def test_TRpower(): assert TRpower(1/sin(x)**2) == 1/sin(x)**2 assert TRpower(cos(x)**3*sin(x/2)**4) == \ (3*cos(x)/4 + cos(3*x)/4)*(-cos(x)/2 + cos(2*x)/8 + Rational(3, 8)) for k in range(2, 8): assert verify_numerically(sin(x)**k, TRpower(sin(x)**k)) assert verify_numerically(cos(x)**k, TRpower(cos(x)**k))
def test_roots_quartic(): assert roots_quartic(Poly(x**4, x)) == [0, 0, 0, 0] assert roots_quartic(Poly(x**4 + x**3, x)) in [ [-1, 0, 0, 0], [0, -1, 0, 0], [0, 0, -1, 0], [0, 0, 0, -1] ] assert roots_quartic(Poly(x**4 - x**3, x)) in [ [1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1] ] lhs = roots_quartic(Poly(x**4 + x, x)) rhs = [Rational(1, 2) + I*sqrt(3)/2, Rational(1, 2) - I*sqrt(3)/2, 0, -1] assert sorted(lhs, key=hash) == sorted(rhs, key=hash) # test of all branches of roots quartic for i, (a, b, c, d) in enumerate([(1, 2, 3, 0), (3, -7, -9, 9), (1, 2, 3, 4), (1, 2, 3, 4), (-7, -3, 3, -6), (-3, 5, -6, -4), (6, -5, -10, -3)]): if i == 2: c = -a*(a**2/Integer(8) - b/Integer(2)) elif i == 3: d = a*(a*(3*a**2/Integer(256) - b/Integer(16)) + c/Integer(4)) eq = x**4 + a*x**3 + b*x**2 + c*x + d ans = roots_quartic(Poly(eq, x)) assert all(eq.subs({x: ai}).evalf(chop=True) == 0 for ai in ans) # not all symbolic quartics are unresolvable eq = Poly(q*x + q/4 + x**4 + x**3 + 2*x**2 - Rational(1, 3), x) sol = roots_quartic(eq) assert all(verify_numerically(eq.subs({x: i}), 0) for i in sol) z = symbols('z', negative=True) eq = x**4 + 2*x**3 + 3*x**2 + x*(z + 11) + 5 zans = roots_quartic(Poly(eq, x)) assert all(verify_numerically(eq.subs({x: i, z: -1}), 0) for i in zans) # but some are (see also issue sympy/sympy#4989) # it's ok if the solution is not Piecewise, but the tests below should pass eq = Poly(y*x**4 + x**3 - x + z, x) ans = roots_quartic(eq) assert all(type(i) == Piecewise for i in ans) reps = ({y: -Rational(1, 3), z: -Rational(1, 4)}, # 4 real {y: -Rational(1, 3), z: -Rational(1, 2)}, # 2 real {y: -Rational(1, 3), z: -2}) # 0 real for rep in reps: sol = roots_quartic(Poly(eq.subs(rep), x)) assert all(verify_numerically(w.subs(rep) - s, 0) for w, s in zip(ans, sol))
def t(fac, arg): g = meijerg([a], [b], [c], [d], arg)*fac subs = {a: randcplx()/10, b: randcplx()/10 + I, c: randcplx(), d: randcplx()} integral = meijerint_indefinite(g, x) assert integral is not None assert verify_numerically(g.subs(subs), integral.diff(x).subs(subs), x)
def mytn(expr1, expr2, expr3, x, d=0): subs = {} for a in expr1.free_symbols: if a != x: subs[a] = random_complex_number() return expr2 == expr3 and verify_numerically(expr1.subs(subs), expr2.subs(subs), x, d=d)
