Python sympy 模块，atan() 实例源码

def real(self, nested_scope=None):
        """Return the correspond floating point number."""
        op = self.children[0].name
        expr = self.children[1]
        dispatch = {
            'sin': sympy.sin,
            'cos': sympy.cos,
            'tan': sympy.tan,
            'asin': sympy.asin,
            'acos': sympy.acos,
            'atan': sympy.atan,
            'exp': sympy.exp,
            'ln': sympy.log,
            'sqrt': sympy.sqrt
        }
        if op in dispatch:
            arg = expr.real(nested_scope)
            return dispatch[op](arg)
        else:
            raise NodeException("internal error: undefined external")

def sym(self, nested_scope=None):
        """Return the corresponding symbolic expression."""
        op = self.children[0].name
        expr = self.children[1]
        dispatch = {
            'sin': sympy.sin,
            'cos': sympy.cos,
            'tan': sympy.tan,
            'asin': sympy.asin,
            'acos': sympy.acos,
            'atan': sympy.atan,
            'exp': sympy.exp,
            'ln': sympy.log,
            'sqrt': sympy.sqrt
        }
        if op in dispatch:
            arg = expr.sym(nested_scope)
            return dispatch[op](arg)
        else:
            raise NodeException("internal error: undefined external")

def test_callback_factory__broadcast():
    args = x, y = symbols('x y')
    expr = x + atan(y)
    cb = _callback_factory(args, [expr], 'numpy', np.float64, 'C')
    inp = np.array([[17, 1], [18, 2]])
    ref = [17 + np.arctan(1), 18 + np.arctan(2)]
    assert np.allclose(cb(inp), ref)

    inp2 = np.array([
        [[17, 1], [18, 2]],
        [[27, 21], [28, 22]]
    ])
    ref2 = [
        [17 + np.arctan(1), 18 + np.arctan(2)],
        [27 + np.arctan(21), 28 + np.arctan(22)]
    ]
    assert np.allclose(cb(inp2), ref2)

def eval(cls, arg):
        from sympy import exp_polar, pi, I, arg as argument
        if arg.is_number:
            ar = argument(arg)
            #if not ar.has(argument) and not ar.has(atan):
            if ar in (0, pi/2, -pi/2, pi):
                return exp_polar(I*ar)*abs(arg)

        if arg.is_Mul:
            args = arg.args
        else:
            args = [arg]
        included = []
        excluded = []
        positive = []
        for arg in args:
            if arg.is_polar:
                included += [arg]
            elif arg.is_positive:
                positive += [arg]
            else:
                excluded += [arg]
        if len(excluded) < len(args):
            if excluded:
                return Mul(*(included + positive))*polar_lift(Mul(*excluded))
            elif included:
                return Mul(*(included + positive))
            else:
                return Mul(*positive)*exp_polar(0)

def eval(cls, ar, period):
        # Our strategy is to evaluate the argument on the riemann surface of the
        # logarithm, and then reduce.
        # NOTE evidently this means it is a rather bad idea to use this with
        # period != 2*pi and non-polar numbers.
        from sympy import ceiling, oo, atan2, atan, polar_lift, pi, Mul
        if not period.is_positive:
            return None
        if period == oo and isinstance(ar, principal_branch):
            return periodic_argument(*ar.args)
        if ar.func is polar_lift and period >= 2*pi:
            return periodic_argument(ar.args[0], period)
        if ar.is_Mul:
            newargs = [x for x in ar.args if not x.is_positive]
            if len(newargs) != len(ar.args):
                return periodic_argument(Mul(*newargs), period)
        unbranched = cls._getunbranched(ar)
        if unbranched is None:
            return None
        if unbranched.has(periodic_argument, atan2, arg, atan):
            return None
        if period == oo:
            return unbranched
        if period != oo:
            n = ceiling(unbranched/period - S(1)/2)*period
            if not n.has(ceiling):
                return unbranched - n

def test_issue_5414():
    assert ratint(1/(x**2 + 16), x) == atan(x/4)/4

def test_issue_5817():
    a, b, c = symbols('a,b,c', positive=True)

    assert simplify(ratint(a/(b*c*x**2 + a**2 + b*a), x)) == \
        sqrt(a)*atan(sqrt(
            b)*sqrt(c)*x/(sqrt(a)*sqrt(a + b)))/(sqrt(b)*sqrt(c)*sqrt(a + b))

def test_issue_5981():
    u = symbols('u')
    assert integrate(1/(u**2 + 1)) == atan(u)

