def test_nsimplify():
x = Symbol("x")
assert nsimplify(0) == 0
assert nsimplify(-1) == -1
assert nsimplify(1) == 1
assert nsimplify(1+x) == 1+x
assert nsimplify(2.7) == Rational(27, 10)
assert nsimplify(1-GoldenRatio) == (1-sqrt(5))/2
assert nsimplify((1+sqrt(5))/4, [GoldenRatio]) == GoldenRatio/2
assert nsimplify(2/GoldenRatio, [GoldenRatio]) == 2*GoldenRatio - 2
assert nsimplify(exp(5*pi*I/3, evaluate=False)) == sympify('1/2 - sqrt(3)*I/2')
assert nsimplify(sin(3*pi/5, evaluate=False)) == sympify('sqrt(sqrt(5)/8 + 5/8)')
assert nsimplify(sqrt(atan('1', evaluate=False))*(2+I), [pi]) == sqrt(pi) + sqrt(pi)/2*I
assert nsimplify(2 + exp(2*atan('1/4')*I)) == sympify('49/17 + 8*I/17')
assert nsimplify(pi, tolerance=0.01) == Rational(22, 7)
assert nsimplify(pi, tolerance=0.001) == Rational(355, 113)
assert nsimplify(0.33333, tolerance=1e-4) == Rational(1, 3)
assert nsimplify(2.0**(1/3.), tolerance=0.001) == Rational(635, 504)
assert nsimplify(2.0**(1/3.), tolerance=0.001, full=True) == 2**Rational(1, 3)
assert nsimplify(x + .5, rational=True) == Rational(1, 2) + x
assert nsimplify(1/.3 + x, rational=True) == Rational(10, 3) + x
assert nsimplify(log(3).n(), rational=True) == \
sympify('109861228866811/100000000000000')
assert nsimplify(Float(0.272198261287950), [pi,log(2)]) == pi*log(2)/8
assert nsimplify(Float(0.272198261287950).n(3), [pi,log(2)]) == \
-pi/4 - log(2) + S(7)/4
assert nsimplify(x/7.0) == x/7
assert nsimplify(pi/1e2) == pi/100
assert nsimplify(pi/1e2, rational=False) == pi/100.0
assert nsimplify(pi/1e-7) == 10000000*pi
def _expr_big_minus(cls, a, z, n):
from sympy import I, pi, exp, sqrt, atan, sin
if n.is_even:
return (1 + z)**a*exp(2*pi*I*n*a)*sqrt(z)*sin(2*a*atan(sqrt(z)))
else:
return (1 + z)**a*exp(2*pi*I*n*a)*sqrt(z) \
*sin(2*a*atan(sqrt(z)) - 2*pi*a)
def log_to_atan(f, g):
"""Convert complex logarithms to real arctangents.
Given a real field K and polynomials f and g in K[x], with g != 0,
returns a sum h of arctangents of polynomials in K[x], such that:
df d f + I g
-- = -- I log( ------- )
dx dx f - I g
"""
if f.degree() < g.degree():
f, g = -g, f
f = f.to_field()
g = g.to_field()
p, q = f.div(g)
if q.is_zero:
return 2*atan(p.as_expr())
else:
s, t, h = g.gcdex(-f)
u = (f*s+g*t).quo(h)
A = 2*atan(u.as_expr())
return A + log_to_atan(s, t)
def test_evalf_integrals():
assert NS(Integral(x, (x, 2, 5)), 15) == '10.5000000000000'
gauss = Integral(exp(-x**2), (x, -oo, oo))
assert NS(gauss, 15) == '1.77245385090552'
assert NS(gauss**2 - pi + E*Rational(1,10**20), 15) in ('2.71828182845904e-20', '2.71828182845905e-20')
# A monster of an integral from http://mathworld.wolfram.com/DefiniteIntegral.html
t = Symbol('t')
a = 8*sqrt(3)/(1+3*t**2)
b = 16*sqrt(2)*(3*t+1)*(4*t**2+t+1)**Rational(3,2)
c = (3*t**2+1)*(11*t**2+2*t+3)**2
d = sqrt(2)*(249*t**2+54*t+65)/(11*t**2+2*t+3)**2
f = a - b/c - d
assert NS(Integral(f, (t, 0, 1)), 50) == NS((3*sqrt(2)-49*pi+162*atan(sqrt(2)))/12,50)
# http://mathworld.wolfram.com/VardisIntegral.html
assert NS(Integral(log(log(1/x))/(1+x+x**2), (x, 0, 1)), 15) == NS('pi/sqrt(3) * log(2*pi**(5/6) / gamma(1/6))', 15)
# http://mathworld.wolfram.com/AhmedsIntegral.html
assert NS(Integral(atan(sqrt(x**2+2))/(sqrt(x**2+2)*(x**2+1)), (x, 0, 1)), 15) == NS(5*pi**2/96, 15)
# http://mathworld.wolfram.com/AbelsIntegral.html
assert NS(Integral(x/((exp(pi*x)-exp(-pi*x))*(x**2+1)), (x, 0, oo)), 15) == NS('log(2)/2-1/4',15)
# Complex part trimming
# http://mathworld.wolfram.com/VardisIntegral.html
assert NS(Integral(log(log(sin(x)/cos(x))), (x, pi/4, pi/2)), 15, chop=True) == \
NS('pi/4*log(4*pi**3/gamma(1/4)**4)', 15)
#
# Endpoints causing trouble (rounding error in integration points -> complex log)
assert NS(2+Integral(log(2*cos(x/2)), (x, -pi, pi)), 17, chop=True) == NS(2, 17)
assert NS(2+Integral(log(2*cos(x/2)), (x, -pi, pi)), 20, chop=True) == NS(2, 20)
assert NS(2+Integral(log(2*cos(x/2)), (x, -pi, pi)), 22, chop=True) == NS(2, 22)
# Needs zero handling
assert NS(pi - 4*Integral('sqrt(1-x**2)', (x, 0, 1)), 15, maxn=30, chop=True) in ('0.0', '0')
# Oscillatory quadrature
a = Integral(sin(x)/x**2, (x, 1, oo)).evalf(maxn=15)
assert 0.49 < a < 0.51
assert NS(Integral(sin(x)/x**2, (x, 1, oo)), quad='osc') == '0.504067061906928'
assert NS(Integral(cos(pi*x+1)/x, (x, -oo, -1)), quad='osc') == '0.276374705640365'
def test_solve_sqrt_3():
R = Symbol("R")
eq = sqrt(2) * R * sqrt(1 / (R + 1)) + (R + 1) * (sqrt(2) * sqrt(1 / (R + 1)) - 1)
sol = solveset_complex(eq, R)
assert sol == FiniteSet(
*[
S(5) / 3 + 4 * sqrt(10) * cos(atan(3 * sqrt(111) / 251) / 3) / 3,
-sqrt(10) * cos(atan(3 * sqrt(111) / 251) / 3) / 3
+ 40 * re(1 / ((-S(1) / 2 - sqrt(3) * I / 2) * (S(251) / 27 + sqrt(111) * I / 9) ** (S(1) / 3))) / 9
+ sqrt(30) * sin(atan(3 * sqrt(111) / 251) / 3) / 3
+ S(5) / 3
+ I
* (
-sqrt(30) * cos(atan(3 * sqrt(111) / 251) / 3) / 3
- sqrt(10) * sin(atan(3 * sqrt(111) / 251) / 3) / 3
+ 40 * im(1 / ((-S(1) / 2 - sqrt(3) * I / 2) * (S(251) / 27 + sqrt(111) * I / 9) ** (S(1) / 3))) / 9
),
]
)
# the number of real roots will depend on the value of m: for m=1 there are 4
# and for m=-1 there are none.
eq = -sqrt((m - q) ** 2 + (-m / (2 * q) + S(1) / 2) ** 2) + sqrt(
(-m ** 2 / 2 - sqrt(4 * m ** 4 - 4 * m ** 2 + 8 * m + 1) / 4 - S(1) / 4) ** 2
+ (m ** 2 / 2 - m - sqrt(4 * m ** 4 - 4 * m ** 2 + 8 * m + 1) / 4 - S(1) / 4) ** 2
)
raises(NotImplementedError, lambda: solveset_real(eq, q))
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:
from sympy.calculus.util import AccumBounds
return S.ImaginaryUnit*AccumBounds(-S.Pi/2, S.Pi/2)
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_2850():
assert manualintegrate(asin(x)*log(x), x) == -x*asin(x) - sqrt(-x**2 + 1) \
+ (x*asin(x) + sqrt(-x**2 + 1))*log(x) - Integral(sqrt(-x**2 + 1)/x, x)
assert manualintegrate(acos(x)*log(x), x) == -x*acos(x) + sqrt(-x**2 + 1) + \
(x*acos(x) - sqrt(-x**2 + 1))*log(x) + Integral(sqrt(-x**2 + 1)/x, x)
assert manualintegrate(atan(x)*log(x), x) == -x*atan(x) + (x*atan(x) - \
log(x**2 + 1)/2)*log(x) + log(x**2 + 1)/2 + Integral(log(x**2 + 1)/x, x)/2
def _expr_big_minus(cls, x, n):
from sympy import atan, sqrt, pi
if n.is_even:
return atan(sqrt(x)) / sqrt(x)
else:
return (atan(sqrt(x)) - pi) / sqrt(x)
def test_heurisch_trigonometric():
assert heurisch(sin(x), x) == -cos(x)
assert heurisch(pi*sin(x) + 1, x) == x - pi*cos(x)
assert heurisch(cos(x), x) == sin(x)
assert heurisch(tan(x), x) in [
log(1 + tan(x)**2)/2,
log(tan(x) + I) + I*x,
log(tan(x) - I) - I*x,
]
assert heurisch(sin(x)*sin(y), x) == -cos(x)*sin(y)
assert heurisch(sin(x)*sin(y), y) == -cos(y)*sin(x)
# gives sin(x) in answer when run via setup.py and cos(x) when run via py.test
assert heurisch(sin(x)*cos(x), x) in [sin(x)**2 / 2, -cos(x)**2 / 2]
assert heurisch(cos(x)/sin(x), x) == log(sin(x))
assert heurisch(x*sin(7*x), x) == sin(7*x) / 49 - x*cos(7*x) / 7
assert heurisch(1/pi/4 * x**2*cos(x), x) == 1/pi/4*(x**2*sin(x) -
2*sin(x) + 2*x*cos(x))
assert heurisch(acos(x/4) * asin(x/4), x) == 2*x - (sqrt(16 - x**2))*asin(x/4) \
+ (sqrt(16 - x**2))*acos(x/4) + x*asin(x/4)*acos(x/4)
assert heurisch(sin(x)/(cos(x)**2+1), x) == -atan(cos(x)) #fixes issue 13723
assert heurisch(1/(cos(x)+2), x) == 2*sqrt(3)*atan(sqrt(3)*tan(x/2)/3)/3
assert heurisch(2*sin(x)*cos(x)/(sin(x)**4 + 1), x) == atan(sqrt(2)*sin(x)
- 1) - atan(sqrt(2)*sin(x) + 1)
assert heurisch(1/cosh(x), x) == 2*atan(tanh(x/2))
def test_atan2_expansion():
assert cancel(atan2(x + 1, x ** 2).diff(x) - atan((x + 1) / x ** 2).diff(x)) == 0
assert cancel(atan(x / y).series(x, 0, 5) - atan2(x, y).series(x, 0, 5) + atan2(0, y) - atan(0)) == O(x ** 5)
assert cancel(atan(x / y).series(y, 1, 4) - atan2(x, y).series(y, 1, 4) + atan2(x, 1) - atan(x)) == O(y ** 4)
assert cancel(
atan((x + y) / y).series(y, 1, 3) - atan2(x + y, y).series(y, 1, 3) + atan2(1 + x, 1) - atan(1 + x)
) == O(y ** 3)
assert Matrix([atan2(x, y)]).jacobian([x, y]) == Matrix([[y / (x ** 2 + y ** 2), -x / (x ** 2 + y ** 2)]])
def test_atan2_expansion():
assert cancel(atan2(x ** 2, x + 1).diff(x) - atan(x ** 2 / (x + 1)).diff(x)) == 0
assert cancel(atan(y / x).series(y, 0, 5) - atan2(y, x).series(y, 0, 5) + atan2(0, x) - atan(0)) == O(y ** 5)
assert cancel(atan(y / x).series(x, 1, 4) - atan2(y, x).series(x, 1, 4) + atan2(y, 1) - atan(y)) == O(x ** 4)
assert cancel(
atan((y + x) / x).series(x, 1, 3) - atan2(y + x, x).series(x, 1, 3) + atan2(1 + y, 1) - atan(1 + y)
) == O(x ** 3)
assert Matrix([atan2(y, x)]).jacobian([y, x]) == Matrix([[x / (y ** 2 + x ** 2), -y / (y ** 2 + x ** 2)]])
def test_atan2():
assert refine(atan2(y, x), Q.real(y) & Q.positive(x)) == atan(y/x)
assert refine(atan2(y, x), Q.negative(y) & Q.positive(x)) == atan(y/x)
assert refine(atan2(y, x), Q.negative(y) & Q.negative(x)) == atan(y/x) - pi
assert refine(atan2(y, x), Q.positive(y) & Q.negative(x)) == atan(y/x) + pi
assert refine(atan2(y, x), Q.zero(y) & Q.negative(x)) == pi
assert refine(atan2(y, x), Q.positive(y) & Q.zero(x)) == pi/2
assert refine(atan2(y, x), Q.negative(y) & Q.zero(x)) == -pi/2
assert refine(atan2(y, x), Q.zero(y) & Q.zero(x)) == nan
def test_manualintegrate_parts():
assert manualintegrate(exp(x) * sin(x), x) == \
(exp(x) * sin(x)) / 2 - (exp(x) * cos(x)) / 2
assert manualintegrate(2*x*cos(x), x) == 2*x*sin(x) + 2*cos(x)
assert manualintegrate(x * log(x), x) == x**2*log(x)/2 - x**2/4
assert manualintegrate(log(x), x) == x * log(x) - x
assert manualintegrate((3*x**2 + 5) * exp(x), x) == \
-6*x*exp(x) + (3*x**2 + 5)*exp(x) + 6*exp(x)
assert manualintegrate(atan(x), x) == x*atan(x) - log(x**2 + 1)/2
def test_atan2():
assert atan2.nargs == FiniteSet(2)
assert atan2(0, 0) == S.NaN
assert atan2(0, 1) == 0
assert atan2(1, 1) == pi/4
assert atan2(1, 0) == pi/2