def test_TR9(): a = Rational(1, 2) b = 3*a assert TR9(a) == a assert TR9(cos(1) + cos(2)) == 2*cos(a)*cos(b) assert TR9(cos(1) - cos(2)) == 2*sin(a)*sin(b) assert TR9(sin(1) - sin(2)) == -2*sin(a)*cos(b) assert TR9(sin(1) + sin(2)) == 2*sin(b)*cos(a) assert TR9(cos(1) + 2*sin(1) + 2*sin(2)) == cos(1) + 4*sin(b)*cos(a) assert TR9(cos(4) + cos(2) + 2*cos(1)*cos(3)) == 4*cos(1)*cos(3) assert TR9((cos(4) + cos(2))/cos(3)/2 + cos(3)) == 2*cos(1)*cos(2) assert TR9(cos(3) + cos(4) + cos(5) + cos(6)) == \ 4*cos(Rational(1, 2))*cos(1)*cos(Rational(9, 2)) assert TR9(cos(3) + cos(3)*cos(2)) == cos(3) + cos(2)*cos(3) assert TR9(-cos(y) + cos(x*y)) == -2*sin(x*y/2 - y/2)*sin(x*y/2 + y/2) assert TR9(-sin(y) + sin(x*y)) == 2*sin(x*y/2 - y/2)*cos(x*y/2 + y/2) c = cos(x) s = sin(x) for si in ((1, 1), (1, -1), (-1, 1), (-1, -1)): for a in ((c, s), (s, c), (cos(x), cos(x*y)), (sin(x), sin(x*y))): args = zip(si, a) ex = Add(*[Mul(*ai) for ai in args]) t = TR9(ex) assert not (a[0].func == a[1].func and ( not verify_numerically(ex, t.expand(trig=True)) or t.is_Add) or a[1].func != a[0].func and ex != t)
def test_TR3(): assert TR3(cos(y - x * (y - x))) == cos(x * (x - y) + y) assert cos(pi / 2 + x) == -sin(x) assert cos(30 * pi / 2 + x) == -cos(x) for f in (cos, sin, tan, cot, csc, sec): i = f(3 * pi / 7) j = TR3(i) assert verify_numerically(i, j) and i.func != j.func
def test_TR3(): assert TR3(cos(y - x*(y - x))) == cos(x*(x - y) + y) assert cos(pi/2 + x) == -sin(x) assert cos(30*pi/2 + x) == -cos(x) for f in (cos, sin, tan, cot, csc, sec): i = f(3*pi/7) j = TR3(i) assert verify_numerically(i, j) and i.func != j.func
def t(a, b, arg, n): m1 = meijerg(a, b, arg) m2 = Mul(*_inflate_g(m1, n)) # NOTE: (the random number)**9 must still be on the principal sheet. # Thus make b&d small to create random numbers of small imaginary part. return verify_numerically(m1.subs(subs), m2.subs(subs), x, b=0.1, d=-0.1)
def t(fac, arg): g = meijerg([a], [b], [c], [d], arg) * fac subs = { a: randcplx() / 10, b: randcplx() / 10 + I, c: randcplx(), d: randcplx() } integral = meijerint_indefinite(g, x) assert integral is not None assert verify_numerically(g.subs(subs), integral.diff(x).subs(subs), x)
def test_reflect(): b = Symbol('b') m = Symbol('m') l = Line((0, b), slope=m) pytest.raises(ValueError, lambda: Line((0, b))) p = Point(x, y) r = p.reflect(l) dp = l.perpendicular_segment(p).length dr = l.perpendicular_segment(r).length assert verify_numerically(dp, dr) t = Triangle((0, 0), (1, 0), (2, 3)) assert t.area == -t.reflect(l).area e = Ellipse((1, 0), 1, 2) assert e.area == -e.reflect(Line((1, 0), slope=0)).area assert e.area == -e.reflect(Line((1, 0), slope=oo)).area pytest.raises(NotImplementedError, lambda: e.reflect(Line( (1, 0), slope=m))) pytest.raises( NotImplementedError, lambda: Ellipse(Point(0, 0), x, 1).reflect( Line(Point(0, 1), Point(1, 0)))) assert Polygon((1, 0), (2, 0), (2, 2)).reflect(Line((3, 0), slope=oo)) \ == Triangle(Point(5, 0), Point(4, 0), Point(4, 2)) assert Polygon((1, 0), (2, 0), (2, 2)).reflect(Line((0, 3), slope=oo)) \ == Triangle(Point(-1, 0), Point(-2, 0), Point(-2, 2)) assert Polygon((1, 0), (2, 0), (2, 2)).reflect(Line((0, 3), slope=0)) \ == Triangle(Point(1, 6), Point(2, 6), Point(2, 4)) assert Polygon((1, 0), (2, 0), (2, 2)).reflect(Line((3, 0), slope=0)) \ == Triangle(Point(1, 