def eval_arctan(integrand, symbol):
    return sympy.atan(symbol)

def eval_trigsubstitution(theta, func, rewritten, substep, integrand, symbol):
    func = func.subs(sympy.sec(theta), 1/sympy.cos(theta))

    trig_function = list(func.find(TrigonometricFunction))
    assert len(trig_function) == 1
    trig_function = trig_function[0]
    relation = sympy.solve(symbol - func, trig_function)
    assert len(relation) == 1
    numer, denom = sympy.fraction(relation[0])

    if isinstance(trig_function, sympy.sin):
        opposite = numer
        hypotenuse = denom
        adjacent = sympy.sqrt(denom**2 - numer**2)
        inverse = sympy.asin(relation[0])
    elif isinstance(trig_function, sympy.cos):
        adjacent = numer
        hypotenuse = denom
        opposite = sympy.sqrt(denom**2 - numer**2)
        inverse = sympy.acos(relation[0])
    elif isinstance(trig_function, sympy.tan):
        opposite = numer
        adjacent = denom
        hypotenuse = sympy.sqrt(denom**2 + numer**2)
        inverse = sympy.atan(relation[0])

    substitution = [
        (sympy.sin(theta), opposite/hypotenuse),
        (sympy.cos(theta), adjacent/hypotenuse),
        (sympy.tan(theta), opposite/adjacent),
        (theta, inverse)
    ]
    return _manualintegrate(substep).subs(substitution).trigsimp()

def eval(cls, arg):
        from sympy import exp_polar, pi, I, arg as argument
        if arg.is_number:
            ar = argument(arg)
            # In general we want to affirm that something is known,
            # e.g. `not ar.has(argument) and not ar.has(atan)`
            # but for now we will just be more restrictive and
            # see that it has evaluated to one of the known values.
            if ar in (0, pi/2, -pi/2, pi):
                return exp_polar(I*ar)*abs(arg)

        if arg.is_Mul:
            args = arg.args
        else:
            args = [arg]
        included = []
        excluded = []
        positive = []
        for arg in args:
            if arg.is_polar:
                included += [arg]
            elif arg.is_positive:
                positive += [arg]
            else:
                excluded += [arg]
        if len(excluded) < len(args):
            if excluded:
                return Mul(*(included + positive))*polar_lift(Mul(*excluded))
            elif included:
                return Mul(*(included + positive))
            else:
                return Mul(*positive)*exp_polar(0)

def eval(cls, ar, period):
        # Our strategy is to evaluate the argument on the Riemann surface of the
        # logarithm, and then reduce.
        # NOTE evidently this means it is a rather bad idea to use this with
        # period != 2*pi and non-polar numbers.
        from sympy import ceiling, oo, atan2, atan, polar_lift, pi, Mul
        if not period.is_positive:
            return None
        if period == oo and isinstance(ar, principal_branch):
            return periodic_argument(*ar.args)
        if ar.func is polar_lift and period >= 2*pi:
            return periodic_argument(ar.args[0], period)
        if ar.is_Mul:
            newargs = [x for x in ar.args if not x.is_positive]
            if len(newargs) != len(ar.args):
                return periodic_argument(Mul(*newargs), period)
        unbranched = cls._getunbranched(ar)
        if unbranched is None:
            return None
        if unbranched.has(periodic_argument, atan2, arg, atan):
            return None
        if period == oo:
            return unbranched
        if period != oo:
            n = ceiling(unbranched/period - S(1)/2)*period
            if not n.has(ceiling):
                return unbranched - n

def eval(cls, arg):
        from sympy import atan
        arg = sympify(arg)

        if arg.is_Number:
            if arg is S.NaN:
                return S.NaN
            elif arg is S.Zero:
                return S.Zero
            elif arg is S.One:
                return S.Infinity
            elif arg is S.NegativeOne:
                return S.NegativeInfinity
            elif arg is S.Infinity:
                return -S.ImaginaryUnit * atan(arg)
            elif arg is S.NegativeInfinity:
                return S.ImaginaryUnit * atan(-arg)
            elif arg.is_negative:
                return -cls(-arg)
        else:
            if arg is S.ComplexInfinity:
                return S.NaN

            i_coeff = arg.as_coefficient(S.ImaginaryUnit)

            if i_coeff is not None:
                return S.ImaginaryUnit * atan(i_coeff)
            else:
                if _coeff_isneg(arg):
                    return -cls(-arg)

