def test_hyper_as_trig(): from sympy.simplify.fu import _osborne, _osbornei eq = sinh(x)**2 + cosh(x)**2 t, f = hyper_as_trig(eq) assert f(fu(t)) == cosh(2 * x) e, f = hyper_as_trig(tanh(x + y)) assert f(TR12(e)) == (tanh(x) + tanh(y)) / (tanh(x) * tanh(y) + 1) d = Dummy() assert _osborne(sinh(x), d) == I * sin(x * d) assert _osborne(tanh(x), d) == I * tan(x * d) assert _osborne(coth(x), d) == cot(x * d) / I assert _osborne(cosh(x), d) == cos(x * d) assert _osborne(sech(x), d) == sec(x * d) assert _osborne(csch(x), d) == csc(x * d) / I for func in (sinh, cosh, tanh, coth, sech, csch): h = func(pi) assert _osbornei(_osborne(h, d), d) == h # /!\ the _osborne functions are not meant to work # in the o(i(trig, d), d) direction so we just check # that they work as they are supposed to work assert _osbornei(cos(x * y + z), y) == cosh(x + z * I) assert _osbornei(sin(x * y + z), y) == sinh(x + z * I) / I assert _osbornei(tan(x * y + z), y) == tanh(x + z * I) / I assert _osbornei(cot(x * y + z), y) == coth(x + z * I) * I assert _osbornei(sec(x * y + z), y) == sech(x + z * I) assert _osbornei(csc(x * y + z), y) == csch(x + z * I) * I
def test_simplifications(): x = Symbol('x') assert sinh(asinh(x)) == x assert sinh(acosh(x)) == sqrt(x - 1) * sqrt(x + 1) assert sinh(atanh(x)) == x / sqrt(1 - x**2) assert sinh(acoth(x)) == 1 / (sqrt(x - 1) * sqrt(x + 1)) assert cosh(asinh(x)) == sqrt(1 + x**2) assert cosh(acosh(x)) == x assert cosh(atanh(x)) == 1 / sqrt(1 - x**2) assert cosh(acoth(x)) == x / (sqrt(x - 1) * sqrt(x + 1)) assert tanh(asinh(x)) == x / sqrt(1 + x**2) assert tanh(acosh(x)) == sqrt(x - 1) * sqrt(x + 1) / x assert tanh(atanh(x)) == x assert tanh(acoth(x)) == 1 / x assert coth(asinh(x)) == sqrt(1 + x**2) / x assert coth(acosh(x)) == x / (sqrt(x - 1) * sqrt(x + 1)) assert coth(atanh(x)) == 1 / x assert coth(acoth(x)) == x assert csch(asinh(x)) == 1 / x assert csch(acosh(x)) == 1 / (sqrt(x - 1) * sqrt(x + 1)) assert csch(atanh(x)) == sqrt(1 - x**2) / x assert csch(acoth(x)) == sqrt(x - 1) * sqrt(x + 1) assert sech(asinh(x)) == 1 / sqrt(1 + x**2) assert sech(acosh(x)) == 1 / x assert sech(atanh(x)) == sqrt(1 - x**2) assert sech(acoth(x)) == sqrt(x - 1) * sqrt(x + 1) / x
def test_csch_rewrite(): x = Symbol('x') assert csch(x).rewrite(exp) == 1 / (exp(x)/2 - exp(-x)/2) \ == csch(x).rewrite('tractable') assert csch(x).rewrite(cosh) == I / cosh(x + I * pi / 2) tanh_half = tanh(S.Half * x) assert csch(x).rewrite(tanh) == (1 - tanh_half**2) / (2 * tanh_half) coth_half = coth(S.Half * x) assert csch(x).rewrite(coth) == (coth_half**2 - 1) / (2 * coth_half)
