def test_asec_is_real(): assert asec(S(1)/2).is_real is False n = Symbol('n', positive=True, integer=True) assert asec(n).is_real is True assert asec(x).is_real is None assert asec(r).is_real is None t = Symbol('t', real=False) assert asec(t).is_real is False
def test_atan_rewrite(): assert atan(x).rewrite(log) == I*log((1 - I*x)/(1 + I*x))/2 assert atan(x).rewrite(asin) == (-asin(1/sqrt(x**2 + 1)) + pi/2)*sqrt(x**2)/x assert atan(x).rewrite(acos) == sqrt(x**2)*acos(1/sqrt(x**2 + 1))/x assert atan(x).rewrite(acot) == acot(1/x) assert atan(x).rewrite(asec) == sqrt(x**2)*asec(sqrt(x**2 + 1))/x assert atan(x).rewrite(acsc) == (-acsc(sqrt(x**2 + 1)) + pi/2)*sqrt(x**2)/x
def test_conv7(): x = Symbol("x") y = Symbol("y") assert sin(x/3) == sin(sympy.Symbol("x") / 3) assert cos(x/3) == cos(sympy.Symbol("x") / 3) assert tan(x/3) == tan(sympy.Symbol("x") / 3) assert cot(x/3) == cot(sympy.Symbol("x") / 3) assert csc(x/3) == csc(sympy.Symbol("x") / 3) assert sec(x/3) == sec(sympy.Symbol("x") / 3) assert asin(x/3) == asin(sympy.Symbol("x") / 3) assert acos(x/3) == acos(sympy.Symbol("x") / 3) assert atan(x/3) == atan(sympy.Symbol("x") / 3) assert acot(x/3) == acot(sympy.Symbol("x") / 3) assert acsc(x/3) == acsc(sympy.Symbol("x") / 3) assert asec(x/3) == asec(sympy.Symbol("x") / 3) assert sin(x/3)._sympy_() == sympy.sin(sympy.Symbol("x") / 3) assert sin(x/3)._sympy_() != sympy.cos(sympy.Symbol("x") / 3) assert cos(x/3)._sympy_() == sympy.cos(sympy.Symbol("x") / 3) assert tan(x/3)._sympy_() == sympy.tan(sympy.Symbol("x") / 3) assert cot(x/3)._sympy_() == sympy.cot(sympy.Symbol("x") / 3) assert csc(x/3)._sympy_() == sympy.csc(sympy.Symbol("x") / 3) assert sec(x/3)._sympy_() == sympy.sec(sympy.Symbol("x") / 3) assert asin(x/3)._sympy_() == sympy.asin(sympy.Symbol("x") / 3) assert acos(x/3)._sympy_() == sympy.acos(sympy.Symbol("x") / 3) assert atan(x/3)._sympy_() == sympy.atan(sympy.Symbol("x") / 3) assert acot(x/3)._sympy_() == sympy.acot(sympy.Symbol("x") / 3) assert acsc(x/3)._sympy_() == sympy.acsc(sympy.Symbol("x") / 3) assert asec(x/3)._sympy_() == sympy.asec(sympy.Symbol("x") / 3)
def test_acot_rewrite(): assert acot(x).rewrite(log) == I*log((x - I)/(x + I))/2 assert acot(x).rewrite(asin) == x*(-asin(sqrt(-x**2)/sqrt(-x**2 - 1)) + pi/2)*sqrt(x**(-2)) assert acot(x).rewrite(acos) == x*sqrt(x**(-2))*acos(sqrt(-x**2)/sqrt(-x**2 - 1)) assert acot(x).rewrite(atan) == atan(1/x) assert acot(x).rewrite(asec) == x*sqrt(x**(-2))*asec(sqrt((x**2 + 1)/x**2)) assert acot(x).rewrite(acsc) == x*(-acsc(sqrt((x**2 + 1)/x**2)) + pi/2)*sqrt(x**(-2))
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) assert asin(x).rewrite(acot) == 2*acot((sqrt(-x**2 + 1) + 1)/x) assert asin(x).rewrite(asec) == -asec(1/x) + pi/2 assert asin(x).rewrite(acsc) == acsc(1/x)