0), Point(2, 0), Point(2, -2)) # test entity overrides c = Circle((x, y), 3) cr = c.reflect(l) assert cr == Circle(r, -3) assert c.area == -cr.area pent = RegularPolygon((1, 2), 1, 5) l = Line((0, pi), slope=sqrt(2)) rpent = pent.reflect(l) assert rpent.center == pent.center.reflect(l) assert [w.evalf(3) for w in rpent.vertices] == \ [Point(Float('-0.585815', dps=3), Float('4.27051', dps=3)), Point(Float('-1.69409', dps=3), Float('4.66211', dps=3)), Point(Float('-2.40918', dps=3), Float('3.72949', dps=3)), Point(Float('-1.74292', dps=3), Float('2.76123', dps=3)), Point(Float('-0.615967', dps=3), Float('3.0957', dps=3))] assert pent.area.equals(-rpent.area)
def test_sympyissue_6828(): f = 1/(1.08*x**2 - 4.3) g = integrate(f, x).diff(x) assert verify_numerically(f, g, tol=1e-12)
def test_fresnel(): assert fresnels(0) == 0 assert fresnels(+oo) == Rational(+1, 2) assert fresnels(-oo) == Rational(-1, 2) assert fresnels(z) == fresnels(z) assert fresnels(-z) == -fresnels(z) assert fresnels(I*z) == -I*fresnels(z) assert fresnels(-I*z) == I*fresnels(z) assert conjugate(fresnels(z)) == fresnels(conjugate(z)) assert fresnels(z).diff(z) == sin(pi*z**2/2) assert fresnels(z).rewrite(erf) == (1 + I)/4 * ( erf((1 + I)/2*sqrt(pi)*z) - I*erf((1 - I)/2*sqrt(pi)*z)) assert fresnels(z).rewrite(hyper) == \ pi*z**3/6 * hyper([Rational(3, 4)], [Rational(3, 2), Rational(7, 4)], -pi**2*z**4/16) assert fresnels(z).series(z, n=15) == \ pi*z**3/6 - pi**3*z**7/336 + pi**5*z**11/42240 + O(z**15) assert fresnels(y/z).limit(z, 0) == fresnels(oo*sign(y)) assert fresnels(x).taylor_term(-1, z) == 0 assert fresnels(x).taylor_term(1, z, *(pi*z**3/6,)) == -pi**3*z**7/336 assert fresnels(x).taylor_term(1, z) == -pi**3*z**7/336 assert fresnels(w).is_extended_real is True assert fresnels(z).is_extended_real is None assert fresnels(z).as_real_imag() == \ ((fresnels(re(z) - I*re(z)*Abs(im(z))/Abs(re(z)))/2 + fresnels(re(z) + I*re(z)*Abs(im(z))/Abs(re(z)))/2, I*(fresnels(re(z) - I*re(z)*Abs(im(z))/Abs(re(z))) - fresnels(re(z) + I*re(z)*Abs(im(z))/Abs(re(z)))) * re(z)*Abs(im(z))/(2*im(z)*Abs(re(z))))) assert fresnels(z).as_real_imag(deep=False) == fresnels(z).as_real_imag() assert fresnels(w).as_real_imag() == (fresnels(w), 0) assert fresnels(w).as_real_imag(deep=False) == fresnels(w).as_real_imag() assert (fresnels(I, evaluate=False).as_real_imag() == (0, -erf(sqrt(pi)/2 + I*sqrt(pi)/2)/4 + I*(-erf(sqrt(pi)/2 + I*sqrt(pi)/2) + erf(sqrt(pi)/2 - I*sqrt(pi)/2))/4 - erf(sqrt(pi)/2 - I*sqrt(pi)/2)/4)) assert fresnels(2 + 3*I).as_real_imag() == ( fresnels(2 + 3*I)/2 + fresnels(2 - 3*I)/2, I*(fresnels(2 - 3*I) - fresnels(2 + 3*I))/2 ) assert expand_func(integrate(fresnels(z), z)) == \ z*fresnels(z) + cos(pi*z**2/2)/pi assert fresnels(z).rewrite(meijerg) == sqrt(2)*pi*z**Rational(9, 4) * \ meijerg(((), (1,)), ((Rational(3, 4),), (Rational(1, 4), 0)), -pi**2*z**4/16)/(2*(-z)**Rational(3, 4)*(z**2)**Rational(3, 4)) assert fresnelc(0) == 0 assert