + 20*t - 5
        sage: derivative(f, t)
        -5*t^4 + 20*t^3 - 30*t^2 + 20*t - 5
        sage: derivative(f, t, t)
        -20*t^3 + 60*t^2 - 60*t + 20
        sage: derivative(f, t, 2)
        -20*t^3 + 60*t^2 - 60*t + 20
        sage: derivative(f, 2)
        -20*t^3 + 60*t^2 - 60*t + 20
    
    We differentiate a symbolic expression::
    
        sage: var('a x')
        (a, x)
        sage: f = exp(sin(a - x^2))/x
        sage: derivative(f, x)
        -2*e^(sin(-x^2 + a))*cos(-x^2 + a) - e^(sin(-x^2 + a))/x^2
        sage: derivative(f, a)
        e^(sin(-x^2 + a))*cos(-x^2 + a)/x
    
    Syntax for repeated differentiation::
    
        sage: R.<u, v> = PolynomialRing(QQ)
        sage: f = u^4*v^5
        sage: derivative(f, u)
        4*u^3*v^5
        sage: f.derivative(u)   # can always use method notation too
        4*u^3*v^5
    
    ::
    
        sage: derivative(f, u, u)
        12*u^2*v^5
        sage: derivative(f, u, u, u)
        24*u*v^5
        sage: derivative(f, u, 3)
        24*u*v^5
    
    ::
    
        sage: derivative(f, u, v)
        20*u^3*v^4
        sage: derivative(f, u, 2, v)
        60*u^2*v^4
        sage: derivative(f, u, v, 2)
        80*u^3*v^3
        sage: derivative(f, [u, v, v])
        80*u^3*v^3
    """
    try:
        return f.derivative(*args, **kwds)
    except AttributeError:
        pass
    if not isinstance(f, Expression):
        f = SR(f)
    return f.derivative(*args, **kwds)

diff = derivative

def integral(f, *args, **kwds):
    r"""
    The integral of `f`.
    
    EXAMPLES::
    
        sage: integral(sin(x), x)
        -cos(x)
        sage: integral(sin(x)^2, x, pi, 123*pi/2)
        121/4*pi
        sage: integral( sin(x), x, 0, pi)
        2
    
    We integrate a symbolic function::
    
        sage: f(x,y,z) = x*y/z + sin(z)
        sage: integral(f, z)
        (x, y, z) |--> x*y*log(z) - cos(z)
    
    ::
    
        sage: var('a,b')
        (a, b)
        sage: assume(b-a>0)
        sage: integral( sin(x), x, a, b)
        cos(a) - cos(b)
        sage: forget()
    
    ::
    
        sage: integral(x/(x^3-1), x)
        1/3*sqrt(3)*arctan(1/3*(2*x + 1)*sqrt(3)) + 1/3*log(x - 1) - 1/6*log(x^2 + x + 1)
    
    ::
    
        sage: integral( exp(-x^2), x )
        1/2*sqrt(pi)*erf(x)
    
    We define the Gaussian, plot and integrate it numerically and
    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: numerical_integral(f, -4, 4)                    # random output
        (0.99993665751633376, 1.1101527003413533e-14)
        sage: integrate(f, x)
        x |--> 1/2*erf(1/2*sqrt(2)*x)
    
    You can have Sage calculate multiple integrals. For example,
    consider the function `exp(y^2)` on the region between the
    lines `x=y`, `x=1`, and `y=0`. We find the
    value of the integral on this region using the command::
    
        sage: area = integral(integral(exp(y^2),x,0,y),y,0,1); area
        1/2*e - 1/2
        sage: float(area)
        0.859140914229522...
    