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) assert acos(x).rewrite(acot) == -2*acot((sqrt(-x**2 + 1) + 1)/x) + pi/2 assert acos(x).rewrite(asec) == asec(1/x) assert acos(x).rewrite(acsc) == -acsc(1/x) + pi/2
def test_branch_cuts(): assert limit(asin(I*x + 2), x, 0) == pi - asin(2) assert limit(asin(I*x + 2), x, 0, '-') == asin(2) assert limit(asin(I*x - 2), x, 0) == -asin(2) assert limit(asin(I*x - 2), x, 0, '-') == -pi + asin(2) assert limit(acos(I*x + 2), x, 0) == -acos(2) assert limit(acos(I*x + 2), x, 0, '-') == acos(2) assert limit(acos(I*x - 2), x, 0) == acos(-2) assert limit(acos(I*x - 2), x, 0, '-') == 2*pi - acos(-2) assert limit(atan(x + 2*I), x, 0) == I*atanh(2) assert limit(atan(x + 2*I), x, 0, '-') == -pi + I*atanh(2) assert limit(atan(x - 2*I), x, 0) == pi - I*atanh(2) assert limit(atan(x - 2*I), x, 0, '-') == -I*atanh(2) assert limit(atan(1/x), x, 0) == pi/2 assert limit(atan(1/x), x, 0, '-') == -pi/2 assert limit(atan(x), x, oo) == pi/2 assert limit(atan(x), x, -oo) == -pi/2 assert limit(acot(x + S(1)/2*I), x, 0) == pi - I*acoth(S(1)/2) assert limit(acot(x + S(1)/2*I), x, 0, '-') == -I*acoth(S(1)/2) assert limit(acot(x - S(1)/2*I), x, 0) == I*acoth(S(1)/2) assert limit(acot(x - S(1)/2*I), x, 0, '-') == -pi + I*acoth(S(1)/2) assert limit(acot(x), x, 0) == pi/2 assert limit(acot(x), x, 0, '-') == -pi/2 assert limit(asec(I*x + S(1)/2), x, 0) == asec(S(1)/2) assert limit(asec(I*x + S(1)/2), x, 0, '-') == -asec(S(1)/2) assert limit(asec(I*x - S(1)/2), x, 0) == 2*pi - asec(-S(1)/2) assert limit(asec(I*x - S(1)/2), x, 0, '-') == asec(-S(1)/2) assert limit(acsc(I*x + S(1)/2), x, 0) == acsc(S(1)/2) assert limit(acsc(I*x + S(1)/2), x, 0, '-') == pi - acsc(S(1)/2) assert limit(acsc(I*x - S(1)/2), x, 0) == -pi + acsc(S(1)/2) assert limit(acsc(I*x - S(1)/2), x, 0, '-') == -acsc(S(1)/2) assert limit(log(I*x - 1), x, 0) == I*pi assert limit(log(I*x - 1), x, 0, '-') == -I*pi assert limit(log(-I*x - 1), x, 0) == -I*pi assert limit(log(-I*x - 1), x, 0, '-') == I*pi assert limit(sqrt(I*x - 1), x, 0) == I assert limit(sqrt(I*x - 1), x, 0, '-') == -I assert limit(sqrt(-I*x - 1), x, 0) == -I assert limit(sqrt(-I*x - 1), x, 0, '-') == I assert limit(cbrt(I*x - 1), x, 0) == (-1)**(S(1)/3) assert limit(cbrt(I*x - 1), x, 0, '-') == -(-1)**(S(2)/3) assert limit(cbrt(-I*x - 1), x, 0) == -(-1)**(S(2)/3) assert limit(cbrt(-I*x - 1), x, 0, '-') == (-1)**(S(1)/3)