assert atan2(1, -1) == 3*pi/4
assert atan2(0, -1) == pi
assert atan2(-1, -1) == -3*pi/4
assert atan2(-1, 0) == -pi/2
assert atan2(-1, 1) == -pi/4
i = symbols('i', imaginary=True)
r = symbols('r', real=True)
eq = atan2(r, i)
ans = -I*log((i + I*r)/sqrt(i**2 + r**2))
reps = ((r, 2), (i, I))
assert eq.subs(reps) == ans.subs(reps)
x = Symbol('x', negative=True)
y = Symbol('y', negative=True)
assert atan2(y, x) == atan(y/x) - pi
y = Symbol('y', nonnegative=True)
assert atan2(y, x) == atan(y/x) + pi
y = Symbol('y')
assert atan2(y, x) == atan2(y, x, evaluate=False)
u = Symbol("u", positive=True)
assert atan2(0, u) == 0
u = Symbol("u", negative=True)
assert atan2(0, u) == pi
assert atan2(y, oo) == 0
assert atan2(y, -oo)== 2*pi*Heaviside(re(y)) - pi
assert atan2(y, x).rewrite(log) == -I*log((x + I*y)/sqrt(x**2 + y**2))
assert atan2(y, x).rewrite(atan) == 2*atan(y/(x + sqrt(x**2 + y**2)))
ex = atan2(y, x) - arg(x + I*y)
assert ex.subs({x:2, y:3}).rewrite(arg) == 0
assert ex.subs({x:2, y:3*I}).rewrite(arg) == -pi - I*log(sqrt(5)*I/5)
assert ex.subs({x:2*I, y:3}).rewrite(arg) == -pi/2 - I*log(sqrt(5)*I)
assert ex.subs({x:2*I, y:3*I}).rewrite(arg) == -pi + atan(2/S(3)) + atan(3/S(2))
i = symbols('i', imaginary=True)
r = symbols('r', real=True)
e = atan2(i, r)
rewrite = e.rewrite(arg)
reps = {i: I, r: -2}
assert rewrite == -I*log(abs(I*i + r)/sqrt(abs(i**2 + r**2))) + arg((I*i + r)/sqrt(i**2 + r**2))
assert (e - rewrite).subs(reps).equals(0)
assert conjugate(atan2(x, y)) == atan2(conjugate(x), conjugate(y))
assert diff(atan2(y, x), x) == -y/(x**2 + y**2)
assert diff(atan2(y, x), y) == x/(x**2 + y**2)
assert simplify(diff(atan2(y, x).rewrite(log), x)) == -y/(x**2 + y**2)
assert simplify(diff(atan2(y, x).rewrite(log), y)) == x/(x**2 + y**2)
def test_invert_real():
x = Symbol('x', real=True)
x = Dummy(real=True)
n = Symbol('n')
d = Dummy()
assert solveset(abs(x) - n, x) == solveset(abs(x) - d, x) == EmptySet()
n = Symbol('n', real=True)
assert invert_real(x + 3, y, x) == (x, FiniteSet(y - 3))
assert invert_real(x*3, y, x) == (x, FiniteSet(y / 3))
assert invert_real(exp(x), y, x) == (x, FiniteSet(log(y)))
assert invert_real(exp(3*x), y, x) == (x, FiniteSet(log(y) / 3))
assert invert_real(exp(x + 3), y, x) == (x, FiniteSet(log(y) - 3))
assert invert_real(exp(x) + 3, y, x) == (x, FiniteSet(log(y - 3)))
assert invert_real(exp(x)*3, y, x) == (x, FiniteSet(log(y / 3)))
assert invert_real(log(x), y, x) == (x, FiniteSet(exp(y)))
assert invert_real(log(3*x), y, x) == (x, FiniteSet(exp(y) / 3))
assert invert_real(log(x + 3), y, x) == (x, FiniteSet(exp(y) - 3))
assert invert_real(Abs(x), y, x) == (x, FiniteSet(-y, y))
assert invert_real(2**x, y, x) == (x, FiniteSet(log(y)/log(2)))
assert invert_real(2**exp(x), y, x) == (x, FiniteSet(log(log(y)/log(2))))
assert invert_real(x**2, y, x) == (x, FiniteSet(sqrt(y), -sqrt(y)))
assert invert_real(x**Rational(1, 2), y, x) == (x, FiniteSet(y**2))
raises(ValueError, lambda: invert_real(x, x, x))
raises(ValueError, lambda: invert_real(x**pi, y, x))
raises(ValueError, lambda: invert_real(S.One, y, x))
assert invert_real(x**31 + x, y, x) == (x**31 + x, FiniteSet(y))
assert invert_real(Abs(x**31 + x + 1), y, x) == (x**31 + x,
FiniteSet(-y - 1, y - 1))
assert invert_real(tan(x), y, x) == \
(x, imageset(Lambda(n, n*pi + atan(y)), S.Integers))
assert invert_real(tan(exp(x)), y, x) == \
(x, imageset(Lambda(n, log(n*pi + atan(y))), S.Integers))
assert invert_real(cot(x), y, x) == \
(x, imageset(Lambda(n, n*pi + acot(y)), S.Integers))
assert invert_real(cot(exp(x)), y, x) == \
(x, imageset(Lambda(n, log(n*pi + acot(y))), S.Integers))
assert invert_real(tan(tan(x)), y, x) == \
(tan(x), imageset(Lambda(n, n*pi + atan(y)), S.Integers))
x = Symbol('x', positive=True)
assert invert_real(x**pi, y, x) == (x, FiniteSet(y**(1/pi)))
def test_nsimplify():
x = Symbol("x")
assert nsimplify(0) == 0
assert nsimplify(-1) == -1
assert nsimplify(1) == 1
assert nsimplify(1 + x) == 1 + x
assert nsimplify(2.7) == Rational(27, 10)
assert nsimplify(1 - GoldenRatio) == (1 - sqrt(5)) / 2
assert nsimplify((1 + sqrt(5)) / 4, [GoldenRatio]) == GoldenRatio / 2
assert nsimplify(2 / GoldenRatio, [GoldenRatio]) == 2 * GoldenRatio - 2
assert nsimplify(exp(5 * pi * I / 3, evaluate=False)) == sympify("1/2 - sqrt(3)*I/2")
assert nsimplify(sin(3 * pi / 5, evaluate=False)) == sympify("sqrt(sqrt(5)/8 + 5/8)")
assert nsimplify(sqrt(atan("1", evaluate=False)) * (2 + I), [pi]) == sqrt(pi) + sqrt(pi) / 2 * I
assert nsimplify(2 + exp(2 * atan("1/4") * I)) == sympify("49/17 + 8*I/17")
assert nsimplify(pi, tolerance=0.01) == Rational(22, 7)
assert nsimplify(pi, tolerance=0.001) == Rational(355, 113)
assert nsimplify(0.33333, tolerance=1e-4) == Rational(1, 3)
assert nsimplify(2.0 ** (1 / 3.0), tolerance=0.001) == Rational(635, 504)
assert nsimplify(2.0 ** (1 / 3.0), tolerance=0.001, full=True) == 2 ** Rational(1, 3)
assert nsimplify(x + 0.5, rational=True) == Rational(1, 2) + x
assert nsimplify(1 / 0.3 + x, rational=True) == Rational(10, 3) + x
assert nsimplify(log(3).n(), rational=True) == sympify("109861228866811/100000000000000")
assert nsimplify(Float(0.272198261287950), [pi, log(2)]) == pi * log(2) / 8
assert nsimplify(Float(0.272198261287950).n(3), [pi, log(2)]) == -pi / 4 - log(2) + S(7) / 4
assert nsimplify(x / 7.0) == x / 7
assert nsimplify(pi / 1e2) == pi / 100
assert nsimplify(pi / 1e2, rational=False) == pi / 100.0
assert nsimplify(pi / 1e-7) == 10000000 * pi
assert not nsimplify(factor(-3.0 * z ** 2 * (z ** 2) ** (-2.5) + 3 * (z ** 2) ** (-1.5))).atoms(Float)
e = x ** 0.0
assert e.is_Pow and nsimplify(x ** 0.0) == 1
assert nsimplify(3.333333, tolerance=0.1, rational=True) == Rational(10, 3)
assert nsimplify(3.333333, tolerance=0.01, rational=True) == Rational(10, 3)
assert nsimplify(3.666666, tolerance=0.1, rational=True) == Rational(11, 3)
assert nsimplify(3.666666, tolerance=0.01, rational=True) == Rational(11, 3)
assert nsimplify(33, tolerance=10, rational=True) == Rational(33)
assert nsimplify(33.33, tolerance=10, rational=True) == Rational(30)
assert nsimplify(37.76, tolerance=10, rational=True) == Rational(40)
assert nsimplify(-203.1) == -S(2031) / 10
assert nsimplify(0.2, tolerance=0) == S.One / 5
assert nsimplify(-0.2, tolerance=0) == -S.One / 5
assert nsimplify(0.2222, tolerance=0) == S(1111) / 5000
assert nsimplify(-0.2222, tolerance=0) == -S(1111) / 5000
# issue 7211, PR 4112
assert nsimplify(S(2e-8)) == S(1) / 50000000
# issue 7322 direct test
assert nsimplify(1e-42, rational=True) != 0
# issue 10336
inf = Float("inf")
infs = (-oo, oo, inf, -inf)
for i in infs:
ans = sign(i) * oo
assert nsimplify(i) == ans
assert nsimplify(i + x) == x + ans
def test_manualintegrate_inversetrig():
# atan
assert manualintegrate(exp(x) / (1 + exp(2*x)), x) == atan(exp(x))
assert manualintegrate(1 / (4 + 9 * x**2), x) == atan(3 * x/2) / 6
assert manualintegrate(1 / (16 + 16 * x**2), x) == atan(x) / 16
assert manualintegrate(1 / (4 + x**2), x) == atan(x / 2) / 2
assert manualintegrate(1 / (1 + 4 * x**2), x) == atan(2*x) / 2
assert manualintegrate(1/(a + b*x**2), x) == \
Piecewise(((sqrt(a/b)*atan(x*sqrt(b/a))/a), And(a > 0, b > 0)))
assert manualintegrate(1/(4 + b*x**2), x) == \
Piecewise((sqrt(1/b)*atan(sqrt(b)*x/2)/2, b > 0))
assert manualintegrate(1/(a + 4*x**2), x) == \
Piecewise((atan(2*x*sqrt(1/a))/(2*sqrt(a)), a > 0))
assert manualintegrate(1/(4 + 4*x**2), x) == atan(x) / 4
# asin
assert manualintegrate(1/sqrt(1-x**2), x) == asin(x)
assert manualintegrate(1/sqrt(4-4*x**2), x) == asin(x)/2
assert manualintegrate(3/sqrt(1-9*x**2), x) == asin(3*x)
assert manualintegrate(1/sqrt(4-9*x**2), x) == asin(3*x/2)/3
# asinh
assert manualintegrate(1/sqrt(x**2 + 1), x) == \
asinh(x)
assert manualintegrate(1/sqrt(x**2 + 4), x) == \
asinh(x/2)
assert manualintegrate(1/sqrt(4*x**2 + 4), x) == \
asinh(x)/2
assert manualintegrate(1/sqrt(4*x**2 + 1), x) == \
asinh(2*x)/2
assert manualintegrate(1/sqrt(a*x**2 + 1), x) == \
Piecewise((sqrt(-1/a)*asin(x*sqrt(-a)), a < 0), (sqrt(1/a)*asinh(sqrt(a)*x), a > 0))
assert manualintegrate(1/sqrt(a + x**2), x) == \
Piecewise((asinh(x*sqrt(1/a)), a > 0), (acosh(x*sqrt(-1/a)), a < 0))
# acosh
assert manualintegrate(1/sqrt(x**2 - 1), x) == \
acosh(x)
assert manualintegrate(1/sqrt(x**2 - 4), x) == \
acosh(x/2)
assert manualintegrate(1/sqrt(4*x**2 - 4), x) == \
acosh(x)/2
assert manualintegrate(1/sqrt(9*x**2 - 1), x) == \
acosh(3*x)/3
assert manualintegrate(1/sqrt(a*x**2 - 4), x) == \
Piecewise((sqrt(1/a)*acosh(sqrt(a)*x/2), a > 0))
assert manualintegrate(1/sqrt(-a + 4*x**2), x) == \
Piecewise((asinh(2*x*sqrt(-1/a))/2, -a > 0), (acosh(2*x*sqrt(1/a))/2, -a < 0))
# piecewise
assert manualintegrate(1/sqrt(a-b*x**2), x) == \
Piecewise((sqrt(a/b)*asin(x*sqrt(b/a))/sqrt(a), And(-b < 0, a > 0)),
(sqrt(-a/b)*asinh(x*sqrt(-b/a))/sqrt(a), And(-b > 0, a > 0)),
(sqrt(a/b)*acosh(x*sqrt(b/a))/sqrt(-a), And(-b > 0, a < 0)))
assert manualintegrate(1/sqrt(a + b*x**2), x) == \
Piecewise((sqrt(-a/b)*asin(x*sqrt(-b/a))/sqrt(a), And(a > 0, b < 0)),
(sqrt(a/b)*asinh(x*sqrt(b/a))/sqrt(a), And(a > 0, b > 0)),
(sqrt(-a/b)*acosh(x*sqrt(-b/a))/sqrt(-a), And(a < 0, b > 0)))
def test_main_trig_functions_numeric(self):
print "\n\n\n" + " Test if sin, cos and tan and inverses Work Numerically ".center(75, "#")
from sympy import symbols, sin, cos, tan, asin, acos, atan
x, y = symbols('x,y')
test_expr = sin(x) + cos(x) + tan(x) + asin(y) + acos(y) + atan(y)
target_expr = sin(x) + cos(x) + tan(x) + asin(y) + acos(y) + atan(y)
print "Target expression: '%s'" % target_expr
print "Test expression: '%s'" % test_expr
equal = api.numeric_equality(test_expr, target_expr)
self.assertTrue(equal, "Expected expressions to be found numerically equal!")