def test_issue_5414():
    assert ratint(1/(x**2 + 16), x) == atan(x/4)/4

def test_issue_5817():
    a, b, c = symbols('a,b,c', positive=True)

    assert simplify(ratint(a/(b*c*x**2 + a**2 + b*a), x)) == \
        sqrt(a)*atan(sqrt(
            b)*sqrt(c)*x/(sqrt(a)*sqrt(a + b)))/(sqrt(b)*sqrt(c)*sqrt(a + b))

def test_log_to_atan():
    f, g = (Poly(x + S(1)/2, x, domain='QQ'), Poly(sqrt(3)/2, x, domain='EX'))
    fg_ans = 2*atan(2*sqrt(3)*x/3 + sqrt(3)/3)
    assert log_to_atan(f, g) == fg_ans
    assert log_to_atan(g, f) == -fg_ans

def eval_arctan(a, b, c, integrand, symbol):
    return a / b * 1 / sympy.sqrt(c / b) * sympy.atan(symbol / sympy.sqrt(c / b))

def eval_trigsubstitution(theta, func, rewritten, substep, restriction, integrand, symbol):
    func = func.subs(sympy.sec(theta), 1/sympy.cos(theta))

    trig_function = list(func.find(TrigonometricFunction))
    assert len(trig_function) == 1
    trig_function = trig_function[0]
    relation = sympy.solve(symbol - func, trig_function)
    assert len(relation) == 1
    numer, denom = sympy.fraction(relation[0])

    if isinstance(trig_function, sympy.sin):
        opposite = numer
        hypotenuse = denom
        adjacent = sympy.sqrt(denom**2 - numer**2)
        inverse = sympy.asin(relation[0])
    elif isinstance(trig_function, sympy.cos):
        adjacent = numer
        hypotenuse = denom
        opposite = sympy.sqrt(denom**2 - numer**2)
        inverse = sympy.acos(relation[0])
    elif isinstance(trig_function, sympy.tan):
        opposite = numer
        adjacent = denom
        hypotenuse = sympy.sqrt(denom**2 + numer**2)
        inverse = sympy.atan(relation[0])

    substitution = [
        (sympy.sin(theta), opposite/hypotenuse),
        (sympy.cos(theta), adjacent/hypotenuse),
        (sympy.tan(theta), opposite/adjacent),
        (theta, inverse)
    ]
    return sympy.Piecewise(
        (_manualintegrate(substep).subs(substitution).trigsimp(), restriction)
    )

def reflect(self, line):
        from sympy import atan, Point, Dummy, oo

        g = self
        l = line
        o = Point(0, 0)
        if l.slope == 0:
            y = l.args[0].y
            if not y:  # x-axis
                return g.scale(y=-1)
            reps = [(p, p.translate(y=2*(y - p.y))) for p in g.atoms(Point)]
        elif l.slope == oo:
            x = l.args[0].x
            if not x:  # y-axis
                return g.scale(x=-1)
            reps = [(p, p.translate(x=2*(x - p.x))) for p in g.atoms(Point)]
        else:
            if not hasattr(g, 'reflect') and not all(
                    isinstance(arg, Point) for arg in g.args):
                raise NotImplementedError(
                    'reflect undefined or non-Point args in %s' % g)
            a = atan(l.slope)
            c = l.coefficients
            d = -c[-1]/c[1]  # y-intercept
            # apply the transform to a single point
            x, y = Dummy(), Dummy()
            xf = Point(x, y)
            xf = xf.translate(y=-d).rotate(-a, o).scale(y=-1
                ).rotate(a, o).translate(y=d)
            # replace every point using that transform
            reps = [(p, xf.xreplace({x: p.x, y: p.y})) for p in g.atoms(Point)]
        return g.xreplace(dict(reps))

def test_callback_factory():
    args = x, y = symbols('x y')
    expr = x + atan(y)
    cb = _callback_factory(args, [expr], 'numpy', np.float64, 'C')
    inp = np.array([17, 1])
    ref = 17 + np.arctan(1)
    assert np.allclose(cb(inp), ref)

def test_callback_factory__numba():
    args = x, y = symbols('x y')
    expr = x + atan(y)
    cb = _callback_factory(args, [expr], 'numpy', np.float64, 'C', use_numba=True)
    n = 500
    inp = np.empty((n, 2))
    inp[:, 0] = np.linspace(0, 1, n)
    inp[:, 1] = np.linspace(-10, 10, n)
    assert np.allclose(cb(inp), inp[:, 0] + np.arctan(inp[:, 1]))

