def test_simplifications():
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_simplifications():
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_sech_rewrite():
assert sech(x).rewrite(exp) == 1 / (exp(x)/2 + exp(-x)/2) \
== sech(x).rewrite('tractable')
assert sech(x).rewrite(sinh) == I / sinh(x + I * pi / 2)
tanh_half = tanh(S.Half * x)**2
assert sech(x).rewrite(tanh) == (1 - tanh_half) / (1 + tanh_half)
coth_half = coth(S.Half * x)**2
assert sech(x).rewrite(coth) == (coth_half - 1) / (coth_half + 1)
def test_derivs():
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)
def test_derivs():
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)
def test_complex():
a, b = symbols('a,b', extended_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_complex():
a, b = symbols('a,b', extended_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_sech_rewrite():
assert sech(x).rewrite(exp) == 1 / (exp(x)/2 + exp(-x)/2) \
== sech(x).rewrite('tractable')
assert sech(x).rewrite(sinh) == I / sinh(x + I * pi / 2)
assert sech(x).rewrite(tanh) == (1 - tanh(x / 2)**2) / (1 + tanh(x / 2)**2)
assert sech(x).rewrite(coth) == (coth(x / 2)**2 - 1) / (coth(x / 2)**2 + 1)
def test_sech_series():
assert sech(x).series(x, 0, 10) == \
1 - x**2/2 + 5*x**4/24 - 61*x**6/720 + 277*x**8/8064 + O(x**10)
def test_sech():
k = Symbol('k', integer=True)
n = Symbol('n', positive=True)
assert sech(nan) == nan
assert sech(zoo) == nan
assert sech(oo) == 0
assert sech(-oo) == 0
assert sech(0) == 1
assert sech(-1) == sech(1)
assert sech(-x) == sech(x)
assert sech(pi * I) == sec(pi)
assert sech(-pi * I) == sec(pi)
assert sech(-2**1024 * E) == sech(2**1024 * E)
assert sech(pi * I / 2) == zoo
assert sech(-pi * I / 2) == zoo
assert sech((-3 * 10**73 + 1) * pi * I / 2) == zoo
assert sech((7 * 10**103 + 1) * pi * I / 2) == zoo
assert sech(pi * I) == -1
assert sech(-pi * I) == -1
assert sech(5 * pi * I) == -1
assert sech(8 * pi * I) == 1
assert sech(pi * I / 3) == 2
assert sech(-2 * pi * I / 3) == -2
assert sech(pi * I / 4) == sqrt(2)
assert sech(-pi * I / 4) == sqrt(2)
assert sech(5 * pi * I / 4) == -sqrt(2)
assert sech(-5 * pi * I / 4) == -sqrt(2)
assert sech(pi * I / 6) == 2 / sqrt(3)
assert sech(-pi * I / 6) == 2 / sqrt(3)
assert sech(7 * pi * I / 6) == -2 / sqrt(3)
assert sech(-5 * pi * I / 6) == -2 / sqrt(3)
assert sech(pi * I / 105) == 1 / cos(pi / 105)
assert sech(-pi * I / 105) == 1 / cos(pi / 105)
assert sech(x * I) == 1 / cos(x)
assert sech(k * pi * I) == 1 / cos(k * pi)
assert sech(17 * k * pi * I) == 1 / cos(17 * k * pi)
assert sech(n).is_extended_real is True
assert csch(n).is_finite is None
pytest.raises(ArgumentIndexError, lambda: sech(x).fdiff(2))
def test_Function():
assert mathematica_code(f(x, y, z)) == 'f[x, y, z]'
assert mathematica_code(sin(x)**cos(x)) == 'Sin[x]^Cos[x]'
assert mathematica_code(sign(x)) == 'Sign[x]'
assert mathematica_code(atanh(x),
user_functions={'atanh':
'ArcTanh'}) == 'ArcTanh[x]'
assert (mathematica_code(meijerg(
((1, 1), (3, 4)), ((1, ), ()),
x)) == 'MeijerG[{{1, 1}, {3, 4}}, {{1}, {}}, x]')
assert (mathematica_code(hyper(
(1, 2, 3), (3, 4), x)) == 'HypergeometricPFQ[{1, 2, 3}, {3, 4}, x]')
assert mathematica_code(Min(x, y)) == 'Min[x, y]'
assert mathematica_code(Max(x, y)) == 'Max[x, y]'
assert mathematica_code(Max(x,
2)) == 'Max[2, x]' # issue sympy/sympy#15344
assert mathematica_code(binomial(x, y)) == 'Binomial[x, y]'
assert mathematica_code(log(x)) == 'Log[x]'
assert mathematica_code(tan(x)) == 'Tan[x]'
assert mathematica_code(cot(x)) == 'Cot[x]'
assert mathematica_code(asin(x)) == 'ArcSin[x]'
assert mathematica_code(acos(x)) == 'ArcCos[x]'
assert mathematica_code(atan(x)) == 'ArcTan[x]'
assert mathematica_code(acot(x)) == 'ArcCot[x]'
assert mathematica_code(sinh(x)) == 'Sinh[x]'
assert mathematica_code(cosh(x)) == 'Cosh[x]'
assert mathematica_code(tanh(x)) == 'Tanh[x]'
assert mathematica_code(coth(x)) == 'Coth[x]'