    We compute the line integral of `\sin(x)` along the arc of
    the curve `x=y^4` from `(1,-1)` to
    `(1,1)`::
    
        sage: t = var('t')
        sage: (x,y) = (t^4,t)
        sage: (dx,dy) = (diff(x,t), diff(y,t))
        sage: integral(sin(x)*dx, t,-1, 1)
        0
        sage: restore('x,y')   # restore the symbolic variables x and y
    
    Sage is unable to do anything with the following integral::
    
        sage: integral( exp(-x^2)*log(x), x )
        integrate(e^(-x^2)*log(x), x)
    
    Note, however, that::
    
        sage: integral( exp(-x^2)*ln(x), x, 0, oo)
        -1/4*(euler_gamma + 2*log(2))*sqrt(pi)
    
    This definite integral is easy::
    
        sage: integral( ln(x)/x, x, 1, 2)
        1/2*log(2)^2
    
    Sage can't do this elliptic integral (yet)::
    
        sage: integral(1/sqrt(2*t^4 - 3*t^2 - 2), t, 2, 3)
        integrate(1/sqrt(2*t^4 - 3*t^2 - 2), t, 2, 3)
    
    A double integral::
    
        sage: y = var('y')
        sage: integral(integral(x*y^2, x, 0, y), y, -2, 2)
        32/5
    
    This illustrates using assumptions::
    
        sage: integral(abs(x), x, 0, 5)
        25/2
        sage: a = var("a")
        sage: integral(abs(x), x, 0, a)
        1/2*a*abs(a)
        sage: integral(abs(x)*x, x, 0, a)
        Traceback (most recent call last):
        ...
        ValueError: Computation failed since Maxima requested additional
        constraints; using the 'assume' command before integral evaluation
        *may* help (example of legal syntax is 'assume(a>0)',
        see `assume?` for more details)
        Is  a  positive, negative, or zero?
        sage: assume(a>0)
        sage: integral(abs(x)*x, x, 0, a)
        1/3*a^3
        sage: forget()      # forget the assumptions.
    
    We integrate and differentiate a huge mess::
    
        sage: f = (x^2-1+3*(1+x^2)^(1/3))/(1+x^2)^(2/3)*x/(x^2+2)^2
        sage: g = integral(f, x)
        sage: h = f - diff(g, x)
    
    ::
    
        sage: [float(h(i)) for i in range(5)] #random
        
        [0.0,
         -1.1102230246251565e-16,
         -5.5511151231257827e-17,
         -5.5511151231257827e-17,
         -6.9388939039072284e-17]
        sage: h.factor()
        0
        sage: bool(h == 0)
        True
    """
    try:
        return f.integral(*args, **kwds)
    except AttributeError:
        pass

    if not isinstance(f, Expression):
        f = SR(f)
    return f.integral(*args, **kwds)

integrate = integral

def limit(f, dir=None, taylor=False, **argv):
    r"""
    Return the limit as the variable `v` approaches `a`
    from the given direction.
    
    ::
    
                limit(expr, x = a)
                limit(expr, x = a, dir='above')
                
    
    INPUT:
    
    - ``dir`` - (default: None); dir may have the value
       'plus' (or 'above') for a limit from above, 'minus' (or 'below')
       for a limit from below, or may be omitted (implying a two-sided
       limit is to be computed).
    
    - ``taylor`` - (default: False); if True, use Taylor
       series, which allows more limits to be computed (but may also
       crash in some obscure cases due to bugs in Maxima).
    
    - ``\*\*argv`` - 1 named parameter
    
    ALIAS: You can also use lim instead of limit.
    
    EXAMPLES::
    
        sage: limit(sin(x)/x, x=0)
        1
        sage: limit(exp(x), x=oo)
        +Infinity
        sage: lim(exp(x), x=-oo)
        0
        sage: lim(1/x, x=0)
        Infinity
        sage: limit(sqrt(x^2+x+1)+x, taylor=True, x=-oo)
        -1/2
        sage: limit((tan(sin(x)) - sin(tan(x)))/x^7, taylor=True, x=0)
        1/30
    
    Sage does not know how to do this limit (which is 0), so it returns
    it unevaluated::
    
        sage: lim(exp(x^2)*(1-erf(x)), x=infinity)
        -limit((erf(x) - 1)*e^(x^2), x, +Infinity)
    """
    if not isinstance(f, Expression):
        f = SR(f)
    return f.limit(dir=dir, taylor=taylor, **argv)

lim = limit

def taylor(f, *args):
    """
    Expands self in a truncated Taylor or Laurent series in the
    variable `v` around the point `a`, containing terms
    through `(x - a)^n`. Functions in more variables are also
    supported.
    