def find_substitutions(integrand, symbol, u_var):
    results = []

    def test_subterm(u, u_diff):
        substituted = integrand / u_diff
        if symbol not in substituted.free_symbols:
            # replaced everything already
            return False

        substituted = substituted.subs(u, u_var).cancel()
        if symbol not in substituted.free_symbols:
            return substituted.as_independent(u_var, as_Add=False)

        return False

    def possible_subterms(term):
        if isinstance(term, (TrigonometricFunction,
                             sympy.asin, sympy.acos, sympy.atan,
                             sympy.exp, sympy.log, sympy.Heaviside)):
            return [term.args[0]]
        elif isinstance(term, sympy.Mul):
            r = []
            for u in term.args:
                r.append(u)
                r.extend(possible_subterms(u))
            return r
        elif isinstance(term, sympy.Pow):
            if term.args[1].is_constant(symbol):
                return [term.args[0]]
            elif term.args[0].is_constant(symbol):
                return [term.args[1]]
        elif isinstance(term, sympy.Add):
            r = []
            for arg in term.args:
                r.append(arg)
                r.extend(possible_subterms(arg))
            return r
        return []

    for u in possible_subterms(integrand):
        if u == symbol:
            continue
        u_diff = manual_diff(u, symbol)
        new_integrand = test_subterm(u, u_diff)
        if new_integrand is not False:
            constant, new_integrand = new_integrand
            substitution = (u, constant, new_integrand)
            if substitution not in results:
                results.append(substitution)

    return results

def _parts_rule(integrand, symbol):
    # LIATE rule:
    # log, inverse trig, algebraic (polynomial), trigonometric, exponential
    def pull_out_polys(integrand):
        integrand = integrand.together()
        polys = [arg for arg in integrand.args if arg.is_polynomial(symbol)]
        if polys:
            u = sympy.Mul(*polys)
            dv = integrand / u
            return u, dv

    def pull_out_u(*functions):
        def pull_out_u_rl(integrand):
            if any([integrand.has(f) for f in functions]):
                args = [arg for arg in integrand.args
                        if any(isinstance(arg, cls) for cls in functions)]
                if args:
                    u = reduce(lambda a,b: a*b, args)
                    dv = integrand / u
                    return u, dv

        return pull_out_u_rl

    liate_rules = [pull_out_u(sympy.log), pull_out_u(sympy.atan, sympy.asin, sympy.acos),
                   pull_out_polys, pull_out_u(sympy.sin, sympy.cos),
                   pull_out_u(sympy.exp)]


    dummy = sympy.Dummy("temporary")
    # we can integrate log(x) and atan(x) by setting dv = 1
    if isinstance(integrand, (sympy.log, sympy.atan, sympy.asin, sympy.acos)):
        integrand = dummy * integrand

    for index, rule in enumerate(liate_rules):
        result = rule(integrand)

        if result:
            u, dv = result

            # Don't pick u to be a constant if possible
            if symbol not in u.free_symbols and not u.has(dummy):
                return

            u = u.subs(dummy, 1)
            dv = dv.subs(dummy, 1)

            for rule in liate_rules[index + 1:]:
                r = rule(integrand)
                # make sure dv is amenable to integration
                if r and r[0].subs(dummy, 1) == dv:
                    du = u.diff(symbol)
                    v_step = integral_steps(dv, symbol)
                    v = _manualintegrate(v_step)

                    return u, dv, v, du, v_step

def manualintegrate(f, var):
    """manualintegrate(f, var)

    Compute indefinite integral of a single variable using an algorithm that
    resembles what a student would do by hand.

    Unlike ``integrate``, var can only be a single symbol.

    Examples
    ========

    >>> from sympy import sin, cos, tan, exp, log, integrate
    >>> from sympy.integrals.manualintegrate import manualintegrate
    >>> from sympy.abc import x
    >>> manualintegrate(1 / x, x)
    log(x)
    >>> integrate(1/x)
    log(x)
    >>> manualintegrate(log(x), x)
    x*log(x) - x
    >>> integrate(log(x))
    x*log(x) - x
    >>> manualintegrate(exp(x) / (1 + exp(2 * x)), x)
    atan(exp(x))
    >>> integrate(exp(x) / (1 + exp(2 * x)))
    RootSum(4*_z**2 + 1, Lambda(_i, _i*log(2*_i + exp(x))))
    >>> manualintegrate(cos(x)**4 * sin(x), x)
    -cos(x)**5/5
    >>> integrate(cos(x)**4 * sin(x), x)
    -cos(x)**5/5
    >>> manualintegrate(cos(x)**4 * sin(x)**3, x)
    cos(x)**7/7 - cos(x)**5/5
    >>> integrate(cos(x)**4 * sin(x)**3, x)
    cos(x)**7/7 - cos(x)**5/5
    >>> manualintegrate(tan(x), x)
    -log(cos(x))
    >>> integrate(tan(x), x)
    -log(sin(x)**2 - 1)/2