print " PASS ".center(75, "#")
def test_fps__hyper():
f = sin(x)
assert fps(f, x).truncate() == x - x ** 3 / 6 + x ** 5 / 120 + O(x ** 6)
f = cos(x)
assert fps(f, x).truncate() == 1 - x ** 2 / 2 + x ** 4 / 24 + O(x ** 6)
f = exp(x)
assert fps(f, x).truncate() == 1 + x + x ** 2 / 2 + x ** 3 / 6 + x ** 4 / 24 + x ** 5 / 120 + O(x ** 6)
f = atan(x)
assert fps(f, x).truncate() == x - x ** 3 / 3 + x ** 5 / 5 + O(x ** 6)
f = exp(acos(x))
assert fps(f, x).truncate() == (
exp(pi / 2)
- x * exp(pi / 2)
+ x ** 2 * exp(pi / 2) / 2
- x ** 3 * exp(pi / 2) / 3
+ 5 * x ** 4 * exp(pi / 2) / 24
- x ** 5 * exp(pi / 2) / 6
+ O(x ** 6)
)
f = exp(acosh(x))
assert fps(f, x).truncate() == I + x - I * x ** 2 / 2 - I * x ** 4 / 8 + O(x ** 6)
f = atan(1 / x)
assert fps(f, x).truncate() == pi / 2 - x + x ** 3 / 3 - x ** 5 / 5 + O(x ** 6)
f = x * atan(x) - log(1 + x ** 2) / 2
assert fps(f, x, rational=False).truncate() == x ** 2 / 2 - x ** 4 / 12 + O(x ** 6)
f = log(1 + x)
assert fps(f, x, rational=False).truncate() == x - x ** 2 / 2 + x ** 3 / 3 - x ** 4 / 4 + x ** 5 / 5 + O(x ** 6)
f = airyai(x ** 2)
assert fps(f, x).truncate() == (
3 ** Rational(5, 6) * gamma(Rational(1, 3)) / (6 * pi)
- 3 ** Rational(2, 3) * x ** 2 / (3 * gamma(Rational(1, 3)))
+ O(x ** 6)
)
f = exp(x) * sin(x)
assert fps(f, x).truncate() == x + x ** 2 + x ** 3 / 3 - x ** 5 / 30 + O(x ** 6)
f = exp(x) * sin(x) / x
assert fps(f, x).truncate() == 1 + x + x ** 2 / 3 - x ** 4 / 30 - x ** 5 / 90 + O(x ** 6)
f = sin(x) * cos(x)
assert fps(f, x).truncate() == x - 2 * x ** 3 / 3 + 2 * x ** 5 / 15 + O(x ** 6)
def test_rational_algorithm():
f = 1 / ((x - 1) ** 2 * (x - 2))
assert rational_algorithm(f, x, k) == (-2 ** (-k - 1) + 1 - (factorial(k + 1) / factorial(k)), 0, 0)
f = (1 + x + x ** 2 + x ** 3) / ((x - 1) * (x - 2))
assert rational_algorithm(f, x, k) == (-15 * 2 ** (-k - 1) + 4, x + 4, 0)
f = z / (y * m - m * x - y * x + x ** 2)
assert rational_algorithm(f, x, k) == (((-y ** (-k - 1) * z) / (y - m)) + ((m ** (-k - 1) * z) / (y - m)), 0, 0)
f = x / (1 - x - x ** 2)
assert rational_algorithm(f, x, k) is None
assert rational_algorithm(f, x, k, full=True) == (
((-Rational(1, 2) + sqrt(5) / 2) ** (-k - 1) * (-sqrt(5) / 10 + Rational(1, 2)))
+ ((-sqrt(5) / 2 - Rational(1, 2)) ** (-k - 1) * (sqrt(5) / 10 + Rational(1, 2))),
0,
0,
)
f = 1 / (x ** 2 + 2 * x + 2)
assert rational_algorithm(f, x, k) is None
assert rational_algorithm(f, x, k, full=True) == (
(I * (-1 + I) ** (-k - 1)) / 2 - (I * (-1 - I) ** (-k - 1)) / 2,
0,
0,
)
f = log(1 + x)
assert rational_algorithm(f, x, k) == (-(-1) ** (-k) / k, 0, 1)
f = atan(x)
assert rational_algorithm(f, x, k) is None
assert rational_algorithm(f, x, k, full=True) == (((I * I ** (-k)) / 2 - (I * (-I) ** (-k)) / 2) / k, 0, 1)
f = x * atan(x) - log(1 + x ** 2) / 2
assert rational_algorithm(f, x, k) is None
assert rational_algorithm(f, x, k, full=True) == (
((I * I ** (-k + 1)) / 2 - (I * (-I) ** (-k + 1)) / 2) / (k * (k - 1)),
0,
2,
)
f = log((1 + x) / (1 - x)) / 2 - atan(x)
assert rational_algorithm(f, x, k) is None
assert rational_algorithm(f, x, k, full=True) == (
(-(-1) ** (-k) / 2 - (I * I ** (-k)) / 2 + (I * (-I) ** (-k)) / 2 + Rational(1, 2)) / k,
0,
1,
)
assert rational_algorithm(cos(x), x, k) is None
def test_fps__rational():
assert fps(1/x) == (1/x)
assert fps((x**2 + x + 1) / x**3, dir=-1) == (x**2 + x + 1) / x**3
f = 1 / ((x - 1)**2 * (x - 2))
assert fps(f, x).truncate() == \
(-Rational(1, 2) - 5*x/4 - 17*x**2/8 - 49*x**3/16 - 129*x**4/32 -
321*x**5/64 + O(x**6))
f = (1 + x + x**2 + x**3) / ((x - 1) * (x - 2))
assert fps(f, x).truncate() == \
(Rational(1, 2) + 5*x/4 + 17*x**2/8 + 49*x**3/16 + 113*x**4/32 +
241*x**5/64 + O(x**6))
f = x / (1 - x - x**2)
assert fps(f, x, full=True).truncate() == \
x + x**2 + 2*x**3 + 3*x**4 + 5*x**5 + O(x**6)
f = 1 / (x**2 + 2*x + 2)
assert fps(f, x, full=True).truncate() == \
Rational(1, 2) - x/2 + x**2/4 - x**4/8 + x**5/8 + O(x**6)
f = log(1 + x)
assert fps(f, x).truncate() == \
x - x**2/2 + x**3/3 - x**4/4 + x**5/5 + O(x**6)
assert fps(f, x, dir=1).truncate() == fps(f, x, dir=-1).truncate()
assert fps(f, x, 2).truncate() == \
(log(3) - Rational(2, 3) - (x - 2)**2/18 + (x - 2)**3/81 -
(x - 2)**4/324 + (x - 2)**5/1215 + x/3 + O((x - 2)**6, (x, 2)))
assert fps(f, x, 2, dir=-1).truncate() == \
(log(3) - Rational(2, 3) - (-x + 2)**2/18 - (-x + 2)**3/81 -
(-x + 2)**4/324 - (-x + 2)**5/1215 + x/3 + O((x - 2)**6, (x, 2)))
f = atan(x)
assert fps(f, x, full=True).truncate() == x - x**3/3 + x**5/5 + O(x**6)
assert fps(f, x, full=True, dir=1).truncate() == \
fps(f, x, full=True, dir=-1).truncate()
assert fps(f, x, 2, full=True).truncate() == \
(atan(2) - Rational(2, 5) - 2*(x - 2)**2/25 + 11*(x - 2)**3/375 -
6*(x - 2)**4/625 + 41*(x - 2)**5/15625 + x/5 + O((x - 2)**6, (x, 2)))
assert fps(f, x, 2, full=True, dir=-1).truncate() == \
(atan(2) - Rational(2, 5) - 2*(-x + 2)**2/25 - 11*(-x + 2)**3/375 -
6*(-x + 2)**4/625 - 41*(-x + 2)**5/15625 + x/5 + O((x - 2)**6, (x, 2)))
f = x*atan(x) - log(1 + x**2) / 2
assert fps(f, x, full=True).truncate() == x**2/2 - x**4/12 + O(x**6)
f = log((1 + x) / (1 - x)) / 2 - atan(x)
assert fps(f, x, full=True).truncate(n=10) == 2*x**3/3 + 2*x**7/7 + O(x**10)
def test_atan():
assert atan(nan) == nan
assert atan(oo) == pi/2
assert atan(-oo) == -pi/2
assert atan(0) == 0
assert atan(1) == pi/4
assert atan(sqrt(3)) == pi/3
assert atan(oo) == pi/2
assert atan(x).diff(x) == 1/(1 + x**2)
assert atan(r).is_real is True
assert atan(-2*I) == -I*atanh(2)
def test_issue_1572_1364_1368():
assert solve((sqrt(x**2 - 1) - 2)) in ([sqrt(5), -sqrt(5)],
[-sqrt(5), sqrt(5)])
assert set(solve((2**exp(y**2/x) + 2)/(x**2 + 15), y)) == set([
-sqrt(x)*sqrt(-log(log(2)) + log(log(2) + I*pi)),
sqrt(x)*sqrt(-log(log(2)) + log(log(2) + I*pi))])
C1, C2 = symbols('C1 C2')
f = Function('f')
assert solve(C1 + C2/x**2 - exp(-f(x)), f(x)) == [log(x**2/(C1*x**2 + C2))]
a = Symbol('a')
E = S.Exp1
assert solve(1 - log(a + 4*x**2), x) in (
[-sqrt(-a + E)/2, sqrt(-a + E)/2],
[sqrt(-a + E)/2, -sqrt(-a + E)/2]
)
assert solve(log(a**(-3) - x**2)/a, x) in (
[-sqrt(-1 + a**(-3)), sqrt(-1 + a**(-3))],
[sqrt(-1 + a**(-3)), -sqrt(-1 + a**(-3))],)
assert solve(1 - log(a + 4*x**2), x) in (
[-sqrt(-a + E)/2, sqrt(-a + E)/2],
[sqrt(-a + E)/2, -sqrt(-a + E)/2],)
assert set(solve((
a**2 + 1) * (sin(a*x) + cos(a*x)), x)) == set([-pi/(4*a), 3*pi/(4*a)])
assert solve(3 - (sinh(a*x) + cosh(a*x)), x) == [2*atanh(S.Half)/a]
assert set(solve(3 - (sinh(a*x) + cosh(a*x)**2), x)) == \
set([
2*atanh(-1 + sqrt(2))/a,
2*atanh(S(1)/2 + sqrt(5)/2)/a,
2*atanh(-sqrt(2) - 1)/a,
2*atanh(-sqrt(5)/2 + S(1)/2)/a
])
assert solve(atan(x) - 1) == [tan(1)]
def test_acos_rewrite():
assert acos(x).rewrite(log) == pi/2 + I*log(I*x + sqrt(1 - x**2))
assert acos(x).rewrite(atan) == \
atan(sqrt(1 - x**2)/x) + (pi/2)*(1 - x*sqrt(1/x**2))
assert acos(0).rewrite(atan) == S.Pi/2
assert acos(0.5).rewrite(atan) == acos(0.5).rewrite(log)
assert acos(x).rewrite(asin) == S.Pi/2 - asin(x)
def test_atan2():
assert atan2(0, 0) == S.NaN
assert atan2(0, 1) == 0
assert atan2(1, 1) == pi/4
assert atan2(1, 0) == pi/2
assert atan2(1, -1) == 3*pi/4
assert atan2(0, -1) == pi
assert atan2(-1, -1) == -3*pi/4
assert atan2(-1, 0) == -pi/2
assert atan2(-1, 1) == -pi/4
u = Symbol("u", positive=True)
assert atan2(0, u) == 0
u = Symbol("u", negative=True)
assert atan2(0, u) == pi
assert atan2(y, oo) == 0
assert atan2(y, -oo)== 2*pi*Heaviside(re(y)) - pi
assert atan2(y, x).rewrite(log) == -I*log((x + I*y)/sqrt(x**2 + y**2))
assert atan2(y, x).rewrite(atan) == 2*atan(y/(x + sqrt(x**2 + y**2)))
assert diff(atan2(y, x), x) == -y/(x**2 + y**2)
assert diff(atan2(y, x), y) == x/(x**2 + y**2)
assert simplify(diff(atan2(y, x).rewrite(log), x)) == -y/(x**2 + y**2)
assert simplify(diff(atan2(y, x).rewrite(log), y)) == x/(x**2 + y**2)
assert isinstance(atan2(2, 3*I).n(), atan2)
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 test_issue_6746():
y = Symbol('y')
n = Symbol('n')
assert manualintegrate(y**x, x) == \
Piecewise((x, Eq(log(y), 0)), (y**x/log(y), True))
assert manualintegrate(y**(n*x), x) == \
Piecewise(
(x, Eq(n, 0)),
(Piecewise(
(n*x, Eq(log(y), 0)),
(y**(n*x)/log(y), True))/n, True))
assert manualintegrate(exp(n*x), x) == \
Piecewise((x, Eq(n, 0)), (exp(n*x)/n, True))
y = Symbol('y', positive=True)
assert manualintegrate((y + 1)**x, x) == (y + 1)**x/log(y + 1)
y = Symbol('y', zero=True)
assert manualintegrate((y + 1)**x, x) == x
y = Symbol('y')
n = Symbol('n', nonzero=True)
assert manualintegrate(y**(n*x), x) == \
Piecewise((n*x, Eq(log(y), 0)), (y**(n*x)/log(y), True))/n
y = Symbol('y', positive=True)
assert manualintegrate((y + 1)**(n*x), x) == \
(y + 1)**(n*x)/(n*log(y + 1))
a = Symbol('a', negative=True)
assert manualintegrate(1 / (a + b*x**2), x) == \
Piecewise((atan(x/sqrt(a/b))/(b*sqrt(a/b)), a/b > 0), \
(-acoth(x/sqrt(-a/b))/(b*sqrt(-a/b)), And(a/b < 0, x**2 > -a/b)), \
(-atanh(x/sqrt(-a/b))/(b*sqrt(-a/b)), And(a/b < 0, x**2 < -a/b)))
def test_intrinsic_math1_codegen():
# not included: log10
from sympy import acos, asin, atan, ceiling, cos, cosh, floor, log, ln, \
sin, sinh, sqrt, tan, tanh, N
x = symbols('x')
name_expr = [
("test_fabs", abs(x)),
("test_acos", acos(x)),
("test_asin", asin(x)),
("test_atan", atan(x)),
("test_cos", cos(x)),
("test_cosh", cosh(x)),
("test_log", log(x)),
("test_ln", ln(x)),
("test_sin", sin(x)),
("test_sinh", sinh(x)),
("test_sqrt", sqrt(x)),
("test_tan", tan(x)),
("test_tanh", tanh(x)),
]
numerical_tests = []
for name, expr in name_expr:
for xval in 0.2, 0.5, 0.8:
expected = N(expr.subs(x, xval))
numerical_tests.append((name, (xval,), expected, 1e-14))
for lang, commands in valid_lang_commands:
if lang == "C":
name_expr_C = [("test_floor", floor(x)), ("test_ceil", ceiling(x))]
else:
name_expr_C = []
run_test("intrinsic_math1", name_expr + name_expr_C, numerical_tests, lang, commands)
def test_messy():
from sympy import (laplace_transform, Si, Shi, Chi, atan, Piecewise,
acoth, E1, besselj, acosh, asin, And, re,
fourier_transform, sqrt)
assert laplace_transform(Si(x), x, s) == ((-atan(s) + pi/2)/s, 0, True)
assert laplace_transform(Shi(x), x, s) == (acoth(s)/s, 1, True)