def test_conv7b(): x = sympy.Symbol("x") y = sympy.Symbol("y") assert sympify(sympy.sin(x/3)) == sin(Symbol("x") / 3) assert sympify(sympy.sin(x/3)) != cos(Symbol("x") / 3) assert sympify(sympy.cos(x/3)) == cos(Symbol("x") / 3) assert sympify(sympy.tan(x/3)) == tan(Symbol("x") / 3) assert sympify(sympy.cot(x/3)) == cot(Symbol("x") / 3) assert sympify(sympy.csc(x/3)) == csc(Symbol("x") / 3) assert sympify(sympy.sec(x/3)) == sec(Symbol("x") / 3) assert sympify(sympy.asin(x/3)) == asin(Symbol("x") / 3) assert sympify(sympy.acos(x/3)) == acos(Symbol("x") / 3) assert sympify(sympy.atan(x/3)) == atan(Symbol("x") / 3) assert sympify(sympy.acot(x/3)) == acot(Symbol("x") / 3) assert sympify(sympy.acsc(x/3)) == acsc(Symbol("x") / 3) assert sympify(sympy.asec(x/3)) == asec(Symbol("x") / 3)
def test_conv7b(): x = sympy.Symbol("x") y = sympy.Symbol("y") assert sympify(sympy.sin(x / 3)) == sin(Symbol("x") / 3) assert sympify(sympy.sin(x / 3)) != cos(Symbol("x") / 3) assert sympify(sympy.cos(x / 3)) == cos(Symbol("x") / 3) assert sympify(sympy.tan(x / 3)) == tan(Symbol("x") / 3) assert sympify(sympy.cot(x / 3)) == cot(Symbol("x") / 3) assert sympify(sympy.csc(x / 3)) == csc(Symbol("x") / 3) assert sympify(sympy.sec(x / 3)) == sec(Symbol("x") / 3) assert sympify(sympy.asin(x / 3)) == asin(Symbol("x") / 3) assert sympify(sympy.acos(x / 3)) == acos(Symbol("x") / 3) assert sympify(sympy.atan(x / 3)) == atan(Symbol("x") / 3) assert sympify(sympy.acot(x / 3)) == acot(Symbol("x") / 3) assert sympify(sympy.acsc(x / 3)) == acsc(Symbol("x") / 3) assert sympify(sympy.asec(x / 3)) == asec(Symbol("x") / 3)
def test_acsc(): assert acsc(nan) == nan assert acsc(1) == pi/2 assert acsc(-1) == -pi/2 assert acsc(oo) == 0 assert acsc(-oo) == 0 assert acsc(zoo) == 0 assert acsc(x).diff(x) == -1/(x**2*sqrt(1 - 1/x**2)) assert acsc(x).as_leading_term(x) == log(x) assert acsc(x).rewrite(log) == -I*log(sqrt(1 - 1/x**2) + I/x) assert acsc(x).rewrite(asin) == asin(1/x) assert acsc(x).rewrite(acos) == -acos(1/x) + pi/2 assert acsc(x).rewrite(atan) == (-atan(sqrt(x**2 - 1)) + pi/2)*sqrt(x**2)/x assert acsc(x).rewrite(acot) == (-acot(1/sqrt(x**2 - 1)) + pi/2)*sqrt(x**2)/x assert acsc(x).rewrite(asec) == -asec(x) + pi/2
def test_sec(): x = symbols('x', real=True) z = symbols('z') assert sec.nargs == FiniteSet(1) assert sec(0) == 1 assert sec(pi) == -1 assert sec(pi/2) == zoo assert sec(-pi/2) == zoo assert sec(pi/6) == 2*sqrt(3)/3 assert sec(pi/3) == 2 assert sec(5*pi/2) == zoo assert sec(9*pi/7) == -sec(2*pi/7) assert sec(3*pi/4) == -sqrt(2) # issue 8421 assert sec(I) == 1/cosh(1) assert sec(x*I) == 1/cosh(x) assert sec(-x) == sec(x) assert sec(asec(x)) == x assert sec(x).rewrite(exp) == 1/(exp(I*x)/2 + exp(-I*x)/2) assert sec(x).rewrite(sin) == sec(x) assert sec(x).rewrite(cos) == 1/cos(x) assert sec(x).rewrite(tan) == (tan(x/2)**2 + 1)/(-tan(x/2)**2 + 