    See Also
    ========

    sympy.integrals.integrals.integrate
    sympy.integrals.integrals.Integral.doit
    sympy.integrals.integrals.Integral
    """
    return _manualintegrate(integral_steps(f, var))

def find_substitutions(integrand, symbol, u_var):
    results = []

    def test_subterm(u, u_diff):
        substituted = integrand / u_diff
        if symbol not in substituted.free_symbols:
            # replaced everything already
            return False

        substituted = substituted.subs(u, u_var).cancel()
        if symbol not in substituted.free_symbols:
            return substituted.as_independent(u_var, as_Add=False)

        return False

    def possible_subterms(term):
        if isinstance(term, (TrigonometricFunction,
                             sympy.asin, sympy.acos, sympy.atan,
                             sympy.exp, sympy.log, sympy.Heaviside)):
            return [term.args[0]]
        elif isinstance(term, sympy.Mul):
            r = []
            for u in term.args:
                r.append(u)
                r.extend(possible_subterms(u))
            return r
        elif isinstance(term, sympy.Pow):
            if term.args[1].is_constant(symbol):
                return [term.args[0]]
            elif term.args[0].is_constant(symbol):
                return [term.args[1]]
        elif isinstance(term, sympy.Add):
            r = []
            for arg in term.args:
                r.append(arg)
                r.extend(possible_subterms(arg))
            return r
        return []

    for u in possible_subterms(integrand):
        if u == symbol:
            continue
        u_diff = manual_diff(u, symbol)
        new_integrand = test_subterm(u, u_diff)
        if new_integrand is not False:
            constant, new_integrand = new_integrand
            if new_integrand == integrand.subs(symbol, u_var):
                continue
            substitution = (u, constant, new_integrand)
            if substitution not in results:
                results.append(substitution)

    return results

def _parts_rule(integrand, symbol):
    # LIATE rule:
    # log, inverse trig, algebraic, trigonometric, exponential
    def pull_out_algebraic(integrand):
        integrand = integrand.cancel().together()
        algebraic = [arg for arg in integrand.args if arg.is_algebraic_expr(symbol)]
        if algebraic:
            u = sympy.Mul(*algebraic)
            dv = (integrand / u).cancel()
            if not u.is_polynomial() and isinstance(dv, sympy.exp):
                return
            return u, dv

    def pull_out_u(*functions):
        def pull_out_u_rl(integrand):
            if any([integrand.has(f) for f in functions]):
                args = [arg for arg in integrand.args
                        if any(isinstance(arg, cls) for cls in functions)]
                if args:
                    u = reduce(lambda a,b: a*b, args)
                    dv = integrand / u
                    return u, dv

        return pull_out_u_rl

    liate_rules = [pull_out_u(sympy.log), pull_out_u(sympy.atan, sympy.asin, sympy.acos),
                   pull_out_algebraic, pull_out_u(sympy.sin, sympy.cos),
                   pull_out_u(sympy.exp)]


    dummy = sympy.Dummy("temporary")
    # we can integrate log(x) and atan(x) by setting dv = 1
    if isinstance(integrand, (sympy.log, sympy.atan, sympy.asin, sympy.acos)):
        integrand = dummy * integrand

    for index, rule in enumerate(liate_rules):
        result = rule(integrand)

        if result:
            u, dv = result

            # Don't pick u to be a constant if possible
            if symbol not in u.free_symbols and not u.has(dummy):
                return

            u = u.subs(dummy, 1)
            dv = dv.subs(dummy, 1)

            for rule in liate_rules[index + 1:]:
                r = rule(integrand)
                # make sure dv is amenable to integration
                if r and sympy.simplify(r[0].subs(dummy, 1)) == sympy.simplify(dv):
                    du = u.diff(symbol)
                    v_step = integral_steps(sympy.simplify(dv), symbol)
                    v = _manualintegrate(v_step)

                    return u, dv, v, du, v_step