# where should the logs be simplified?
assert laplace_transform(Chi(x), x, s) == \
((log(s**(-2)) - log((s**2 - 1)/s**2))/(2*s), 1, True)
# TODO maybe simplify the inequalities?
assert laplace_transform(besselj(a, x), x, s)[1:] == \
(0, And(S(0) < re(a/2) + S(1)/2, S(0) < re(a/2) + 1))
# NOTE s < 0 can be done, but argument reduction is not good enough yet
assert fourier_transform(besselj(1, x)/x, x, s, noconds=False) == \
(Piecewise((0, 4*abs(pi**2*s**2) > 1),
(2*sqrt(-4*pi**2*s**2 + 1), True)), s > 0)
# TODO FT(besselj(0,x)) - conditions are messy (but for acceptable reasons)
# - folding could be better
assert integrate(E1(x)*besselj(0, x), (x, 0, oo), meijerg=True) == \
log(1 + sqrt(2))
assert integrate(E1(x)*besselj(1, x), (x, 0, oo), meijerg=True) == \
log(S(1)/2 + sqrt(2)/2)
assert integrate(1/x/sqrt(1 - x**2), x, meijerg=True) == \
Piecewise((-acosh(1/x), 1 < abs(x**(-2))), (I*asin(1/x), True))
def test_heuristic():
x = Symbol("x", real=True)
assert heuristics(sin(1 / x) + atan(x), x, 0, '+') == AccumBounds(-1, 1)
assert limit(log(2 + sqrt(atan(x)) * sqrt(sin(1 / x))), x, 0) == log(2)
def eval_arctan(a, b, c, integrand, symbol):
return a / b * 1 / sympy.sqrt(c / b) * sympy.atan(
symbol / sympy.sqrt(c / b))
def test_heuristic_function_sum():
eq = f(x).diff(x) - (3*(1 + x**2/f(x)**2)*atan(f(x)/x) + (1 - 2*f(x))/x +
(1 - 3*f(x))*(x/f(x)**2))
i = infinitesimals(eq, hint='function_sum')
assert i == [{eta(x, f(x)): f(x)**(-2) + x**(-2), xi(x, f(x)): 0}]
assert checkinfsol(eq, i)[0]
def test_Abs():
raises(TypeError, lambda: Abs(Interval(2, 3))) # issue 8717
x, y = symbols('x,y')
assert sign(sign(x)) == sign(x)
assert sign(x*y).func is sign
assert Abs(0) == 0
assert Abs(1) == 1
assert Abs(-1) == 1
assert Abs(I) == 1
assert Abs(-I) == 1
assert Abs(nan) == nan
assert Abs(zoo) == oo
assert Abs(I * pi) == pi
assert Abs(-I * pi) == pi
assert Abs(I * x) == Abs(x)
assert Abs(-I * x) == Abs(x)
assert Abs(-2*x) == 2*Abs(x)
assert Abs(-2.0*x) == 2.0*Abs(x)
assert Abs(2*pi*x*y) == 2*pi*Abs(x*y)
assert Abs(conjugate(x)) == Abs(x)
assert conjugate(Abs(x)) == Abs(x)
assert Abs(x).expand(complex=True) == sqrt(re(x)**2 + im(x)**2)
a = Symbol('a', positive=True)
assert Abs(2*pi*x*a) == 2*pi*a*Abs(x)
assert Abs(2*pi*I*x*a) == 2*pi*a*Abs(x)
x = Symbol('x', real=True)
n = Symbol('n', integer=True)
assert Abs((-1)**n) == 1
assert x**(2*n) == Abs(x)**(2*n)
assert Abs(x).diff(x) == sign(x)
assert abs(x) == Abs(x) # Python built-in
assert Abs(x)**3 == x**2*Abs(x)
assert Abs(x)**4 == x**4
assert (
Abs(x)**(3*n)).args == (Abs(x), 3*n) # leave symbolic odd unchanged
assert (1/Abs(x)).args == (Abs(x), -1)
assert 1/Abs(x)**3 == 1/(x**2*Abs(x))
assert Abs(x)**-3 == Abs(x)/(x**4)
assert Abs(x**3) == x**2*Abs(x)
assert Abs(I**I) == exp(-pi/2)
assert Abs((4 + 5*I)**(6 + 7*I)) == 68921*exp(-7*atan(S(5)/4))
y = Symbol('y', real=True)
assert Abs(I**y) == 1
y = Symbol('y')
assert Abs(I**y) == exp(-pi*im(y)/2)
x = Symbol('x', imaginary=True)
assert Abs(x).diff(x) == -sign(x)
eq = -sqrt(10 + 6*sqrt(3)) + sqrt(1 + sqrt(3)) + sqrt(3 + 3*sqrt(3))
# if there is a fast way to know when you can and when you cannot prove an
# expression like this is zero then the equality to zero is ok
assert abs(eq).func is Abs or abs(eq) == 0
# but sometimes it's hard to do this so it's better not to load
# abs down with tests that will be very slow
q = 1 + sqrt(2) - 2*sqrt(3) + 1331*sqrt(6)
p = expand(q**3)**Rational(1, 3)
d = p - q
assert abs(d).func is Abs or abs(d) == 0
assert Abs(4*exp(pi*I/4)) == 4
assert Abs(3**(2 + I)) == 9
assert Abs((-3)**(1 - I)) == 3*exp(pi)
assert Abs(oo) is oo
assert Abs(-oo) is oo
assert Abs(oo + I) is oo
assert Abs(oo + I*oo) is oo
a = Symbol('a', algebraic=True)
t = Symbol('t', transcendental=True)
x = Symbol('x')
assert re(a).is_algebraic
assert re(x).is_algebraic is None
assert re(t).is_algebraic is False
assert Abs(x).fdiff() == sign(x)
raises(ArgumentIndexError, lambda: Abs(x).fdiff(2))
# doesn't have recursion error
arg = sqrt(acos(1 - I)*acos(1 + I))
assert abs(arg) == arg
# special handling to put Abs in denom
assert abs(1/x) == 1/Abs(x)
e = abs(2/x**2)
assert e.is_Mul and e == 2/Abs(x**2)
assert unchanged(Abs, y/x)
assert unchanged(Abs, x/(x + 1))
assert unchanged(Abs, x*y)
p = Symbol('p', positive=True)
assert abs(x/p) == abs(x)/p
# coverage
assert unchanged(Abs, Symbol('x', real=True)**y)
def test_asin_rewrite():
assert asin(x).rewrite(log) == -I * log(I * x + sqrt(1 - x**2))
assert asin(x).rewrite(atan) == 2 * atan(x / (1 + sqrt(1 - x**2)))
assert asin(x).rewrite(acos) == S.Pi / 2 - acos(x)
def test_atan_rewrite():
assert atan(x).rewrite(log) == I * log((1 - I * x) / (1 + I * x)) / 2
def test_nsimplify():
x = Symbol("x")
assert nsimplify(0) == 0
assert nsimplify(-1) == -1
assert nsimplify(1) == 1
assert nsimplify(1 + x) == 1 + x
assert nsimplify(2.7) == Rational(27, 10)
assert nsimplify(1 - GoldenRatio) == (1 - sqrt(5)) / 2
assert nsimplify((1 + sqrt(5)) / 4, [GoldenRatio]) == GoldenRatio / 2
assert nsimplify(2 / GoldenRatio, [GoldenRatio]) == 2 * GoldenRatio - 2
assert nsimplify(exp(5*pi*I/3, evaluate=False)) == \
sympify('1/2 - sqrt(3)*I/2')
assert nsimplify(sin(3*pi/5, evaluate=False)) == \
sympify('sqrt(sqrt(5)/8 + 5/8)')
assert nsimplify(sqrt(atan('1', evaluate=False))*(2 + I), [pi]) == \
sqrt(pi) + sqrt(pi)/2*I
assert nsimplify(2 + exp(2 * atan('1/4') * I)) == sympify('49/17 + 8*I/17')
assert nsimplify(pi, tolerance=0.01) == Rational(22, 7)
assert nsimplify(pi, tolerance=0.001) == Rational(355, 113)
assert nsimplify(0.33333, tolerance=1e-4) == Rational(1, 3)
assert nsimplify(2.0**(1 / 3.), tolerance=0.001) == Rational(635, 504)
assert nsimplify(2.0**(1/3.), tolerance=0.001, full=True) == \
2**Rational(1, 3)
assert nsimplify(x + .5, rational=True) == Rational(1, 2) + x
assert nsimplify(1 / .3 + x, rational=True) == Rational(10, 3) + x
assert nsimplify(log(3).n(), rational=True) == \
sympify('109861228866811/100000000000000')
assert nsimplify(Float(0.272198261287950), [pi, log(2)]) == pi * log(2) / 8
assert nsimplify(Float(0.272198261287950).n(3), [pi, log(2)]) == \
-pi/4 - log(2) + S(7)/4
assert nsimplify(x / 7.0) == x / 7
assert nsimplify(pi / 1e2) == pi / 100
assert nsimplify(pi / 1e2, rational=False) == pi / 100.0
assert nsimplify(pi / 1e-7) == 10000000 * pi
assert not nsimplify(
factor(-3.0 * z**2 * (z**2)**(-2.5) + 3 * (z**2)**(-1.5))).atoms(Float)
e = x**0.0
assert e.is_Pow and nsimplify(x**0.0) == 1
assert nsimplify(3.333333, tolerance=0.1, rational=True) == Rational(10, 3)
assert nsimplify(3.333333, tolerance=0.01,
rational=True) == Rational(10, 3)
assert nsimplify(3.666666, tolerance=0.1, rational=True) == Rational(11, 3)
assert nsimplify(3.666666, tolerance=0.01,
rational=True) == Rational(11, 3)
assert nsimplify(33, tolerance=10, rational=True) == Rational(33)
assert nsimplify(33.33, tolerance=10, rational=True) == Rational(30)
assert nsimplify(37.76, tolerance=10, rational=True) == Rational(40)
assert nsimplify(-203.1) == -S(2031) / 10
assert nsimplify(.2, tolerance=0) == S.One / 5
assert nsimplify(-.2, tolerance=0) == -S.One / 5
assert nsimplify(.2222, tolerance=0) == S(1111) / 5000
assert nsimplify(-.2222, tolerance=0) == -S(1111) / 5000
# issue 7211, PR 4112
assert nsimplify(S(2e-8)) == S(1) / 50000000
# issue 7322 direct test
assert nsimplify(1e-42, rational=True) != 0
# issue 10336
inf = Float('inf')
infs = (-oo, oo, inf, -inf)
for i in infs:
ans = sign(i) * oo
assert nsimplify(i) == ans
assert nsimplify(i + x) == x + ans
assert nsimplify(0.33333333, rational=True,
rational_conversion='exact') == Rational(0.33333333)
# Make sure nsimplify on expressions uses full precision
assert nsimplify(
pi.evalf(100) * x,
rational_conversion='exact').evalf(100) == pi.evalf(100) * x
def _ex_atan(self, e):
return sp.atan(self.ex(e[0]))
def test_aseries_trig():
assert cancel(
gruntz(1 / log(atan(x)), x, oo) - 1 / (log(pi) + log(S(1) / 2))) == 0
assert gruntz(1 / acot(x), x, -oo) == -oo
def test_issue689():
assert integrate(1 / (1 + x**2), x) == atan(x)
def AXIMOCdesigner(p_e, F_T, gamma, M_e, T_c, R, r_c, num_of_char_lines, numiter, theta_max_rad, L, pressure_type, q_factor):
global outbox
fig = Figure(figsize=(8, 5), dpi=100)
c_v = float(R / (gamma - 1))
c_p = float(c_v + R)
p_c = float(p_e/(1+((gamma-1)/2)*M_e**2)**(-gamma/(gamma-1)))
rho_c = float(p_c/(R*T_c))
rho_e = float(rho_c*(1+((gamma-1)/2)*M_e**2)**(-1/(gamma-1)))
p_t = float(p_c * ((2 / (gamma + 1)) ** (gamma / (gamma - 1))))
T_e = float(T_c/(p_c/p_e)**((gamma-1)/gamma))
u_e = M_e * sp.sqrt(gamma * R * T_e)
A_e = float(F_T / (u_e ** 2 * rho_e))
M_t = float(1.0001)
A_t = float(A_e/sp.sqrt((1/M_e**2)*((2/(gamma+1))*(1+((gamma-1)/2)*M_e**2))**((gamma+1)/(gamma-1))))
r_e = float(sp.sqrt(A_e/sp.pi))
r_t = float(sp.sqrt(A_t/sp.pi))
alpha = float(sp.sqrt(2 / ((gamma + 1) * 1.5 * r_t * r_t)))
x_t = float(0)
print(x_t)
if r_cset.get() == 1:
r_c = float(sp.sqrt(3)*r_t)
dec3 = Decimal(10) ** -3
dec4 = Decimal(10) ** -4
dec1 = Decimal(10) ** -2
warnings.filterwarnings("ignore")
# _________________________________MOC_________________________________________
book = openpyxl.load_workbook('export.xlsx')
sheet = book.active
v_PMerad = float(sp.sqrt((gamma + 1) / (gamma - 1))) * sp.atan(