1) assert sec(x).rewrite(pow) == sec(x) assert sec(x).rewrite(sqrt) == sec(x) assert sec(z).rewrite(cot) == (cot(z/2)**2 + 1)/(cot(z/2)**2 - 1) assert sec(z).conjugate() == sec(conjugate(z)) assert (sec(z).as_real_imag() == (cos(re(z))*cosh(im(z))/(sin(re(z))**2*sinh(im(z))**2 + cos(re(z))**2*cosh(im(z))**2), sin(re(z))*sinh(im(z))/(sin(re(z))**2*sinh(im(z))**2 + cos(re(z))**2*cosh(im(z))**2))) assert sec(x).expand(trig=True) == 1/cos(x) assert sec(2*x).expand(trig=True) == 1/(2*cos(x)**2 - 1) assert sec(x).is_real == True assert sec(z).is_real == None assert sec(a).is_algebraic is None assert sec(na).is_algebraic is False assert sec(x).as_leading_term() == sec(x) assert sec(0).is_finite == True assert sec(x).is_finite == None assert sec(pi/2).is_finite == False assert series(sec(x), x, x0=0, n=6) == 1 + x**2/2 + 5*x**4/24 + O(x**6) # https://github.com/sympy/sympy/issues/7166 assert series(sqrt(sec(x))) == 1 + x**2/4 + 7*x**4/96 + O(x**6) # https://github.com/sympy/sympy/issues/7167 assert (series(sqrt(sec(x)), x, x0=pi*3/2, n=4) == 1/sqrt(x - 3*pi/2) + (x - 3*pi/2)**(S(3)/2)/12 + (x - 3*pi/2)**(S(7)/2)/160 + O((x - 3*pi/2)**4, (x, 3*pi/2))) assert sec(x).diff(x) == tan(x)*sec(x) # Taylor Term checks assert sec(z).taylor_term(4, z) == 5*z**4/24 assert sec(z).taylor_term(6, z) == 61*z**6/720 assert sec(z).taylor_term(5, z) == 0
def test_asec(): z = Symbol('z', zero=True) assert asec(z) == nan assert asec(nan) == nan assert asec(1) == 0 assert asec(-1) == pi assert asec(oo) == pi/2 assert asec(-oo) == pi/2 assert asec(zoo) == pi/2 assert asec(x).diff(x) == 1/(x**2*sqrt(1 - 1/x**2)) assert asec(x).as_leading_term(x) == log(x) assert asec(x).rewrite(log) == I*log(sqrt(1 - 1/x**2) + I/x) + pi/2 assert asec(x).rewrite(asin) == -asin(1/x) + pi/2 assert asec(x).rewrite(acos) == acos(1/x) assert asec(x).rewrite(atan) == (2*atan(x + sqrt(x**2 - 1)) - pi/2)*sqrt(x**2)/x assert asec(x).rewrite(acot) == (2*acot(x - sqrt(x**2 - 1)) - pi/2)*sqrt(x**2)/x assert asec(x).rewrite(acsc) == -acsc(x) + pi/2
def test_sec(): x = symbols('x', real=True) z = symbols('z') assert sec.nargs == FiniteSet(1) assert sec(0) == 1 assert sec(pi) == -1 assert sec(pi / 2) == zoo assert sec(-pi / 2) == zoo assert sec(pi / 6) == 2 * sqrt(3) / 3 assert sec(pi / 3) == 2 assert sec(5 * pi / 2) == zoo assert sec(9 * pi / 7) == -sec(2 * pi / 7) assert sec(3 * pi / 4) == -sqrt(2) # issue 8421 assert sec(I) == 1 / cosh(1) assert sec(x * I) == 1 / cosh(x) assert sec(-x) == sec(x) assert sec(asec(x)) == x assert sec(x).rewrite(exp) == 1 / (exp(I * x) / 2 + exp(-I * x) / 2) assert sec(x).rewrite(sin) == sec(x) assert sec(x).rewrite(cos) == 1 / cos(x) assert sec(x).rewrite(tan) == (tan(x / 2)**2 + 1) / (-tan(x / 2)**2 + 1) assert sec(x).rewrite(pow) == sec(x) assert sec(x).rewrite(sqrt) == sec(x) assert sec(z).rewrite(cot) == (cot(z / 2)**2 + 1) / (cot(z / 2)**2 - 1) assert sec(z).conjugate() == sec(conjugate(z)) assert (sec(z).as_real_imag() == ( cos(re(z)) * cosh(im(z)) / (sin(re(z))**2 * sinh(im(z))**2 + cos(re(z))**2 * cosh(im(z))**2), sin(re(z)) * sinh(im(z)) / (sin(re(z))**2 * sinh(im(z))**2 + cos(re(z))**2 * cosh(im(z))**2))) assert sec(x).expand(trig=True) == 1 / cos(x) assert sec(2 * x).expand(trig=True) == 1 / (2 * cos(x)**2 - 1) assert sec(x).is_real == True assert sec(z).is_real == None assert sec(a).is_algebraic is None assert sec(na).is_algebraic is False assert sec(x).as_leading_term() == sec(x) assert sec(0).is_finite == True assert sec(x).is_finite == None assert sec(pi / 2).is_finite == False assert series(sec(x), x, x0=0, n=6) == 1 + x**2 / 2 + 5 * x**4 / 24 + O(x**6) # https://github.com/sympy/sympy/issues/7166 assert series(sqrt(sec(x))) == 1 + x**2 / 4 + 7 * x**4 / 96 + O(x**6) # https://github.com/sympy/sympy/issues/7167 assert (series(sqrt(sec(x)), x, x0=pi * 3 / 2, n=4) == 1 / sqrt(x - 3 * pi / 2) + (x - 3 * pi / 2)**(S(3) / 2) / 12 + (x - 3 * pi / 2)**(S(7) / 2) / 160 + O((x - 3 * pi / 2)**4, (x, 3 * pi / 2))) assert sec(x).diff(x) == tan(x) * sec(x) # Taylor Term checks assert sec(z).taylor_term(4, z) == 5 * z**4 / 24 assert sec(z).taylor_term(6, z) == 61 * z**6 / 720 assert sec(z).taylor_term(5, z) == 0
def test_invert_real(): x = Symbol('x', real=True) y = Symbol('y') n = Symbol('n') def ireal(x, s=S.Reals): return Intersection(s, x) 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) assert invert_real(exp(x), y, x) == (x, ireal(FiniteSet(log(y)))) y = Symbol('y', positive=True) 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, ireal(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, ireal(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(-oo, -3), FiniteSet(-a + b - 3), imageset(Lambda(n, n - a - 3), Interval(0, oo))) h2 = Intersection(Interval(-3, oo), FiniteSet(a - b - 3), imageset(Lambda(n, -n + a - 3), Interval(0, oo))) assert invert_real(Abs(Abs(x + 3) - a) - b, 0, x) == (x, Union(h1, h2))
sp.rad(num) # Convertir ángulos de grados a radianes. # Funciones para trigonometría (Angulos en radianes) sp.sin(num) # Seno sp.cos(num) # Coseno sp.tan(num) # tangente sp.cot(num) # cotangente sp.sec(num) # secante sp.csc(num) # cosecante sp.asin(num) # Arcoseno sp.acos(num) # Arcocoseno sp.atan(num) # Arcotangente sp.atan2(catetoY, catetoX) # Arcotangente de un triangulo segun los catetos (Angulo) sp.acot(num) # Arcocotangente sp.asec(num) # Arcosecante sp.acsc(num) # Arcocosecante # Funciones hiperbólicas (Angulos en radianes) sp.sinh(num) # Seno sp.cosh(num) # Coseno sp.tanh(num) # tangente sp.coth(num) # cotangente sp.sech(num) # secante sp.csch(num) # cosecante sp.asinh(num) # Arcoseno sp.acosh(num) # Arcocoseno sp.atanh(num) # Arcotangente sp.acoth(num) # Arcocotangente sp.asech(num) # Arcosecante sp.acsch(num) # Arcocosecante