fresnelc(+oo) == Rational(+1, 2) assert fresnelc(-oo) == Rational(-1, 2) assert fresnelc(z) == fresnelc(z) assert fresnelc(-z) == -fresnelc(z) assert fresnelc(I*z) == I*fresnelc(z) assert fresnelc(-I*z) == -I*fresnelc(z) assert conjugate(fresnelc(z)) == fresnelc(conjugate(z)) assert fresnelc(z).diff(z) == cos(pi*z**2/2) pytest.raises(ArgumentIndexError, lambda: fresnels(z).fdiff(2)) pytest.raises(ArgumentIndexError, lambda: fresnelc(z).fdiff(2)) assert fresnelc(z).rewrite(erf) == (1 - I)/4 * ( erf((1 + I)/2*sqrt(pi)*z) + I*erf((1 - I)/2*sqrt(pi)*z)) assert fresnelc(z).rewrite(hyper) == \ z * hyper([Rational(1, 4)], [Rational(1, 2), Rational(5, 4)], -pi**2*z**4/16) assert fresnelc(x).taylor_term(-1, z) == 0 assert fresnelc(x).taylor_term(1, z, *(z,)) == -pi**2*z**5/40 assert fresnelc(x).taylor_term(1, z) == -pi**2*z**5/40 assert fresnelc(z).series(z, n=15) == \ z - pi**2*z**5/40 + pi**4*z**9/3456 - pi**6*z**13/599040 + O(z**15) assert fresnelc(y/z).limit(z, 0) == fresnelc(oo*sign(y)) # issue sympy/sympy#6510 assert fresnels(z).series(z, oo) == \ (-1/(pi**2*z**3) + O(z**(-6), (z, oo)))*sin(pi*z**2/2) + \ (3/(pi**3*z**5) - 1/(pi*z) + O(z**(-6), (z, oo)))*cos(pi*z**2/2) + Rational(1, 2) assert fresnelc(z).series(z, oo) == \ (-1/(pi**2*z**3) + O(z**(-6), (z, oo)))*cos(pi*z**2/2) + \ (-3/(pi**3*z**5) + 1/(pi*z) + O(z**(-6), (z, oo)))*sin(pi*z**2/2) + Rational(1, 2) assert fresnels(1/z).series(z) == \ (-z**3/pi**2 + O(z**6))*sin(pi/(2*z**2)) + (-z/pi + 3*z**5/pi**3 + O(z**6))*cos(pi/(2*z**2)) + Rational(1, 2) assert fresnelc(1/z).series(z) == \ (-z**3/pi**2 + O(z**6))*cos(pi/(2*z**2)) + (z/pi - 3*z**5/pi**3 + O(z**6))*sin(pi/(2*z**2)) + Rational(1, 2) assert fresnelc(w).is_extended_real is True assert fresnelc(z).as_real_imag() == \ ((fresnelc(re(z) - I*re(z)*Abs(im(z))/Abs(re(z)))/2 + fresnelc(re(z) + I*re(z)*Abs(im(z))/Abs(re(z)))/2, I*(fresnelc(re(z) - I*re(z)*Abs(im(z))/Abs(re(z))) - fresnelc(re(z) + I*re(z)*Abs(im(z))/Abs(re(z)))) * re(z)*Abs(im(z))/(2*im(z)*Abs(re(z))))) assert fresnelc(2 + 3*I).as_real_imag() == ( fresnelc(2 - 3*I)/2 + fresnelc(2 + 3*I)/2, I*(fresnelc(2 - 3*I) - fresnelc(2 + 3*I))/2 ) assert expand_func(integrate(fresnelc(z), z)) == \ z*fresnelc(z) - sin(pi*z**2/2)/pi assert fresnelc(z).rewrite(meijerg) == sqrt(2)*pi*z**Rational(3, 4) * \ meijerg(((), (1,)), ((Rational(1, 4),), (Rational(3, 4), 0)), -pi**2*z**4/16)/(2*root(-z, 4)*root(z**2, 4)) verify_numerically(re(fresnels(z)), fresnels(z).as_real_imag()[0], z) verify_numerically(im(fresnels(z)), fresnels(z).as_real_imag()[1], z) verify_numerically(fresnels(z), fresnels(z).rewrite(hyper), z) verify_numerically(fresnels(z), fresnels(z).rewrite(meijerg), z) verify_numerically(re(fresnelc(z)), fresnelc(z).as_real_imag()[0], z) verify_numerically(im(fresnelc(z)), fresnelc(z).as_real_imag()[1], z) verify_numerically(fresnelc(z), fresnelc(z).rewrite(hyper), z) verify_numerically(fresnelc(z), fresnelc(z).rewrite(meijerg), z)