sp.sqrt((gamma - 1) * (M_e ** 2 - 1) / (gamma + 1))) - sp.atan(sp.sqrt(M_e ** 2 - 1))
v_PMedeg = float(math.degrees(v_PMerad))
theta_max_deg = float(math.degrees(theta_max_rad))
first_point = x_t
# _____________________________________________________________________________
z = 0
x_list = []
y_list = []
p_xx_list = []
M_xx_list = []
v_PM_xx_rad_list = []
T_xx_list = []
V_xx_list = []
mu_xx_rad_list = []
theta_xx_rad_list = []
K_neg_xx_rad_list = []
K_pos_xx_rad_list = []
K_neg_xx_deg_list = []
K_pos_xx_deg_list = []
theta_xx_deg_list = []
v_PM_xx_deg_list = []
mu_xx_deg_list = []
xerr_list = []
yerr_list = []
thetaerr_list = []
Verr_list = []
iteration_list = []
sheet.cell(row=1, column=1).value = "K_neg"
sheet.cell(row=1, column=2).value = "K_pos"
sheet.cell(row=1, column=3).value = "Theta"
sheet.cell(row=1, column=4).value = "v"
sheet.cell(row=1, column=5).value = "M"
sheet.cell(row=1, column=6).value = "mu"
sheet.cell(row=1, column=7).value = "V"
sheet.cell(row=1, column=8).value = "x"
sheet.cell(row=1, column=9).value = "y"
sheet.cell(row=1, column=10).value = "p"
sheet.cell(row=1, column=11).value = "T"
if num_of_char_lines > 80:
outbox = Text(tab1, borderwidth=2, width=90, height=10, fg="yellow")
outbox.grid(row=26, column=0, columnspan=100, padx=5, pady=40)
outbox.insert(END,
"CAUTION: For number of characteristic lines above 80, the plot window and plot resizing functions may take longer to respond. It is recommended that the user is patient." + '\n')
outbox.see("end")
for x in np.linspace(first_point, L, num=num_of_char_lines, endpoint=True):
x_list.append(x)
y = float(0)
y_list.append(y)
if pressure_type == 'parabolic':
p_xx = float(mp.e ** ((L ** (-2)) * (x - L) ** 2 * (sp.ln(p_t) - sp.ln(p_e)) + ln(p_e)))
if pressure_type == 'cubic':
p_xx = float(sp.exp(((q_factor * L + 2 * (sp.ln(p_t) - sp.ln(p_e))) / L ** 3) * x ** 3 - ((2 * q_factor * L + 3 * (sp.ln(p_t) - sp.ln(p_e))) / L ** 2) * x ** 2 + q_factor * x + sp.ln(p_t)))
p_xx_list.append(p_xx)
M_xx = float(sp.sqrt(((p_c / p_xx) ** ((gamma - 1) / gamma) - 1) * (2 / (gamma - 1))))
M_xx_list.append(M_xx)
v_PM_xx_rad = float((math.sqrt((gamma + 1) / (gamma - 1))) * sp.atan(
math.sqrt((gamma - 1) * (M_xx ** 2 - 1) / (gamma + 1))) - sp.atan(math.sqrt(M_xx ** 2 - 1)))
v_PM_xx_rad_list.append(v_PM_xx_rad)
T_xx = T_c * (p_xx / p_c) ** ((gamma - 1) / gamma)
T_xx_list.append(T_xx)
V_xx = float(M_xx * sp.sqrt(gamma * R * T_xx))
V_xx_list.append(V_xx)
mu_xx_rad = float(sp.asin(1 / M_xx))
mu_xx_rad_list.append(mu_xx_rad)
theta_xx_rad = float(0)
theta_xx_rad_list.append(theta_xx_rad)
K_neg_xx_rad = float(theta_xx_rad + v_PM_xx_rad)
K_neg_xx_rad_list.append(K_neg_xx_rad)
K_pos_xx_rad = float(theta_xx_rad - v_PM_xx_rad)
K_pos_xx_rad_list.append(K_pos_xx_rad)
plt.plot(x_list, y_list, 'k.', markersize=1)
z += 1
w = 1
for g in range(0, z, 1):
sheet.cell(row=w + 1, column=1).value = math.degrees(K_neg_xx_rad_list[g])
sheet.cell(row=w + 1, column=2).value = math.degrees(K_pos_xx_rad_list[g])
sheet.cell(row=w + 1, column=3).value = math.degrees(theta_xx_rad_list[g])
sheet.cell(row=w + 1, column=4).value = math.degrees(v_PM_xx_rad_list[g])
sheet.cell(row=w + 1, column=5).value = M_xx_list[g]
sheet.cell(row=w + 1, column=6).value = math.degrees(mu_xx_rad_list[g])
sheet.cell(row=w + 1, column=7).value = V_xx_list[g]
sheet.cell(row=w + 1, column=8).value = x_list[g]
sheet.cell(row=w + 1, column=9).value = y_list[g]
sheet.cell(row=w + 1, column=10).value = p_xx_list[g]
sheet.cell(row=w + 1, column=11).value = T_xx_list[g]
w = w + z + 1 - g
for j in range(z - 1, 0, -1):
f = 0
progress['value'] = 0
perbox.delete(0, END)
tab1.update_idletasks()
percentage = int(round(float(abs((j/(z)*100)-100))))
percentagestr = str(percentage)
strtotal = str(percentagestr+str('%'))
perbox.insert(0, strtotal)
progress['value'] = percentage
tab1.update_idletasks()
root.update()
if j == 1:
perbox.delete(0, END)
progress['value'] = 100
perbox.insert(0, 'Done')
strtotal = str('')
tab1.update_idletasks()
root.update()
remaining.delete(0, j)
remaining.insert(0, j)
for i in range(0, j, 1):
if j == z - 1:
x = float((y_list[i] - y_list[i + 1] + x_list[i + 1] * sp.tan(
theta_xx_rad_list[i + 1] - mu_xx_rad_list[i + 1]) - x_list[i] * sp.tan(
theta_xx_rad_list[i] + mu_xx_rad_list[i])) / (
sp.tan(theta_xx_rad_list[i + 1] - mu_xx_rad_list[i + 1]) - sp.tan(
theta_xx_rad_list[i] + mu_xx_rad_list[i])))
y = float(y_list[i] + (x - x_list[i]) * sp.tan(theta_xx_rad_list[i] + mu_xx_rad_list[i]))
A = float((sp.tan(mu_xx_rad_list[i]) * sp.sin(mu_xx_rad_list[i]) * sp.sin(theta_xx_rad_list[i])) / (
y * sp.cos(theta_xx_rad_list[i] + mu_xx_rad_list[i])))
B = float((sp.tan(mu_xx_rad_list[i + 1]) * sp.sin(mu_xx_rad_list[i + 1]) * sp.sin(
theta_xx_rad_list[i + 1])) / (y * sp.cos(theta_xx_rad_list[i + 1] - mu_xx_rad_list[i + 1])))
V = float(((sp.cot(mu_xx_rad_list[i]) * (1 + (A * (x - x_list[i])))) + (
sp.cot(mu_xx_rad_list[i + 1]) * (1 + (B * (x - x_list[i + 1])))) + theta_xx_rad_list[
i + 1] - theta_xx_rad_list[i]) / ((sp.cot(mu_xx_rad_list[i]) / V_xx_list[i]) + (
sp.cot(mu_xx_rad_list[i + 1]) / V_xx_list[i + 1])))
theta = float(theta_xx_rad_list[i] + sp.cot(mu_xx_rad_list[i]) * (
((V - V_xx_list[i]) / V_xx_list[i]) - (A * (x - x_list[i]))))
T = float(T_c - (V ** 2 / (2 * c_p)))
a = float(sp.sqrt(gamma * R * T))
M = float(V / a)
mu = float(sp.asin(1 / M))
v = float((math.sqrt((gamma + 1) / (gamma - 1))) * sp.atan(
math.sqrt((gamma - 1) * (M ** 2 - 1) / (gamma + 1))) - sp.atan(math.sqrt(M ** 2 - 1)))
K_neg = float(theta + v)
K_pos = float(theta - v)
counting = 1
# --------------------iterations
for iteration in range(1, numiter + 1, 1):
x_2 = float((y_list[i] - y_list[i + 1] + x_list[i + 1] * 0.5 * (
sp.tan(theta_xx_rad_list[i + 1] - mu_xx_rad_list[i + 1]) + sp.tan(theta - mu)) -
x_list[i] * 0.5 * (sp.tan(theta_xx_rad_list[i] + mu_xx_rad_list[i]) + sp.tan(
theta + mu))) / (0.5 * (
sp.tan(theta_xx_rad_list[i + 1] - mu_xx_rad_list[i + 1]) - sp.tan(
theta_xx_rad_list[i] + mu_xx_rad_list[i]) + sp.tan(theta - mu) - sp.tan(theta + mu))))
y_2 = float(y_list[i] + (x - x_list[i]) * 0.5 * (
sp.tan(theta_xx_rad_list[i] + mu_xx_rad_list[i]) + sp.tan(theta + mu)))
A_2A = float(
(sp.tan(mu_xx_rad_list[i]) * sp.sin(mu_xx_rad_list[i]) * sp.sin(theta_xx_rad_list[i])) / (
y * sp.cos(theta_xx_rad_list[i] + mu_xx_rad_list[i])))
A_2C = float((sp.tan(mu) * sp.sin(mu) * sp.sin(theta)) / (y * sp.cos(theta + mu)))
B_2B = float((sp.tan(mu_xx_rad_list[i + 1]) * sp.sin(mu_xx_rad_list[i + 1]) * sp.sin(
theta_xx_rad_list[i + 1])) / (y * sp.cos(theta_xx_rad_list[i + 1] - mu_xx_rad_list[i + 1])))
B_2C = float((sp.tan(mu) * sp.sin(mu) * sp.sin(theta)) / (y * sp.cos(theta - mu)))
A_2 = float((2 / (sp.tan(mu_xx_rad_list[i]) + sp.tan(mu))) * (
((2 * V_xx_list[i]) / (V_xx_list[i] + V)) + ((A_2A + A_2C) / 2) * (x - x_list[i])))
B_2 = float((2 / (sp.tan(mu_xx_rad_list[i + 1]) + sp.tan(mu))) * (
((2 * V_xx_list[i + 1]) / (V_xx_list[i + 1] + V)) + ((B_2B + B_2C) / 2) * (
x - x_list[i + 1])))
V_2 = float((A_2 + B_2 + theta_xx_rad_list[i + 1] - theta_xx_rad_list[i]) / (4 * (
(V_xx_list[i] + V) ** (-1) * (sp.tan(mu_xx_rad_list[i]) + sp.tan(mu)) ** (-1) + (
V_xx_list[i + 1] + V) ** (-1) * (
sp.tan(mu_xx_rad_list[i + 1]) + sp.tan(mu)) ** (-1))))
theta_2 = float(theta_xx_rad_list[i] + (2 / (sp.tan(mu_xx_rad_list[i]) + sp.tan(mu))) * (
((2 * (V_2 - V_xx_list[i])) / (V + V_xx_list[i])) - ((A_2A + A_2C) / 2) * (
x - x_list[i])))
T_2 = float(T_c - (V_2 ** 2 / (2 * c_p)))
a_2 = float(sp.sqrt(gamma * R * T_2))
M_2 = float(V_2 / a_2)
mu_2 = float(sp.asin(1 / M_2))
v_2 = float((math.sqrt((gamma + 1) / (gamma - 1))) * sp.atan(
math.sqrt((gamma - 1) * (M_2 ** 2 - 1) / (gamma + 1))) - sp.atan(math.sqrt(M_2 ** 2 - 1)))
K_neg_2 = float(theta_2 + v_2)
K_pos_2 = float(theta_2 - v_2)
xdiff = float(x - x_2)
if xdiff == 0:
xerr = float(0)
else:
xerr = float(x_2 / xdiff)
ydiff = float(y - y_2)
if ydiff == 0:
yerr = float(0)
else:
yerr = float(y_2 / ydiff)
thetadiff = float(theta - theta_2)
if thetadiff == 0:
thetaerr = float(0)
else:
thetaerr = float(theta_2 / thetadiff)
Vdiff = float(V - V_2)
if Vdiff == 0:
Verr = float(0)
else:
Verr = float(V_2 / Vdiff)
x = x_2
y = y_2
V = V_2
theta = theta_2
T = T_2
a = a_2
M = M_2
mu = mu_2
v = v_2
K_neg = K_neg_2
K_pos = K_pos_2
counting += 1
else:
x = float((y_list[i] - y_list[i + 1] + x_list[i + 1] * sp.tan(
theta_xx_rad_list[i + 1] - mu_xx_rad_list[i + 1]) - x_list[i] * sp.tan(
theta_xx_rad_list[i] + mu_xx_rad_list[i])) / (
sp.tan(theta_xx_rad_list[i + 1] - mu_xx_rad_list[i + 1]) - sp.tan(
theta_xx_rad_list[i] + mu_xx_rad_list[i])))
y = float(y_list[i] + (x - x_list[i]) * sp.tan(theta_xx_rad_list[i] + mu_xx_rad_list[i]))
A = float((sp.tan(mu_xx_rad_list[i]) * sp.sin(mu_xx_rad_list[i]) * sp.sin(theta_xx_rad_list[i])) / (
y_list[i] * sp.cos(theta_xx_rad_list[i] + mu_xx_rad_list[i])))
B = float((sp.tan(mu_xx_rad_list[i + 1]) * sp.sin(mu_xx_rad_list[i + 1]) * sp.sin(
theta_xx_rad_list[i + 1])) / (
y_list[i + 1] * sp.cos(theta_xx_rad_list[i + 1] - mu_xx_rad_list[i + 1])))
V = float(((sp.cot(mu_xx_rad_list[i]) * (1 + (A * (x - x_list[i])))) + (
sp.cot(mu_xx_rad_list[i + 1]) * (1 + (B * (x - x_list[i + 1])))) + theta_xx_rad_list[