assert mathematica_code(asinh(x)) == 'ArcSinh[x]'
assert mathematica_code(acosh(x)) == 'ArcCosh[x]'
assert mathematica_code(atanh(x)) == 'ArcTanh[x]'
assert mathematica_code(acoth(x)) == 'ArcCoth[x]'
assert mathematica_code(sech(x)) == 'Sech[x]'
assert mathematica_code(csch(x)) == 'Csch[x]'
assert mathematica_code(erf(x)) == 'Erf[x]'
assert mathematica_code(erfi(x)) == 'Erfi[x]'
assert mathematica_code(erfc(x)) == 'Erfc[x]'
assert mathematica_code(conjugate(x)) == 'Conjugate[x]'
assert mathematica_code(re(x)) == 'Re[x]'
assert mathematica_code(im(x)) == 'Im[x]'
assert mathematica_code(polygamma(x, y)) == 'PolyGamma[x, y]'
assert mathematica_code(factorial(x)) == 'Factorial[x]'
assert mathematica_code(factorial2(x)) == 'Factorial2[x]'
assert mathematica_code(rf(x, y)) == 'Pochhammer[x, y]'
assert mathematica_code(gamma(x)) == 'Gamma[x]'
assert mathematica_code(zeta(x)) == 'Zeta[x]'
assert mathematica_code(Heaviside(x)) == 'UnitStep[x]'
assert mathematica_code(fibonacci(x)) == 'Fibonacci[x]'
assert mathematica_code(polylog(x, y)) == 'PolyLog[x, y]'
assert mathematica_code(loggamma(x)) == 'LogGamma[x]'
assert mathematica_code(uppergamma(x, y)) == 'Gamma[x, y]'
class MyFunc1(Function):
@classmethod
def eval(cls, x):
pass
class MyFunc2(Function):
@classmethod
def eval(cls, x, y):
pass
pytest.raises(
ValueError,
lambda: mathematica_code(MyFunc1(x),
user_functions={'MyFunc1': ['Myfunc1']}))
assert mathematica_code(MyFunc1(x),
user_functions={'MyFunc1':
'Myfunc1'}) == 'Myfunc1[x]'
assert mathematica_code(
MyFunc2(x, y),
user_functions={'MyFunc2':
[(lambda *x: False, 'Myfunc2')]}) == 'MyFunc2[x, y]'
def test_sech_rewrite():
assert sech(x).rewrite(exp) == 1 / (exp(x)/2 + exp(-x)/2) \
== sech(x).rewrite('tractable')
assert sech(x).rewrite(sinh) == I/sinh(x + I*pi/2)
assert sech(x).rewrite(tanh) == (1 - tanh(x/2)**2)/(1 + tanh(x/2)**2)
assert sech(x).rewrite(coth) == (coth(x/2)**2 - 1)/(coth(x/2)**2 + 1)
def test_sech_series():
assert sech(x).series(x, 0, 10) == \
1 - x**2/2 + 5*x**4/24 - 61*x**6/720 + 277*x**8/8064 + O(x**10)
def test_sech():
k = Symbol('k', integer=True)
n = Symbol('n', positive=True)
assert sech(nan) == nan
assert sech(zoo) == nan
assert sech(oo) == 0
assert sech(-oo) == 0
assert sech(0) == 1
assert sech(-1) == sech(1)
assert sech(-x) == sech(x)
assert sech(pi*I) == sec(pi)
assert sech(-pi*I) == sec(pi)
assert sech(-2**1024 * E) == sech(2**1024 * E)
assert sech(pi*I/2) == zoo
assert sech(-pi*I/2) == zoo
assert sech((-3*10**73 + 1)*pi*I/2) == zoo
assert sech((7*10**103 + 1)*pi*I/2) == zoo
assert sech(pi*I) == -1
assert sech(-pi*I) == -1
assert sech(5*pi*I) == -1
assert sech(8*pi*I) == 1
assert sech(pi*I/3) == 2
assert sech(-2*pi*I/3) == -2
assert sech(pi*I/4) == sqrt(2)
assert sech(-pi*I/4) == sqrt(2)
assert sech(5*pi*I/4) == -sqrt(2)
assert sech(-5*pi*I/4) == -sqrt(2)
assert sech(pi*I/6) == 2/sqrt(3)
assert sech(-pi*I/6) == 2/sqrt(3)
assert sech(7*pi*I/6) == -2/sqrt(3)
assert sech(-5*pi*I/6) == -2/sqrt(3)
assert sech(pi*I/105) == 1/cos(pi/105)
assert sech(-pi*I/105) == 1/cos(pi/105)
assert sech(x*I) == 1/cos(x)
assert sech(k*pi*I) == 1/cos(k*pi)
assert sech(17*k*pi*I) == 1/cos(17*k*pi)
assert sech(n).is_extended_real is True
assert csch(n).is_finite is None
pytest.raises(ArgumentIndexError, lambda: sech(x).fdiff(2))
def test_sec():
x = symbols('x', extended_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 sympy/sympy#8421
assert sec(I) == sech(1)
assert sec(x*I) == sech(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(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(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
assert sec(x).is_finite is None
assert sec(pi/2).is_finite is 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)**Rational(3, 2)/12 +
(x - 3*pi/2)**Rational(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
pytest.raises(ArgumentIndexError, lambda: sec(x).fdiff(2))