    INPUT:
    
    - ``*args`` - the following notation is supported
    
    - ``x, a, n`` - variable, point, degree
    
    - ``(x, a), (y, b), ..., n`` - variables with points, degree of polynomial
    
    EXAMPLES::
    
        sage: var('x,k,n')
        (x, k, n)
        sage: taylor (sqrt (1 - k^2*sin(x)^2), x, 0, 6)
        -1/720*(45*k^6 - 60*k^4 + 16*k^2)*x^6 - 1/24*(3*k^4 - 4*k^2)*x^4 - 1/2*k^2*x^2 + 1
        
    ::

        sage: taylor ((x + 1)^n, x, 0, 4)
        1/24*(n^4 - 6*n^3 + 11*n^2 - 6*n)*x^4 + 1/6*(n^3 - 3*n^2 + 2*n)*x^3 + 1/2*(n^2 - n)*x^2 + n*x + 1
        
    ::

        sage: taylor ((x + 1)^n, x, 0, 4)
        1/24*(n^4 - 6*n^3 + 11*n^2 - 6*n)*x^4 + 1/6*(n^3 - 3*n^2 + 2*n)*x^3 + 1/2*(n^2 - n)*x^2 + n*x + 1
        
    Taylor polynomial in two variables::

        sage: x,y=var('x y'); taylor(x*y^3,(x,1),(y,-1),4)
        (y + 1)^3*(x - 1) + (y + 1)^3 - 3*(y + 1)^2*(x - 1) - 3*(y + 1)^2 + 3*(y + 1)*(x - 1) - x + 3*y + 3 
    """
    if not isinstance(f, Expression):
        f = SR(f)
    return f.taylor(*args)

def expand(x, *args, **kwds):
    """
    EXAMPLES::
    
        sage: a = (x-1)*(x^2 - 1); a
        (x - 1)*(x^2 - 1)
        sage: expand(a)
        x^3 - x^2 - x + 1
    
    You can also use expand on polynomial, integer, and other
    factorizations::
    
        sage: x = polygen(ZZ)
        sage: F = factor(x^12 - 1); F
        (x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + 1) * (x^2 + x + 1) * (x^4 - x^2 + 1)
        sage: expand(F)
        x^12 - 1
        sage: F.expand()
        x^12 - 1
        sage: F = factor(2007); F
        3^2 * 223
        sage: expand(F)
        2007
    
    Note: If you want to compute the expanded form of a polynomial
    arithmetic operation quickly and the coefficients of the polynomial
    all lie in some ring, e.g., the integers, it is vastly faster to
    create a polynomial ring and do the arithmetic there.
    
    ::
    
        sage: x = polygen(ZZ)      # polynomial over a given base ring.
        sage: f = sum(x^n for n in range(5))
        sage: f*f                  # much faster, even if the degree is huge
        x^8 + 2*x^7 + 3*x^6 + 4*x^5 + 5*x^4 + 4*x^3 + 3*x^2 + 2*x + 1
    
    TESTS::
    
        sage: t1 = (sqrt(3)-3)*(sqrt(3)+1)/6;
        sage: tt1 = -1/sqrt(3);
        sage: t2 = sqrt(3)/6;
        sage: float(t1)
        -0.577350269189625...
        sage: float(tt1)
        -0.577350269189625...
        sage: float(t2)
        0.28867513459481287
        sage: float(expand(t1 + t2))
        -0.288675134594812...
        sage: float(expand(tt1 + t2))
        -0.288675134594812...
    """
    try:
        return x.expand(*args, **kwds)
    except AttributeError:
        return x