i + 1] - theta_xx_rad_list[i]) / ((sp.cot(mu_xx_rad_list[i]) / V_xx_list[i]) + (
sp.cot(mu_xx_rad_list[i + 1]) / V_xx_list[i + 1])))
theta = float(theta_xx_rad_list[i] + sp.cot(mu_xx_rad_list[i]) * (
((V - V_xx_list[i]) / V_xx_list[i]) - (A * (x - x_list[i]))))
T = float(T_c - (V ** 2 / (2 * c_p)))
a = float(sp.sqrt(gamma * R * T))
M = float(V / a)
v = float((math.sqrt((gamma + 1) / (gamma - 1))) * sp.atan(
math.sqrt((gamma - 1) * (M ** 2 - 1) / (gamma + 1))) - sp.atan(math.sqrt(M ** 2 - 1)))
K_neg = float(theta + v)
K_pos = float(theta - v)
mu = float(sp.asin(1 / M))
kkk = 0
# --------------------iterations
for iteration in range(1, numiter + 1, 1):
if iteration == numiter + 1:
kkk = kkk + numiter + 1
iteration_num = kkk + iteration
x_2 = float((y_list[i] - y_list[i + 1] + x_list[i + 1] * 0.5 * (
sp.tan(theta_xx_rad_list[i + 1] - mu_xx_rad_list[i + 1]) + sp.tan(theta - mu)) -
x_list[i] * 0.5 * (sp.tan(theta_xx_rad_list[i] + mu_xx_rad_list[i]) + sp.tan(
theta + mu))) / (0.5 * (
sp.tan(theta_xx_rad_list[i + 1] - mu_xx_rad_list[i + 1]) - sp.tan(
theta_xx_rad_list[i] + mu_xx_rad_list[i]) + sp.tan(theta - mu) - sp.tan(theta + mu))))
y_2 = float(y_list[i] + (x - x_list[i]) * 0.5 * (
sp.tan(theta_xx_rad_list[i] + mu_xx_rad_list[i]) + sp.tan(theta + mu)))
A_2A = float(
(sp.tan(mu_xx_rad_list[i]) * sp.sin(mu_xx_rad_list[i]) * sp.sin(theta_xx_rad_list[i])) / (
y_list[i] * sp.cos(theta_xx_rad_list[i] + mu_xx_rad_list[i])))
A_2C = float((sp.tan(mu) * sp.sin(mu) * sp.sin(theta)) / (y * sp.cos(theta + mu)))
B_2B = float((sp.tan(mu_xx_rad_list[i + 1]) * sp.sin(mu_xx_rad_list[i + 1]) * sp.sin(
theta_xx_rad_list[i + 1])) / (y_list[i + 1] * sp.cos(
theta_xx_rad_list[i + 1] - mu_xx_rad_list[i + 1])))
B_2C = float((sp.tan(mu) * sp.sin(mu) * sp.sin(theta)) / (y * sp.cos(theta - mu)))
A_2 = float((2 / (sp.tan(mu_xx_rad_list[i]) + sp.tan(mu))) * (
((2 * V_xx_list[i]) / (V_xx_list[i] + V)) + ((A_2A + A_2C) / 2) * (x - x_list[i])))
B_2 = float((2 / (sp.tan(mu_xx_rad_list[i + 1]) + sp.tan(mu))) * (
((2 * V_xx_list[i + 1]) / (V_xx_list[i + 1] + V)) + ((B_2B + B_2C) / 2) * (
x - x_list[i + 1])))
V_2 = float((A_2 + B_2 + theta_xx_rad_list[i + 1] - theta_xx_rad_list[i]) / (4 * (
(V_xx_list[i] + V) ** (-1) * (sp.tan(mu_xx_rad_list[i]) + sp.tan(mu)) ** (-1) + (
V_xx_list[i + 1] + V) ** (-1) * (
sp.tan(mu_xx_rad_list[i + 1]) + sp.tan(mu)) ** (-1))))
theta_2 = float(theta_xx_rad_list[i] + (2 / (sp.tan(mu_xx_rad_list[i]) + sp.tan(mu))) * (
((2 * (V_2 - V_xx_list[i])) / (V + V_xx_list[i])) - ((A_2A + A_2C) / 2) * (
x - x_list[i])))
T_2 = float(T_c - (V_2 ** 2 / (2 * c_p)))
a_2 = float(sp.sqrt(gamma * R * T_2))
M_2 = float(V_2 / a_2)
mu_2 = float(sp.asin(1 / M_2))
v_2 = float((math.sqrt((gamma + 1) / (gamma - 1))) * sp.atan(
math.sqrt((gamma - 1) * (M_2 ** 2 - 1) / (gamma + 1))) - sp.atan(math.sqrt(M_2 ** 2 - 1)))
K_neg_2 = float(theta_2 + v_2)
K_pos_2 = float(theta_2 - v_2)
xdiff = float(x - x_2)
if xdiff == 0:
xerr = float(0)
else:
xerr = float(x_2 / xdiff)
ydiff = float(y - y_2)
if ydiff == 0:
yerr = float(0)
else:
yerr = float(y_2 / ydiff)
thetadiff = float(theta - theta_2)
if thetadiff == 0:
thetaerr = float(0)
else:
thetaerr = float(theta_2 / thetadiff)
Vdiff = float(V - V_2)
if Vdiff == 0:
Verr = float(0)
else:
Verr = float(V_2 / Vdiff)
x = x_2
y = y_2
V = V_2
theta = theta_2
T = T_2
a = a_2
M = M_2
mu = mu_2
v = v_2
K_neg = K_neg_2
K_pos = K_pos_2
x_list[i] = x
y_list[i] = y
V_xx_list[i] = V
theta_xx_rad_list[i] = theta
v_PM_xx_rad_list[i] = v
K_neg_xx_rad_list[i] = K_neg
K_pos_xx_rad_list[i] = K_pos
M_xx_list[i] = M
mu_xx_rad_list[i] = mu
T_xx_list[i] = T
plt.plot(x_list, y_list, 'k.', markersize=1)
n = (z - j + 1) + f
f = f + z + 1 - i
sheet.cell(row=n + 1, column=1).value = math.degrees(K_neg_xx_rad_list[i])
sheet.cell(row=n + 1, column=2).value = math.degrees(K_pos_xx_rad_list[i])
sheet.cell(row=n + 1, column=3).value = math.degrees(theta_xx_rad_list[i])
sheet.cell(row=n + 1, column=4).value = math.degrees(v_PM_xx_rad_list[i])
sheet.cell(row=n + 1, column=5).value = M_xx_list[i]
sheet.cell(row=n + 1, column=6).value = math.degrees(mu_xx_rad_list[i])
sheet.cell(row=n + 1, column=7).value = V_xx_list[i]
sheet.cell(row=n + 1, column=8).value = x_list[i]
sheet.cell(row=n + 1, column=9).value = y_list[i]
sheet.cell(row=n + 1, column=11).value = T_xx_list[i]
counter = 1
sheet.cell(row=1, column=12).value = "x nozzle wall"
sheet.cell(row=1, column=13).value = "y nozzle wall"
d = 0
for q in range(0, z, 1):
if q == 0:
x = float((-x_list[q] * sp.tan(theta_xx_rad_list[q] + mu_xx_rad_list[q]) + y_list[q] - r_t) / (
sp.tan(theta_max_rad) - sp.tan(theta_xx_rad_list[q] + mu_xx_rad_list[q])))
y = float(sp.tan(theta_xx_rad_list[q] + mu_xx_rad_list[q]) * (x - x_list[q]) + y_list[q])
theta = float((theta_max_rad + theta_xx_rad_list[q]) / 2)
else:
x = float((-x_list[q] * sp.tan(theta_xx_rad_list[q] + mu_xx_rad_list[q]) + y_list[q] + x_list[
q - 1] * sp.tan(theta) - y_list[q - 1]) / (
sp.tan(theta) - sp.tan(theta_xx_rad_list[q] + mu_xx_rad_list[q])))
y = float(sp.tan(theta_xx_rad_list[q] + mu_xx_rad_list[q]) * (x - x_list[q]) + y_list[q])
theta = float((theta + theta_xx_rad_list[q]) / 2)
x_list[q] = x
y_list[q] = y
d = d + z + 1 - q
sheet.cell(row=d + 1, column=1).value = math.degrees(K_neg_xx_rad_list[q])
sheet.cell(row=d + 1, column=2).value = math.degrees(K_pos_xx_rad_list[q])
sheet.cell(row=d + 1, column=3).value = math.degrees(theta_xx_rad_list[q])
sheet.cell(row=d + 1, column=4).value = math.degrees(v_PM_xx_rad_list[q])
sheet.cell(row=d + 1, column=5).value = M_xx_list[q]
sheet.cell(row=d + 1, column=6).value = math.degrees(mu_xx_rad_list[q])
sheet.cell(row=d + 1, column=7).value = V_xx_list[q]
sheet.cell(row=d + 1, column=8).value = x_list[q]
sheet.cell(row=d + 1, column=9).value = y_list[q]
sheet.cell(row=counter + 1, column=12).value = x
sheet.cell(row=counter + 1, column=13).value = y
counter += 1
sheet.cell(row=1, column=15).value = "x_conv"
sheet.cell(row=1, column=16).value = "y_conv"
x_conv_list = []
y_conv_list = []
counter = 1
for beta in np.arange(-90, -136, -1):
x_conv = float(1.5 * r_t * sp.cos(math.radians(beta)))
y_conv = float(1.5 * r_t * sp.sin(math.radians(beta)) + 1.5 * r_t + r_t)
sheet.cell(row=counter + 1, column=15).value = x_conv
sheet.cell(row=counter + 1, column=16).value = y_conv
x_conv_list.append(x_conv)
y_conv_list.append(y_conv)
counter += 1
outbox = Text(tab1, borderwidth=2, width=90, height=10, fg="black")
outbox.grid(row=26, column=0, columnspan=100, padx=5, pady=40)
outbox.insert(END,"All data from calculation has been exported in 'export1.xlsx'. If another calculation is initiated, the data from the previous calculation will be overwritten." + '\n')
outbox.see("end")
x_line = ((r_c - y_conv) / sp.tan(math.radians(-45))) + x_conv
x_conv_list.insert(45, x_line)
y_conv_list.insert(45, r_c)
plt.plot(x_conv_list, y_conv_list, 'ko', linewidth=2, linestyle='-', markersize=2)
x_list.insert(0, 0)
y_list.insert(0, r_t)
plt.plot(x_list, y_list, 'ko', linewidth=2, linestyle='-', markersize=4)
plt.ylabel("Radius [m]")
plt.xlabel("Length [m]")
plt.axis('equal')
rerr = float((r_e - y_list[z]) / y_list[z] * 100)
exhaust_radius.insert(0, y_list[z])
nozzle_length.insert(0, x_list[z])
isen_exhaust_radius.insert(0, r_e)
error.insert(0, rerr)
textstr = "Combustion Pressure: "+str(p_c)+" [Pa]\n" \
"Combusiton Temperature: "+str(T_c)+" [K]\n" \
"Ratio of Spec. Heat: "+str(gamma)+" [-]\n" \
"Specific Gas Constant: "+str(R)+" [J/kgK]\n" \
"Throat Radius: "+str(r_t)+" [m]\n" \
"Combustion Chamber Radius: "+str(r_c)+" [m]\n" \
"Number of Char. Lines: "+str(num_of_char_lines)+" \n" \
"Number of Iterations: "+str(numiter)+" \n" \
"Initial Expansion Angle: "+str(theta_max_deg)+" [deg]\n" \
"Characteristic Net Length: "+str(L)+" [m]\n" \
"Pressure Distribution Type: "+str(pressure_type)+" \n" \
"Cubic q-factor: "+str(q_factor_value)+" [-]\n"
exhauststr = "Exhaust Mach Number: " + str(M_e) + " [-]\n" \
"Exhaust velocity: " + str(u_e) + " [m/s]\n" \
"Exhaust Pressure: " + str(p_e) + " [Pa]\n" \
"Exhaust Temperature: " + str(T_e) + " [K]\n" \
"Exhaust Density: " + str(rho_e) + " [kg/m^3]\n" \
"MOC Exahust Radius: " + str(y_list[z]) + " [m]\n" \
"Isentropic Exhaust Radius: " + str(r_e) + " [m]\n" \
T_t = float(T_c*(2/(gamma+1)))
rho_t = float(rho_c*(2/(gamma+1))**(1/(gamma-1)))
throatstr = "Throat Mach Number: "+str(M_t)+" [-]\n" \
"Throat velocity: " + str(float(sp.sqrt(gamma*R*T_t))) + " [m/s]\n" \
"Throat Pressure: " + str(p_t) + " [Pa]\n" \
"Throat Temperature: " + str(T_t) + " [K]\n" \
"Throat Density: " + str(rho_t) + " [kg/m^3]\n" \
"Initial Expansion Angle: " + str(theta_max_deg) + " [deg]\n" \
"Isentropic Throat Radius: " + str(r_t) + " [m]\n" \
combstr = "Combustion Pressure: " + str(p_c) + " [Pa]\n" \
"Combustion Temperature: " + str(T_c) + " [K]\n" \
"Combustion Density: " + str(rho_c) + " [kg/m^3]\n" \
"Combustion Chamber Radius: " + str(r_c) + " [m]\n" \
lengthstr = "Characteristic Net Length: "+str(L)+" [m] \n" \
"Nozzle Length: "+str(x_list[z])+" [m]"
plt.title("AXIMOC Nozzle with "+str(num_of_char_lines)+" char. lines; "+str(numiter)+" iterations. Pressure type: "+str(pressure_type)+"; Expansion angle: "+str(theta_max_deg)+" [deg].")