def u(expr, x): r = _rewrite_single(expr, x) e = Add(*[res[0] * res[2] for res in r[0]]).replace(exp_polar, exp) # XXX Hack? assert verify_numerically(e, expr, x)
def test_roots_quartic(): assert roots_quartic((x**4).as_poly()) == [0, 0, 0, 0] assert roots_quartic((x**4 + x**3).as_poly()) in [[-1, 0, 0, 0], [0, -1, 0, 0], [0, 0, -1, 0], [0, 0, 0, -1]] assert roots_quartic( (x**4 - x**3).as_poly()) in [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]] lhs = roots_quartic((x**4 + x).as_poly()) rhs = [ Rational(1, 2) + I * sqrt(3) / 2, Rational(1, 2) - I * sqrt(3) / 2, 0, -1 ] assert sorted(lhs, key=hash) == sorted(rhs, key=hash) # test of all branches of roots quartic for i, (a, b, c, d) in enumerate([(1, 2, 3, 0), (3, -7, -9, 9), (1, 2, 3, 4), (1, 2, 3, 4), (-7, -3, 3, -6), (-3, 5, -6, -4), (6, -5, -10, -3)]): if i == 2: c = -a * (a**2 / Integer(8) - b / Integer(2)) elif i == 3: d = a * (a * (3 * a**2 / Integer(256) - b / Integer(16)) + c / Integer(4)) eq = x**4 + a * x**3 + b * x**2 + c * x + d ans = roots_quartic(eq.as_poly()) assert all(eq.subs({x: ai}).evalf(chop=True) == 0 for ai in ans) # not all symbolic quartics are unresolvable eq = (q * x + q / 4 + x**4 + x**3 + 2 * x**2 - Rational(1, 3)).as_poly(x) sol = roots_quartic(eq) assert all(verify_numerically(eq.subs({x: i}), 0) for i in sol) z = symbols('z', negative=True) eq = x**4 + 2 * x**3 + 3 * x**2 + x * (z + 11) + 5 zans = roots_quartic(eq.as_poly(x)) assert all(verify_numerically(eq.subs({x: i, z: -1}), 0) for i in zans) # but some are (see also issue sympy/sympy#4989) # it's ok if the solution is not Piecewise, but the tests below should pass eq = (y * x**4 + x**3 - x + z).as_poly(x) ans = roots_quartic(eq) assert all(type(i) == Piecewise for i in ans) reps = ( { y: -Rational(1, 3), z: -Rational(1, 4) }, # 4 real { y: -Rational(1, 3), z: -Rational(1, 2) }, # 2 real { y: -Rational(1, 3), z: -2 }) # 0 real for rep in reps: sol = roots_quartic(eq.subs(rep).as_poly(x)) assert all( verify_numerically(w.subs(rep) - s, 0) for w, s in zip(ans, sol))
def test_sympyissue_6828(): f = 1 / (1.08 * x**2 - 4.3) g = integrate(f, x).diff(x) assert verify_numerically(f, g, tol=1e-12)
def ok(a, b, n): e = (a + I * b)**n return verify_numerically(e, expand_multinomial(e))
def test_fresnel(): assert fresnels(0) == 0 assert fresnels(oo) == S.Half assert fresnels(-oo) == -S.Half assert fresnels(z) == fresnels(z) assert fresnels(-z) == -fresnels(z) assert fresnels(I*z) == -I*fresnels(z) assert fresnels(-I*z) == I*fresnels(z) assert conjugate(fresnels(z)) == fresnels(conjugate(z)) assert fresnels(z).diff(z) == sin(pi*z**2/2) assert fresnels(z).rewrite(erf) == (S.One + I)/4 * ( erf((S.One + I)/2*sqrt(pi)*z) - I*erf((S.One - I)/2*sqrt(pi)*z)) assert fresnels(z).rewrite(hyper) == \ pi*z**3/6 * hyper([Rational(3, 4)], [Rational(3, 2), Rational(7, 4)], -pi**2*z**4/16) assert fresnels(z).series(z, n=15) == \ pi*z**3/6 - pi**3*z**7/336 + pi**5*z**11/42240 + O(z**15) assert fresnels(w).is_extended_real is True assert