def test_derivs(): x = Symbol('x') assert coth(x).diff(x) == -sinh(x)**(-2) assert sinh(x).diff(x) == cosh(x) assert cosh(x).diff(x) == sinh(x) assert tanh(x).diff(x) == -tanh(x)**2 + 1 assert csch(x).diff(x) == -coth(x) * csch(x) assert sech(x).diff(x) == -tanh(x) * sech(x) assert acoth(x).diff(x) == 1 / (-x**2 + 1) assert asinh(x).diff(x) == 1 / sqrt(x**2 + 1) assert acosh(x).diff(x) == 1 / sqrt(x**2 - 1) assert atanh(x).diff(x) == 1 / (-x**2 + 1) assert asech(x).diff(x) == -1 / (x * sqrt(1 - x**2)) assert acsch(x).diff(x) == -1 / (x**2 * sqrt(1 + x**(-2)))
def test_sign_assumptions(): p = Symbol('p', positive=True) n = Symbol('n', negative=True) assert sinh(n).is_negative is True assert sinh(p).is_positive is True assert cosh(n).is_positive is True assert cosh(p).is_positive is True assert tanh(n).is_negative is True assert tanh(p).is_positive is True assert csch(n).is_negative is True assert csch(p).is_positive is True assert sech(n).is_positive is True assert sech(p).is_positive is True assert coth(n).is_negative is True assert coth(p).is_positive is True
def test_manualintegrate_trigonometry(): assert manualintegrate(sin(x), x) == -cos(x) assert manualintegrate(tan(x), x) == -log(cos(x)) assert manualintegrate(sec(x), x) == log(sec(x) + tan(x)) assert manualintegrate(csc(x), x) == -log(csc(x) + cot(x)) assert manualintegrate(sin(x) * cos(x), x) in [sin(x)**2 / 2, -cos(x)**2 / 2] assert manualintegrate(-sec(x) * tan(x), x) == -sec(x) assert manualintegrate(csc(x) * cot(x), x) == -csc(x) assert manualintegrate(sec(x)**2, x) == tan(x) assert manualintegrate(csc(x)**2, x) == -cot(x) assert manualintegrate(x * sec(x**2), x) == log(tan(x**2) + sec(x**2)) / 2 assert manualintegrate(cos(x) * csc(sin(x)), x) == -log(cot(sin(x)) + csc(sin(x))) assert manualintegrate(cos(3 * x) * sec(x), x) == -x + sin(2 * x) assert manualintegrate(sin(3*x)*sec(x), x) == \ -3*log(cos(x)) + 2*log(cos(x)**2) - 2*cos(x)**2 assert_is_integral_of(sinh(2 * x), cosh(2 * x) / 2) assert_is_integral_of(x * cosh(x**2), sinh(x**2) / 2) assert_is_integral_of(tanh(x), log(cosh(x))) assert_is_integral_of(coth(x), log(sinh(x))) f, F = sech(x), 2 * atan(tanh(x / 2)) assert manualintegrate(f, x) == F assert (F.diff(x) - f).rewrite(exp).simplify() == 0 # todo: equals returns None f, F = csch(x), log(tanh(x / 2)) assert manualintegrate(f, x) == F assert (F.diff(x) - f).rewrite(exp).simplify() == 0
def test_real_assumptions(): z = Symbol('z', real=False) assert sinh(z).is_real is None assert cosh(z).is_real is None assert tanh(z).is_real is None assert sech(z).is_real is None assert csch(z).is_real is None assert coth(z).is_real is None
def f(rv): if not isinstance(rv, TrigonometricFunction): return rv const, x = rv.args[0].as_independent(d, as_Add=True) a = x.xreplace({d: S.One}) + const * I if isinstance(rv, sin): return sinh(a) / I elif isinstance(rv, cos): return cosh(a) elif isinstance(rv, tan): return tanh(a) / I elif isinstance(rv, cot): return coth(a) * I elif isinstance(rv, sec): return sech(a) elif isinstance(rv, csc): return csch(a) * I else: raise NotImplementedError('unhandled %s' % rv.func)