m_dot = float(rho_e*u_e*A_e)
print(m_dot)
plt.text(x_list[z-round(num_of_char_lines/4)], y_list[z], exhauststr, fontsize=8)
plt.text(x_list[z-round(num_of_char_lines)], y_list[z], throatstr, fontsize=8)
plt.text(x_line, -0.06, combstr, fontsize=8)
plt.text(x_list[round(z/3)], -0.025, lengthstr, fontsize=8)
book.save('export1.xlsx')
calculatebutton = Button(tab1, text="Calculate", padx=35, pady=10, command=calculate).grid(row=11, column=3,
rowspan=2,
columnspan=10,
sticky=W)
return plt.show()
def test_issue_4403_2():
assert integrate(sqrt(-x**2 - 4), x) == \
-2*atan(x/sqrt(-4 - x**2)) + x*sqrt(-4 - x**2)/2
def test_invert_real():
x = Dummy(real=True)
n = Symbol('n')
minus_n = Intersection(Interval(-oo, 0), FiniteSet(-n))
plus_n = Intersection(Interval(0, oo), FiniteSet(n))
assert solveset(abs(x) - n, x, S.Reals) == Union(minus_n, plus_n)
n = Symbol('n', real=True)
assert invert_real(x + 3, y, x) == (x, FiniteSet(y - 3))
assert invert_real(x * 3, y, x) == (x, FiniteSet(y / 3))
assert invert_real(exp(x), y, x) == (x, FiniteSet(log(y)))
assert invert_real(exp(3 * x), y, x) == (x, FiniteSet(log(y) / 3))
assert invert_real(exp(x + 3), y, x) == (x, FiniteSet(log(y) - 3))
assert invert_real(exp(x) + 3, y, x) == (x, FiniteSet(log(y - 3)))
assert invert_real(exp(x) * 3, y, x) == (x, FiniteSet(log(y / 3)))
assert invert_real(log(x), y, x) == (x, FiniteSet(exp(y)))
assert invert_real(log(3 * x), y, x) == (x, FiniteSet(exp(y) / 3))
assert invert_real(log(x + 3), y, x) == (x, FiniteSet(exp(y) - 3))
minus_y = Intersection(Interval(-oo, 0), FiniteSet(-y))
plus_y = Intersection(Interval(0, oo), FiniteSet(y))
assert invert_real(Abs(x), y, x) == (x, Union(minus_y, plus_y))
assert invert_real(2**x, y, x) == (x, FiniteSet(log(y) / log(2)))
assert invert_real(2**exp(x), y, x) == (x, FiniteSet(log(log(y) / log(2))))
assert invert_real(x**2, y, x) == (x, FiniteSet(sqrt(y), -sqrt(y)))
assert invert_real(x**Rational(1, 2), y, x) == (x, FiniteSet(y**2))
raises(ValueError, lambda: invert_real(x, x, x))
raises(ValueError, lambda: invert_real(x**pi, y, x))
raises(ValueError, lambda: invert_real(S.One, y, x))
assert invert_real(x**31 + x, y, x) == (x**31 + x, FiniteSet(y))
y_1 = Intersection(Interval(-1, oo), FiniteSet(y - 1))
y_2 = Intersection(Interval(-oo, -1), FiniteSet(-y - 1))
assert invert_real(Abs(x**31 + x + 1), y,
x) == (x**31 + x, Union(y_1, y_2))
assert invert_real(sin(x), y, x) == \
(x, imageset(Lambda(n, n*pi + (-1)**n*asin(y)), S.Integers))
assert invert_real(sin(exp(x)), y, x) == \
(x, imageset(Lambda(n, log((-1)**n*asin(y) + n*pi)), S.Integers))
assert invert_real(csc(x), y, x) == \
(x, imageset(Lambda(n, n*pi + (-1)**n*acsc(y)), S.Integers))
assert invert_real(csc(exp(x)), y, x) == \
(x, imageset(Lambda(n, log((-1)**n*acsc(y) + n*pi)), S.Integers))
assert invert_real(cos(x), y, x) == \
(x, Union(imageset(Lambda(n, 2*n*pi + acos(y)), S.Integers), \
imageset(Lambda(n, 2*n*pi - acos(y)), S.Integers)))
assert invert_real(cos(exp(x)), y, x) == \
(x, Union(imageset(Lambda(n, log(2*n*pi + acos(y))), S.Integers), \
imageset(Lambda(n, log(2*n*pi - acos(y))), S.Integers)))
assert invert_real(sec(x), y, x) == \
(x, Union(imageset(Lambda(n, 2*n*pi + asec(y)), S.Integers), \
imageset(Lambda(n, 2*n*pi - asec(y)), S.Integers)))
assert invert_real(sec(exp(x)), y, x) == \
(x, Union(imageset(Lambda(n, log(2*n*pi + asec(y))), S.Integers), \
imageset(Lambda(n, log(2*n*pi - asec(y))), S.Integers)))
assert invert_real(tan(x), y, x) == \
(x, imageset(Lambda(n, n*pi + atan(y)), S.Integers))
assert invert_real(tan(exp(x)), y, x) == \
(x, imageset(Lambda(n, log(n*pi + atan(y))), S.Integers))
assert invert_real(cot(x), y, x) == \
(x, imageset(Lambda(n, n*pi + acot(y)), S.Integers))
assert invert_real(cot(exp(x)), y, x) == \
(x, imageset(Lambda(n, log(n*pi + acot(y))), S.Integers))
assert invert_real(tan(tan(x)), y, x) == \
(tan(x), imageset(Lambda(n, n*pi + atan(y)), S.Integers))
x = Symbol('x', positive=True)
assert invert_real(x**pi, y, x) == (x, FiniteSet(y**(1 / pi)))
# Test for ``set_h`` containing information about the domain
n = Dummy('n')
x = Symbol('x')
h1 = Intersection(Interval(-3, oo), FiniteSet(a + b - 3),
imageset(Lambda(n, -n + a - 3), Interval(-oo, 0)))
h2 = Intersection(Interval(-oo, -3), FiniteSet(-a + b - 3),
imageset(Lambda(n, n - a - 3), Interval(0, oo)))
h3 = Intersection(Interval(-3, oo), FiniteSet(a - b - 3),
imageset(Lambda(n, -n + a - 3), Interval(0, oo)))
h4 = Intersection(Interval(-oo, -3), FiniteSet(-a - b - 3),
imageset(Lambda(n, n - a - 3), Interval(-oo, 0)))
assert invert_real(Abs(Abs(x + 3) - a) - b, 0,
x) == (x, Union(h1, h2, h3, h4))
def test_cos():
x, y = symbols('x y')
assert cos.nargs == FiniteSet(1)
assert cos(nan) == nan
assert cos(oo * I) == oo
assert cos(-oo * I) == oo
assert cos(0) == 1
assert cos(acos(x)) == x
assert cos(atan(x)) == 1 / sqrt(1 + x**2)
assert cos(asin(x)) == sqrt(1 - x**2)
assert cos(acot(x)) == 1 / sqrt(1 + 1 / x**2)
assert cos(atan2(y, x)) == x / sqrt(x**2 + y**2)
assert cos(pi * I) == cosh(pi)
assert cos(-pi * I) == cosh(pi)
assert cos(-2 * I) == cosh(2)
assert cos(pi / 2) == 0
assert cos(-pi / 2) == 0
assert cos(pi / 2) == 0
assert cos(-pi / 2) == 0
assert cos((-3 * 10**73 + 1) * pi / 2) == 0
assert cos((7 * 10**103 + 1) * pi / 2) == 0
n = symbols('n', integer=True)
assert cos(pi * n / 2) == 0
assert cos(pi) == -1
assert cos(-pi) == -1
assert cos(2 * pi) == 1
assert cos(5 * pi) == -1
assert cos(8 * pi) == 1
assert cos(pi / 3) == S.Half
assert cos(-2 * pi / 3) == -S.Half
assert cos(pi / 4) == S.Half * sqrt(2)
assert cos(-pi / 4) == S.Half * sqrt(2)
assert cos(11 * pi / 4) == -S.Half * sqrt(2)
assert cos(-3 * pi / 4) == -S.Half * sqrt(2)
assert cos(pi / 6) == S.Half * sqrt(3)
assert cos(-pi / 6) == S.Half * sqrt(3)
assert cos(7 * pi / 6) == -S.Half * sqrt(3)
assert cos(-5 * pi / 6) == -S.Half * sqrt(3)
assert cos(1 * pi / 5) == (sqrt(5) + 1) / 4
assert cos(2 * pi / 5) == (sqrt(5) - 1) / 4
assert cos(3 * pi / 5) == -cos(2 * pi / 5)
assert cos(4 * pi / 5) == -cos(1 * pi / 5)
assert cos(6 * pi / 5) == -cos(1 * pi / 5)
assert cos(8 * pi / 5) == cos(2 * pi / 5)
assert cos(-1273 * pi / 5) == -cos(2 * pi / 5)
assert cos(pi / 8) == sqrt((2 + sqrt(2)) / 4)
assert cos(104 * pi / 105) == -cos(pi / 105)
assert cos(106 * pi / 105) == -cos(pi / 105)
assert cos(-104 * pi / 105) == -cos(pi / 105)
assert cos(-106 * pi / 105) == -cos(pi / 105)
assert cos(x * I) == cosh(x)
assert cos(k * pi * I) == cosh(k * pi)
assert cos(r).is_real is True
assert cos(k * pi) == (-1)**k
assert cos(2 * k * pi) == 1
for d in list(range(1, 22)) + [60, 85]:
for n in xrange(0, 2 * d + 1):
x = n * pi / d
e = abs(float(cos(x)) - cos(float(x)))
assert e < 1e-12
def log(self):
""" logarithmic map"""
n = sympy.sqrt(sophus.squared_norm(self.q.vec))
return 2 * sympy.atan(n / self.q.real) / n * self.q.vec
def test_sin():
x, y = symbols('x y')
assert sin.nargs == FiniteSet(1)
assert sin(nan) == nan
assert sin(oo * I) == oo * I
assert sin(-oo * I) == -oo * I
assert sin(oo).args[0] == oo
assert sin(0) == 0
assert sin(asin(x)) == x
assert sin(atan(x)) == x / sqrt(1 + x**2)
assert sin(acos(x)) == sqrt(1 - x**2)
assert sin(acot(x)) == 1 / (sqrt(1 + 1 / x**2) * x)
assert sin(atan2(y, x)) == y / sqrt(x**2 + y**2)
assert sin(pi * I) == sinh(pi) * I
assert sin(-pi * I) == -sinh(pi) * I
assert sin(-2 * I) == -sinh(2) * I
assert sin(pi) == 0
assert sin(-pi) == 0
assert sin(2 * pi) == 0
assert sin(-2 * pi) == 0
assert sin(-3 * 10**73 * pi) == 0
assert sin(7 * 10**103 * pi) == 0
assert sin(pi / 2) == 1
assert sin(-pi / 2) == -1
assert sin(5 * pi / 2) == 1
assert sin(7 * pi / 2) == -1
n = symbols('n', integer=True)
assert sin(pi * n / 2) == (-1)**(n / 2 - S.Half)
assert sin(pi / 3) == S.Half * sqrt(3)
assert sin(-2 * pi / 3) == -S.Half * sqrt(3)
assert sin(pi / 4) == S.Half * sqrt(2)
assert sin(-pi / 4) == -S.Half * sqrt(2)
assert sin(17 * pi / 4) == S.Half * sqrt(2)
assert sin(-3 * pi / 4) == -S.Half * sqrt(2)
assert sin(pi / 6) == S.Half
assert sin(-pi / 6) == -S.Half
assert sin(7 * pi / 6) == -S.Half
assert sin(-5 * pi / 6) == -S.Half
assert sin(1 * pi / 5) == sqrt((5 - sqrt(5)) / 8)
assert sin(2 * pi / 5) == sqrt((5 + sqrt(5)) / 8)
assert sin(3 * pi / 5) == sin(2 * pi / 5)
assert sin(4 * pi / 5) == sin(1 * pi / 5)
assert sin(6 * pi / 5) == -sin(1 * pi / 5)
assert sin(8 * pi / 5) == -sin(2 * pi / 5)
assert sin(-1273 * pi / 5) == -sin(2 * pi / 5)
assert sin(pi / 8) == sqrt((2 - sqrt(2)) / 4)
assert sin(104 * pi / 105) == sin(pi / 105)
assert sin(106 * pi / 105) == -sin(pi / 105)
assert sin(-104 * pi / 105) == -sin(pi / 105)
assert sin(-106 * pi / 105) == sin(pi / 105)
assert sin(x * I) == sinh(x) * I
assert sin(k * pi) == 0
assert sin(17 * k * pi) == 0
assert sin(k * pi * I) == sinh(k * pi) * I
assert sin(r).is_real is True
assert isinstance(sin(re(x) - im(y)), sin) is True
assert isinstance(sin(-re(x) + im(y)), sin) is False
for d in list(range(1, 22)) + [60, 85]:
for n in xrange(0, d * 2 + 1):
x = n * pi / d
e = abs(float(sin(x)) - sin(float(x)))
assert e < 1e-12
def eval_arctan(integrand, symbol):
return sympy.atan(symbol)
def test_ratint():
assert ratint(S(0), x) == 0
assert ratint(S(7), x) == 7*x
assert ratint(x, x) == x**2/2
assert ratint(2*x, x) == x**2
assert ratint(-2*x, x) == -x**2
assert ratint(8*x**7+2*x+1, x) == x**8+x**2+x
f = S(1)
g = x + 1