fresnels(z).is_extended_real is None assert fresnels(z).as_real_imag() == \ ((fresnels(re(z) - I*re(z)*Abs(im(z))/Abs(re(z)))/2 + fresnels(re(z) + I*re(z)*Abs(im(z))/Abs(re(z)))/2, I*(fresnels(re(z) - I*re(z)*Abs(im(z))/Abs(re(z))) - fresnels(re(z) + I*re(z)*Abs(im(z))/Abs(re(z)))) * re(z)*Abs(im(z))/(2*im(z)*Abs(re(z))))) assert fresnels(2 + 3*I).as_real_imag() == ( fresnels(2 + 3*I)/2 + fresnels(2 - 3*I)/2, I*(fresnels(2 - 3*I) - fresnels(2 + 3*I))/2 ) assert expand_func(integrate(fresnels(z), z)) == \ z*fresnels(z) + cos(pi*z**2/2)/pi assert fresnels(z).rewrite(meijerg) == sqrt(2)*pi*z**Rational(9, 4) * \ meijerg(((), (1,)), ((Rational(3, 4),), (Rational(1, 4), 0)), -pi**2*z**4/16)/(2*(-z)**Rational(3, 4)*(z**2)**Rational(3, 4)) assert fresnelc(0) == 0 assert fresnelc(oo) == S.Half assert fresnelc(-oo) == -S.Half assert fresnelc(z) == fresnelc(z) assert fresnelc(-z) == -fresnelc(z) assert fresnelc(I*z) == I*fresnelc(z) assert fresnelc(-I*z) == -I*fresnelc(z) assert conjugate(fresnelc(z)) == fresnelc(conjugate(z)) assert fresnelc(z).diff(z) == cos(pi*z**2/2) assert fresnelc(z).rewrite(erf) == (S.One - I)/4 * ( erf((S.One + I)/2*sqrt(pi)*z) + I*erf((S.One - I)/2*sqrt(pi)*z)) assert fresnelc(z).rewrite(hyper) == \ z * hyper([S.One/4], [S.One/2, Rational(5, 4)], -pi**2*z**4/16) assert fresnelc(z).series(z, n=15) == \ z - pi**2*z**5/40 + pi**4*z**9/3456 - pi**6*z**13/599040 + O(z**15) # issue 6510 assert fresnels(z).series(z, S.Infinity) == \ (-1/(pi**2*z**3) + O(z**(-6), (z, oo)))*sin(pi*z**2/2) + \ (3/(pi**3*z**5) - 1/(pi*z) + O(z**(-6), (z, oo)))*cos(pi*z**2/2) + S.Half assert fresnelc(z).series(z, S.Infinity) == \ (-1/(pi**2*z**3) + O(z**(-6), (z, oo)))*cos(pi*z**2/2) + \ (-3/(pi**3*z**5) + 1/(pi*z) + O(z**(-6), (z, oo)))*sin(pi*z**2/2) + S.Half assert fresnels(1/z).series(z) == \ (-z**3/pi**2 + O(z**6))*sin(pi/(2*z**2)) + (-z/pi + 3*z**5/pi**3 + O(z**6))*cos(pi/(2*z**2)) + S.Half assert fresnelc(1/z).series(z) == \ (-z**3/pi**2 + O(z**6))*cos(pi/(2*z**2)) + (z/pi - 3*z**5/pi**3 + O(z**6))*sin(pi/(2*z**2)) + S.Half assert fresnelc(w).is_extended_real is True assert fresnelc(z).as_real_imag() == \ ((fresnelc(re(z) - I*re(z)*Abs(im(z))/Abs(re(z)))/2 + fresnelc(re(z) + I*re(z)*Abs(im(z))/Abs(re(z)))/2, I*(fresnelc(re(z) - I*re(z)*Abs(im(z))/Abs(re(z))) - fresnelc(re(z) + I*re(z)*Abs(im(z))/Abs(re(z)))) * re(z)*Abs(im(z))/(2*im(z)*Abs(re(z))))) assert fresnelc(2 + 3*I).as_real_imag() == ( fresnelc(2 - 3*I)/2 + fresnelc(2 + 3*I)/2, I*(fresnelc(2 - 3*I) - fresnelc(2 + 3*I))/2 ) assert expand_func(integrate(fresnelc(z), z)) == \ z*fresnelc(z) - sin(pi*z**2/2)/pi assert fresnelc(z).rewrite(meijerg) == sqrt(2)*pi*z**Rational(3, 4) * \ meijerg(((), (1,)), ((Rational(1, 4),), (Rational(3, 4), 0)), -pi**2*z**4/16)/(2*(-z)**Rational(1, 4)*(z**2)**Rational(1, 4)) from diofant.utilities.randtest import verify_numerically verify_numerically(re(fresnels(z)), fresnels(z).as_real_imag()[0], z) verify_numerically(im(fresnels(z)), fresnels(z).as_real_imag()[1], z) verify_numerically(fresnels(z), fresnels(z).rewrite(hyper), z) verify_numerically(fresnels(z), fresnels(z).rewrite(meijerg), z) verify_numerically(re(fresnelc(z)), fresnelc(z).as_real_imag()[0], z) verify_numerically(im(fresnelc(z)), fresnelc(z).as_real_imag()[1], z) verify_numerically(fresnelc(z), fresnelc(z).rewrite(hyper), z) verify_numerically(fresnelc(z), fresnelc(z).rewrite(meijerg), z)