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_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)) # From https://www.wikiwand.com/en/List_of_integrals_of_inverse_trigonometric_functions # asin assert manualintegrate(asin(x), x) == x * asin(x) + sqrt(1 - x**2) assert manualintegrate(asin(a * x), x) == Piecewise( ((a * x * asin(a * x) + sqrt(-a**2 * x**2 + 1)) / a, Ne(a, 0)), (0, True)) assert manualintegrate(x * asin(a * x), x) == -a * Integral( x**2 / sqrt(-a**2 * x**2 + 1), x) / 2 + x**2 * asin(a * x) / 2 # acos assert manualintegrate(acos(x), x) == x * acos(x) - sqrt(1 - x**2) assert manualintegrate(acos(a * x), x) == Piecewise( ((a * x * acos(a * x) - sqrt(-a**2 * x**2 + 1)) / a, Ne(a, 0)), (pi * x / 2, True)) assert manualintegrate(x * acos(a * x), x) == a * Integral( x**2 / sqrt(-a**2 * x**2 + 1), x) / 2 + x**2 * acos(a * x) / 2 # atan assert manualintegrate(atan(x), x) == x * atan(x) - log(x**2 + 1) / 2 assert manualintegrate(atan(a * x), x) == Piecewise( ((a * x * atan(a * x) - log(a**2 * x**2 + 1) / 2) / a, Ne(a, 0)), (0, True)) assert manualintegrate( x * atan(a * x), x) == -a * (x / a**2 - atan(x / sqrt(a**(-2))) / (a**4 * sqrt(a**(-2)))) / 2 + x**2 * atan(a * x) / 2 # acsc assert manualintegrate( acsc(x), x) == x * acsc(x) + Integral(1 / (x * sqrt(1 - 1 / x**2)), x) assert manualintegrate( acsc(a * x), x) == x * acsc(a * x) + Integral(1 / (x * sqrt(1 - 1 / (a**2 * x**2))), x) / a assert manualintegrate(x * acsc(a * x), x) == x**2 * acsc(a * x) / 2 + Integral( 1 / sqrt(1 - 1 / (a**2 * x**2)), x) / (2 * a) # asec assert manualintegrate( asec(x), x) == x * asec(x) - Integral(1 / (x * sqrt(1 - 1 / x**2)), x) assert manualintegrate( asec(a * x), x) == x * asec(a * x) - Integral(1 / (x * sqrt(1 - 1 / (a**2 * x**2))), x) / a assert manualintegrate(x * asec(a * x), x) == x**2 * asec(a * x) / 2 - Integral( 1 / sqrt(1 - 1 / (a**2 * x**2)), x) / (2 * a) # acot assert manualintegrate(acot(x), x) == x * acot(x) + log(x**2 + 1) / 2 assert manualintegrate(acot(a * x), x) == Piecewise( ((a * x * acot(a * x) + log(a**2 * x**2 + 1) / 2) / a, Ne(a, 0)), (pi * x / 2, True)) assert manualintegrate( x * acot(a * x), x) == a * (x / a**2 - atan(x / sqrt(a**(-2))) / (a**4 * sqrt(a**(-2)))) / 2 + x**2 * acot(a * x) / 2 # 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)))