assert ratint(f / g, x) == log(x + 1)
assert ratint((f,g), x) == log(x + 1)
f = x**3 - x
g = x - 1
assert ratint(f/g, x) == x**3/3 + x**2/2
f = x
g = (x - a)*(x + a)
assert ratint(f/g, x) == log(x**2 - a**2)/2
f = S(1)
g = x**2 + 1
assert ratint(f/g, x, real=None) == atan(x)
assert ratint(f/g, x, real=True) == atan(x)
assert ratint(f/g, x, real=False) == I*log(x + I)/2 - I*log(x - I)/2
f = S(36)
g = x**5-2*x**4-2*x**3+4*x**2+x-2
assert ratint(f/g, x) == \
-4*log(x + 1) + 4*log(x - 2) + (12*x + 6)/(x**2 - 1)
f = x**4-3*x**2+6
g = x**6-5*x**4+5*x**2+4
assert ratint(f/g, x) == \
atan(x) + atan(x**3) + atan(x/2 - 3*x**S(3)/2 + S(1)/2*x**5)
f = x**7-24*x**4-4*x**2+8*x-8
g = x**8+6*x**6+12*x**4+8*x**2
assert ratint(f/g, x) == \
(4 + 6*x + 8*x**2 + 3*x**3)/(4*x + 4*x**3 + x**5) + log(x)
assert ratint((x**3*f)/(x*g), x) == \
-(12 - 16*x + 6*x**2 - 14*x**3)/(4 + 4*x**2 + x**4) - \
5*sqrt(2)*atan(x*sqrt(2)/2) + S(1)/2*x**2 - 3*log(2 + x**2)
f = x**5-x**4+4*x**3+x**2-x+5
g = x**4-2*x**3+5*x**2-4*x+4
assert ratint(f/g, x) == \
x + S(1)/2*x**2 + S(1)/2*log(2-x+x**2) - (4*x-9)/(14-7*x+7*x**2) + \
13*sqrt(7)*atan(-S(1)/7*sqrt(7) + 2*x*sqrt(7)/7)/49
assert ratint(1/(x**2+x+1), x) == \
2*sqrt(3)*atan(sqrt(3)/3 + 2*x*sqrt(3)/3)/3
assert ratint(1/(x**3+1), x) == \
-log(1 - x + x**2)/6 + log(1 + x)/3 + sqrt(3)*atan(-sqrt(3)/3 + 2*x*sqrt(3)/3)/3
assert ratint(1/(x**2+x+1), x, real=False) == \
-I*3**half*log(half + x - half*I*3**half)/3 + \
I*3**half*log(half + x + half*I*3**half)/3
assert ratint(1/(x**3+1), x, real=False) == log(1 + x)/3 + \
(-S(1)/6 + I*3**half/6)*log(-half + x + I*3**half/2) + \
(-S(1)/6 - I*3**half/6)*log(-half + x - I*3**half/2)
# Issue 1892
assert ratint(1/(x*(a+b*x)**3), x) == \
(sqrt(a**(-6))*log(x + (a - a**4*sqrt(a**(-6)))/(2*b)) +
(3*a + 2*b*x)/(2*a**2*b**2*x**2 + 4*b*x*a**3 + 2*a**4) -
sqrt(a**(-6))*log(x + (a + a**4*sqrt(a**(-6)))/(2*b)))
assert ratint(x/(1 - x**2), x) == -log(x**2 - 1)/2
assert ratint(-x/(1 - x**2), x) == log(x**2 - 1)/2
ans = atan(x)
assert ratint(1/(x**2 + 1), x, symbol=x) == ans
assert ratint(1/(x**2 + 1), x, symbol='x') == ans
assert ratint(1/(x**2 + 1), x, symbol=a) == ans
def test_issue_8520():
assert manualintegrate(x / (x**4 + 1), x) == atan(x**2) / 2
assert manualintegrate(x**2 / (x**6 + 25), x) == atan(x**3 / 5) / 15
f = x / (9 * x**4 + 4)**2
assert manualintegrate(f, x).diff(x).factor() == f
def test_issue_10847_slow():
assert manualintegrate((4*x**4 + 4*x**3 + 16*x**2 + 12*x + 8)
/ (x**6 + 2*x**5 + 3*x**4 + 4*x**3 + 3*x**2 + 2*x + 1), x) == \
2*x/(x**2 + 1) + 3*atan(x) - 1/(x**2 + 1) - 3/(x + 1)
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 test_atan():
x = Symbol("x", real=True)
assert limit(atan(x) * sin(1 / x), x, 0) == 0
assert limit(atan(x) + sqrt(x + 1) - sqrt(x), x, oo) == pi / 2
def test_issue_2882():
u = symbols('u')
assert integrate(1/(u**2 + 1)) == atan(u)
def test_issue_2718():
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_manualintegrate_inversetrig():
# atan
assert manualintegrate(exp(x) / (1 + exp(2 * x)), x) == atan(exp(x))
assert manualintegrate(1 / (4 + 9 * x**2), x) == atan(3 * x / 2) / 6
assert manualintegrate(1 / (16 + 16 * x**2), x) == atan(x) / 16
assert manualintegrate(1 / (4 + x**2), x) == atan(x / 2) / 2
assert manualintegrate(1 / (1 + 4 * x**2), x) == atan(2 * x) / 2
ra = Symbol('a', real=True)
rb = Symbol('b', real=True)
assert manualintegrate(1/(ra + rb*x**2), x) == \
Piecewise((atan(x/sqrt(ra/rb))/(rb*sqrt(ra/rb)), ra/rb > 0),
(-acoth(x/sqrt(-ra/rb))/(rb*sqrt(-ra/rb)), And(ra/rb < 0, x**2 > -ra/rb)),
(-atanh(x/sqrt(-ra/rb))/(rb*sqrt(-ra/rb)), And(ra/rb < 0, x**2 < -ra/rb)))
assert manualintegrate(1/(4 + rb*x**2), x) == \
Piecewise((atan(x/(2*sqrt(1/rb)))/(2*rb*sqrt(1/rb)), 4/rb > 0),
(-acoth(x/(2*sqrt(-1/rb)))/(2*rb*sqrt(-1/rb)), And(4/rb < 0, x**2 > -4/rb)),
(-atanh(x/(2*sqrt(-1/rb)))/(2*rb*sqrt(-1/rb)), And(4/rb < 0, x**2 < -4/rb)))
assert manualintegrate(1/(ra + 4*x**2), x) == \
Piecewise((atan(2*x/sqrt(ra))/(2*sqrt(ra)), ra/4 > 0),
(-acoth(2*x/sqrt(-ra))/(2*sqrt(-ra)), And(ra/4 < 0, x**2 > -ra/4)),
(-atanh(2*x/sqrt(-ra))/(2*sqrt(-ra)), And(ra/4 < 0, x**2 < -ra/4)))
assert manualintegrate(1 / (4 + 4 * x**2), x) == atan(x) / 4
assert manualintegrate(1 / (a + b * x**2),
x) == atan(x / sqrt(a / b)) / (b * sqrt(a / b))
# asin
assert manualintegrate(1 / sqrt(1 - x**2), x) == asin(x)
assert manualintegrate(1 / sqrt(4 - 4 * x**2), x) == asin(x) / 2
assert manualintegrate(3 / sqrt(1 - 9 * x**2), x) == asin(3 * x)
assert manualintegrate(1 / sqrt(4 - 9 * x**2),
x) == asin(x * Rational(3, 2)) / 3
# asinh
assert manualintegrate(1/sqrt(x**2 + 1), x) == \
asinh(x)
assert manualintegrate(1/sqrt(x**2 + 4), x) == \
asinh(x/2)
assert manualintegrate(1/sqrt(4*x**2 + 4), x) == \
asinh(x)/2
assert manualintegrate(1/sqrt(4*x**2 + 1), x) == \
asinh(2*x)/2
assert manualintegrate(1/sqrt(a*x**2 + 1), x) == \
Piecewise((sqrt(-1/a)*asin(x*sqrt(-a)), a < 0), (sqrt(1/a)*asinh(sqrt(a)*x), a > 0))
assert manualintegrate(1/sqrt(a + x**2), x) == \
Piecewise((asinh(x*sqrt(1/a)), a > 0), (acosh(x*sqrt(-1/a)), a < 0))
# acosh
assert manualintegrate(1/sqrt(x**2 - 1), x) == \
acosh(x)
assert manualintegrate(1/sqrt(x**2 - 4), x) == \
acosh(x/2)
assert manualintegrate(1/sqrt(4*x**2 - 4), x) == \
acosh(x)/2
assert manualintegrate(1/sqrt(9*x**2 - 1), x) == \
acosh(3*x)/3
assert manualintegrate(1/sqrt(a*x**2 - 4), x) == \
Piecewise((sqrt(1/a)*acosh(sqrt(a)*x/2), a > 0))
assert manualintegrate(1/sqrt(-a + 4*x**2), x) == \
Piecewise((asinh(2*x*sqrt(-1/a))/2, -a > 0), (acosh(2*x*sqrt(1/a))/2, -a < 0))
# piecewise
assert manualintegrate(1/sqrt(a-b*x**2), x) == \
Piecewise((sqrt(a/b)*asin(x*sqrt(b/a))/sqrt(a), And(-b < 0, a > 0)),
(sqrt(-a/b)*asinh(x*sqrt(-b/a))/sqrt(a), And(-b > 0, a > 0)),
(sqrt(a/b)*acosh(x*sqrt(b/a))/sqrt(-a), And(-b > 0, a < 0)))
assert manualintegrate(1/sqrt(a + b*x**2), x) == \
Piecewise((sqrt(-a/b)*asin(x*sqrt(-b/a))/sqrt(a), And(a > 0, b < 0)),
(sqrt(a/b)*asinh(x*sqrt(b/a))/sqrt(a), And(a > 0, b > 0)),
(sqrt(-a/b)*acosh(x*sqrt(-b/a))/sqrt(-a), And(a < 0, b > 0)))
def test_issue_2315():
assert ratint(1/(x**2 + 16), x) == atan(x/4)/4
from sympy import var, sqrt, sin, pprint, simplify, ccode, atan2, atan, exp, S
var("x y alpha_w alpha_p omega pi x_w y_w x_p y_p epsilon r0")
def diff_f_x(f):
return f.diff(x, 1)
r = sqrt(x**2 + y**2)
r_w = sqrt((x - x_w)**2 + (y - y_w)**2)
r_p_squared = (x - x_p)**2 + (y - y_p)**2
theta = atan(y / x)
u = r**(pi / omega) * sin(theta * pi / omega) + atan(
alpha_w *
(r_w - r0)) + exp(-alpha_p * r_p_squared) + exp(-(1 + y) / epsilon)
params = {
omega: 3 * pi / 2,
x_w: 0,
y_w: -S(3) / 4,
r0: S(3) / 4,
alpha_p: 1000,
alpha_w: 200,
x_p: sqrt(5) / 4,
y_p: -S(1) / 4,
epsilon: S(1) / 100,
}
u = u.subs(params)
lhs = diff_f_x(u)
print "lhs, as a formula:"
pprint(lhs)
def test_issue_15539():
assert series(atan(x), x, -oo) == (-1 / (5 * x**5) + 1 / (3 * x**3) -
1 / x - pi / 2 + O(x**(-6), (x, -oo)))
assert series(atan(x), x, oo) == (-1 / (5 * x**5) + 1 / (3 * x**3) -
1 / x + pi / 2 + O(x**(-6), (x, oo)))
def test_issue_1707():
assert integrate(atan(x)**2, (x, -1, 1)).evalf().round(1) == 0.5
assert atan(0, evaluate=False).n() == 0
'square': lambda x: x**2, 'cube': lambda x: x**3, 'plus': lambda x, y: x + y, 'sub': lambda x, y: x - y, 'neg': lambda x: -x, 'pow': lambda x, y: sympy.sign(x) * abs(x)**y, 'cos': lambda x: sympy.cos(x), 'sin': lambda x: sympy.sin(x), 'tan': lambda x: sympy.tan(x), 'cosh': lambda x: sympy.cosh(x), 'sinh': lambda x: sympy.sinh(x), 'tanh': lambda x: sympy.tanh(x), 'exp': lambda x: sympy.exp(x), 'acos': lambda x: sympy.acos(x), 'asin': lambda x: sympy.asin(x), 'atan': lambda x: sympy.atan(x), 'acosh': lambda x: sympy.acosh(x), 'asinh': lambda x: sympy.asinh(x), 'atanh': lambda x: sympy.atanh(x), 'abs': lambda x: abs(x), 'mod': lambda x, y: sympy.Mod(x, y), 'erf': lambda x: sympy.erf(x), 'erfc': lambda x: sympy.erfc(x), 'logm': lambda x: sympy.log(abs(x)), 'logm10': lambda x: sympy.log10(abs(x)), 'logm2': lambda x: sympy.log2(abs(x)), 'log1p': lambda x: sympy.log(x + 1), 'floor': lambda x: sympy.floor(x), 'ceil': lambda x: sympy.ceil(x), 'sign': lambda x: sympy.sign(x), 'round': lambda x: sympy.round(x),
def test_linearize_pendulum_lagrange_nonminimal():
q1, q2 = dynamicsymbols('q1:3')
q1d, q2d = dynamicsymbols('q1:3', level=1)
L, m, t = symbols('L, m, t')
g = 9.8
# Compose World Frame
N = ReferenceFrame('N')
pN = Point('N*')
pN.set_vel(N, 0)
# A.x is along the pendulum
theta1 = atan(q2/q1)
A = N.orientnew('A', 'axis', [theta1, N.z])
# Create point P, the pendulum mass
P = pN.locatenew('P1', q1*N.x + q2*N.y)
P.set_vel(N, P.pos_from(pN).dt(N))
pP = Particle('pP', P, m)
# Constraint Equations
f_c = Matrix([q1**2 + q2**2 - L**2])
# Calculate the lagrangian, and form the equations of motion
Lag = Lagrangian(N, pP)
LM = LagrangesMethod(Lag, [q1, q2], hol_coneqs=f_c, forcelist=[(P, m*g*N.x)], frame=N)
LM.form_lagranges_equations()
# Compose operating point
op_point = {q1: L, q2: 0, q1d: 0, q2d: 0, q1d.diff(t): 0, q2d.diff(t): 0}
# Solve for multiplier operating point
lam_op = LM.solve_multipliers(op_point=op_point)
op_point.update(lam_op)
# Perform the Linearization
A, B, inp_vec = LM.linearize([q2], [q2d], [q1], [q1d],
op_point=op_point, A_and_B=True)
assert A == Matrix([[0, 1], [-9.8/L, 0]])
assert B == Matrix([])