def ok(a, b, n): e = (a + I*b)**n return verify_numerically(e, expand_multinomial(e))
def test_fresnel(): assert fresnels(0) == 0 assert fresnels(+oo) == Rational(+1, 2) assert fresnels(-oo) == Rational(-1, 2) assert fresnels(z) == fresnels(z) assert fresnels(-z) == -fresnels(z) assert fresnels(I * z) == -I * fresnels(z) assert fresnels(-I * z) == I * fresnels(z) assert conjugate(fresnels(z)) == fresnels(conjugate(z)) assert fresnels(z).diff(z) == sin(pi * z**2 / 2) assert fresnels(z).rewrite(erf) == (1 + I) / 4 * (erf( (1 + I) / 2 * sqrt(pi) * z) - I * erf((1 - I) / 2 * sqrt(pi) * z)) assert fresnels(z).rewrite(hyper) == \ pi*z**3/6 * hyper([Rational(3, 4)], [Rational(3, 2), Rational(7, 4)], -pi**2*z**4/16) assert fresnels(z).series(z, n=15) == \ pi*z**3/6 - pi**3*z**7/336 + pi**5*z**11/42240 + O(z**15) assert fresnels(y / z).limit(z, 0) == fresnels(oo * sign(y)) assert fresnels(x).taylor_term(-1, z) == 0 assert fresnels(x).taylor_term(1, z, *(pi * z**3 / 6, )) == -pi**3 * z**7 / 336 assert fresnels(x).taylor_term(1, z) == -pi**3 * z**7 / 336 assert fresnels(w).is_extended_real is True assert fresnels(z).is_extended_real is None assert fresnels(z).as_real_imag() == \ ((fresnels(re(z) - I*re(z)*abs(im(z))/abs(re(z)))/2 + fresnels(re(z) + I*re(z)*abs(im(z))/abs(re(z)))/2, I*(fresnels(re(z) - I*re(z)*abs(im(z))/abs(re(z))) - fresnels(re(z) + I*re(z)*abs(im(z))/abs(re(z)))) * re(z)*abs(im(z))/(2*im(z)*abs(re(z))))) assert fresnels(z).as_real_imag(deep=False) == fresnels(z).as_real_imag() assert fresnels(w).as_real_imag() == (fresnels(w), 0) assert fresnels(w).as_real_imag(deep=False) == fresnels(w).as_real_imag() assert (fresnels(I, evaluate=False).as_real_imag() == ( 0, -erf(sqrt(pi) / 2 + I * sqrt(pi) / 2) / 4 + I * (-erf(sqrt(pi) / 2 + I * sqrt(pi) / 2) + erf(sqrt(pi) / 2 - I * sqrt(pi) / 2)) / 4 - erf(sqrt(pi) / 2 - I * sqrt(pi) / 2) / 4)) assert fresnels(2 + 3 * I).as_real_imag() == ( fresnels(2 + 3 * I) / 2 + fresnels(2 - 3 * I) / 2, I * (fresnels(2 - 3 * I) - fresnels(2 + 3 * I)) / 2) assert expand_func(integrate(fresnels(z), z)) == \ z*fresnels(z) + cos(pi*z**2/2)/pi assert fresnels(z).rewrite(meijerg) == sqrt(2)*pi*z**Rational(9, 4) * \ meijerg(((), (1,)), ((Rational(3, 4),), (Rational(1, 4), 0)), -pi**2*z**4/16)/(2*(-z)**Rational(3, 4)*(z**2)**Rational(3, 4)) assert fresnelc(0) == 0 assert fresnelc(+oo) == Rational(+1, 2) assert fresnelc(-oo) == Rational(-1, 2) assert fresnelc(z) == fresnelc(z) assert fresnelc(-z) == -fresnelc(z) assert fresnelc(I * z) == I * fresnelc(z) assert