def test_complex(): a, b = symbols('a,b', real=True) z = a + b * I for func in [sinh, cosh, tanh, coth, sech, csch]: assert func(z).conjugate() == func(a - b * I) for deep in [True, False]: assert sinh(z).expand( complex=True, deep=deep) == sinh(a) * cos(b) + I * cosh(a) * sin(b) assert cosh(z).expand( complex=True, deep=deep) == cosh(a) * cos(b) + I * sinh(a) * sin(b) assert tanh(z).expand( complex=True, deep=deep) == sinh(a) * cosh(a) / (cos(b)**2 + sinh( a)**2) + I * sin(b) * cos(b) / (cos(b)**2 + sinh(a)**2) assert coth(z).expand( complex=True, deep=deep) == sinh(a) * cosh(a) / (sin(b)**2 + sinh( a)**2) - I * sin(b) * cos(b) / (sin(b)**2 + sinh(a)**2) assert csch(z).expand(complex=True, deep=deep) == cos(b) * sinh(a) / (sin(b)**2\ *cosh(a)**2 + cos(b)**2 * sinh(a)**2) - I*sin(b) * cosh(a) / (sin(b)**2\ *cosh(a)**2 + cos(b)**2 * sinh(a)**2) assert sech(z).expand(complex=True, deep=deep) == cos(b) * cosh(a) / (sin(b)**2\ *sinh(a)**2 + cos(b)**2 * cosh(a)**2) - I*sin(b) * sinh(a) / (sin(b)**2\ *sinh(a)**2 + cos(b)**2 * cosh(a)**2)
def test_acsch(): x = Symbol('x') assert unchanged(acsch, x) assert acsch(-x) == -acsch(x) # values at fixed points assert acsch(1) == log(1 + sqrt(2)) assert acsch(-1) == -log(1 + sqrt(2)) assert acsch(0) is zoo assert acsch(2) == log((1 + sqrt(5)) / 2) assert acsch(-2) == -log((1 + sqrt(5)) / 2) assert acsch(I) == -I * pi / 2 assert acsch(-I) == I * pi / 2 assert acsch(-I * (sqrt(6) + sqrt(2))) == I * pi / 12 assert acsch(I * (sqrt(2) + sqrt(6))) == -I * pi / 12 assert acsch(-I * (1 + sqrt(5))) == I * pi / 10 assert acsch(I * (1 + sqrt(5))) == -I * pi / 10 assert acsch(-I * 2 / sqrt(2 - sqrt(2))) == I * pi / 8 assert acsch(I * 2 / sqrt(2 - sqrt(2))) == -I * pi / 8 assert acsch(-I * 2) == I * pi / 6 assert acsch(I * 2) == -I * pi / 6 assert acsch(-I * sqrt(2 + 2 / sqrt(5))) == I * pi / 5 assert acsch(I * sqrt(2 + 2 / sqrt(5))) == -I * pi / 5 assert acsch(-I * sqrt(2)) == I * pi / 4 assert acsch(I * sqrt(2)) == -I * pi / 4 assert acsch(-I * (sqrt(5) - 1)) == 3 * I * pi / 10 assert acsch(I * (sqrt(5) - 1)) == -3 * I * pi / 10 assert acsch(-I * 2 / sqrt(3)) == I * pi / 3 assert acsch(I * 2 / sqrt(3)) == -I * pi / 3 assert acsch(-I * 2 / sqrt(2 + sqrt(2))) == 3 * I * pi / 8 assert acsch(I * 2 / sqrt(2 + sqrt(2))) == -3 * I * pi / 8 assert acsch(-I * sqrt(2 - 2 / sqrt(5))) == 2 * I * pi / 5 assert acsch(I * sqrt(2 - 2 / sqrt(5))) == -2 * I * pi / 5 assert acsch(-I * (sqrt(6) - sqrt(2))) == 5 * I * pi / 12 assert acsch(I * (sqrt(6) - sqrt(2))) == -5 * I * pi / 12 assert acsch(nan) is nan # properties # acsch(x) == asinh(1/x) assert acsch(-I * sqrt(2)) == asinh(I / sqrt(2)) assert acsch(-I * 2 / sqrt(3)) == asinh(I * sqrt(3) / 2) # acsch(x) == -I*asin(I/x) assert acsch(-I * sqrt(2)) == -I * asin(-1 / sqrt(2)) assert acsch(-I * 2 / sqrt(3)) == -I * asin(-sqrt(3) / 2) # csch(acsch(x)) / x == 1 assert expand_mul( csch(acsch(-I * (sqrt(6) + sqrt(2)))) / (-I * (sqrt(6) + sqrt(2)))) == 1 assert expand_mul(csch(acsch(I * (1 + sqrt(5)))) / (I * (1 + sqrt(5)))) == 1 assert (csch(acsch(I * sqrt(2 - 2 / sqrt(5)))) / (I * sqrt(2 - 2 / sqrt(5)))).simplify() == 1 assert (csch(acsch(-I * sqrt(2 - 2 / sqrt(5)))) / (-I * sqrt(2 - 2 / sqrt(5)))).simplify() == 1 # numerical evaluation assert str(acsch(5 * I + 1).n(6)) == '0.0391819 - 0.193363*I' assert str(acsch(-5 * I + 1).n(6)) == '0.0391819 + 0.193363*I'
def test_csch_fdiff(): x = Symbol('x') raises(ArgumentIndexError, lambda: csch(x).fdiff(2))
def test_csch_series(): x = Symbol('x') assert csch(x).series(x, 0, 10) == \ 1/ x - x/6 + 7*x**3/360 - 31*x**5/15120 + 127*x**7/604800 \ - 73*x**9/3421440 + O(x**10)
def test_csch(): x, y = symbols('x,y') k = Symbol('k', integer=True) n = Symbol('n', positive=True) assert csch(nan) is nan assert csch(zoo) is nan assert csch(oo) == 0 assert csch(-oo) == 0 assert csch(0) is zoo assert csch(-1) == -csch(1) assert csch(-x) == -csch(x) assert csch(-pi) == -csch(pi) assert csch(-2**1024 * E) == -csch(2**1024 * E) assert csch(pi * I) is zoo assert csch(-pi * I) is zoo assert csch(2 * pi * I) is zoo assert csch(-2 * pi * I) is zoo assert csch(-3 * 10**73 * pi * I) is zoo assert csch(7 * 10**103 * pi * I) is zoo assert csch(pi * I / 2) == -I assert csch(-pi * I / 2) == I assert csch(pi * I * Rational(5, 2)) == -I assert csch(pi * I * Rational(7, 2)) == I assert csch(pi * I / 3) == -2 / sqrt(3) * I assert csch(pi * I * Rational(-2, 3)) == 2 / sqrt(3) * I assert csch(pi * I / 4) == -sqrt(2) * I assert csch(-pi * I / 4) == sqrt(2) * I assert csch(pi * I * Rational(7, 4)) == sqrt(2) * I assert csch(pi * I * Rational(-3, 4)) == sqrt(2) * I assert csch(pi * I / 6) == -2 * I assert csch(-pi * I / 6) == 2 * I assert csch(pi * I * Rational(7, 6)) == 2 * I assert csch(pi * I * Rational(-7, 6)) == -2 * I assert csch(pi * I * Rational(-5, 6)) == 2 * I assert csch(pi * I / 105) == -1 / sin(pi / 105) * I assert csch(-pi * I / 105) == 1 / sin(pi / 105) * I assert csch(x * I) == -1 / sin(x) * I assert csch(k * pi * I) is zoo assert csch(17 * k * pi * I) is zoo assert csch(k * pi * I / 2) == -1 / sin(k * pi / 2) * I assert csch(n).is_real is True assert expand_trig(csch(x + y)) == 1 / (sinh(x) * cosh(y) + cosh(x) * sinh(y))
def test_issue_11254b(): assert not integrate(csch(x), (x, 0, 1)).has(Integral)