fresnelc(-I * z) == -I * fresnelc(z) assert conjugate(fresnelc(z)) == fresnelc(conjugate(z)) assert fresnelc(z).diff(z) == cos(pi * z**2 / 2) pytest.raises(ArgumentIndexError, lambda: fresnels(z).fdiff(2)) pytest.raises(ArgumentIndexError, lambda: fresnelc(z).fdiff(2)) assert fresnelc(z).rewrite(erf) == (1 - I) / 4 * (erf( (1 + I) / 2 * sqrt(pi) * z) + I * erf((1 - I) / 2 * sqrt(pi) * z)) assert fresnelc(z).rewrite(hyper) == \ z * hyper([Rational(1, 4)], [Rational(1, 2), Rational(5, 4)], -pi**2*z**4/16) assert fresnelc(x).taylor_term(-1, z) == 0 assert fresnelc(x).taylor_term(1, z, *(z, )) == -pi**2 * z**5 / 40 assert fresnelc(x).taylor_term(1, z) == -pi**2 * z**5 / 40 assert fresnelc(z).series(z, n=15) == \ z - pi**2*z**5/40 + pi**4*z**9/3456 - pi**6*z**13/599040 + O(z**15) assert fresnelc(y / z).limit(z, 0) == fresnelc(oo * sign(y)) # issue sympy/sympy#6510 assert fresnels(z).series(z, oo) == \ (-1/(pi**2*z**3) + O(z**(-6), (z, oo)))*sin(pi*z**2/2) + \ (3/(pi**3*z**5) - 1/(pi*z) + O(z**(-6), (z, oo)))*cos(pi*z**2/2) + Rational(1, 2) assert fresnelc(z).series(z, oo) == \ (-1/(pi**2*z**3) + O(z**(-6), (z, oo)))*cos(pi*z**2/2) + \ (-3/(pi**3*z**5) + 1/(pi*z) + O(z**(-6), (z, oo)))*sin(pi*z**2/2) + Rational(1, 2) assert fresnels(1/z).series(z) == \ (-z**3/pi**2 + O(z**6))*sin(pi/(2*z**2)) + (-z/pi + 3*z**5/pi**3 + O(z**6))*cos(pi/(2*z**2)) + Rational(1, 2) assert fresnelc(1/z).series(z) == \ (-z**3/pi**2 + O(z**6))*cos(pi/(2*z**2)) + (z/pi - 3*z**5/pi**3 + O(z**6))*sin(pi/(2*z**2)) + Rational(1, 2) assert fresnelc(w).is_extended_real is True assert fresnelc(z).as_real_imag() == \ ((fresnelc(re(z) - I*re(z)*abs(im(z))/abs(re(z)))/2 + fresnelc(re(z) + I*re(z)*abs(im(z))/abs(re(z)))/2, I*(fresnelc(re(z) - I*re(z)*abs(im(z))/abs(re(z))) - fresnelc(re(z) + I*re(z)*abs(im(z))/abs(re(z)))) * re(z)*abs(im(z))/(2*im(z)*abs(re(z))))) assert fresnelc(2 + 3 * I).as_real_imag() == ( fresnelc(2 - 3 * I) / 2 + fresnelc(2 + 3 * I) / 2, I * (fresnelc(2 - 3 * I) - fresnelc(2 + 3 * I)) / 2) assert expand_func(integrate(fresnelc(z), z)) == \ z*fresnelc(z) - sin(pi*z**2/2)/pi assert fresnelc(z).rewrite(meijerg) == sqrt(2)*pi*z**Rational(3, 4) * \ meijerg(((), (1,)), ((Rational(1, 4),), (Rational(3, 4), 0)), -pi**2*z**4/16)/(2*root(-z, 4)*root(z**2, 4)) verify_numerically(re(fresnels(z)), fresnels(z).as_real_imag()[0], z) verify_numerically(im(fresnels(z)), fresnels(z).as_real_imag()[1], z) verify_numerically(fresnels(z), fresnels(z).rewrite(hyper), z) verify_numerically(fresnels(z), fresnels(z).rewrite(meijerg), z) verify_numerically(re(fresnelc(z)), fresnelc(z).as_real_imag()[0], z) verify_numerically(im(fresnelc(z)), fresnelc(z).as_real_imag()[1], z) verify_numerically(fresnelc(z), fresnelc(z).rewrite(hyper), z) verify_numerically(fresnelc(z), fresnelc(z).rewrite(meijerg), z)
def u(expr, x): r = _rewrite_single(expr, x) e = Add(*[res[0]*res[2] for res in r[0]]).replace( exp_polar, exp) # XXX Hack? assert verify_numerically(e, expr, x)