def test_hyperbolic_simp():
x, y = symbols('x,y')
assert trigsimp(sinh(x)**2 + 1) == cosh(x)**2
assert trigsimp(cosh(x)**2 - 1) == sinh(x)**2
assert trigsimp(cosh(x)**2 - sinh(x)**2) == 1
assert trigsimp(1 - tanh(x)**2) == 1 / cosh(x)**2
assert trigsimp(1 - 1 / cosh(x)**2) == tanh(x)**2
assert trigsimp(tanh(x)**2 + 1 / cosh(x)**2) == 1
assert trigsimp(coth(x)**2 - 1) == 1 / sinh(x)**2
assert trigsimp(1 / sinh(x)**2 + 1) == 1 / tanh(x)**2
assert trigsimp(coth(x)**2 - 1 / sinh(x)**2) == 1
assert trigsimp(5 * cosh(x)**2 - 5 * sinh(x)**2) == 5
assert trigsimp(5 * cosh(x / 2)**2 -
2 * sinh(x / 2)**2) == 3 * cosh(x) / 2 + Rational(7, 2)
assert trigsimp(sinh(x) / cosh(x)) == tanh(x)
assert trigsimp(tanh(x)) == trigsimp(sinh(x) / cosh(x))
assert trigsimp(cosh(x) / sinh(x)) == 1 / tanh(x)
assert trigsimp(2 * tanh(x) * cosh(x)) == 2 * sinh(x)
assert trigsimp(coth(x)**3 * sinh(x)**3) == cosh(x)**3
assert trigsimp(y * tanh(x)**2 / sinh(x)**2) == y / cosh(x)**2
assert trigsimp(coth(x) / cosh(x)) == 1 / sinh(x)
for a in (pi / 6 * I, pi / 4 * I, pi / 3 * I):
assert trigsimp(sinh(a) * cosh(x) + cosh(a) * sinh(x)) == sinh(x + a)
assert trigsimp(-sinh(a) * cosh(x) + cosh(a) * sinh(x)) == sinh(x - a)
e = 2 * cosh(x)**2 - 2 * sinh(x)**2
assert trigsimp(log(e)) == log(2)
# issue 19535:
assert trigsimp(sqrt(cosh(x)**2 - 1)) == sqrt(sinh(x)**2)
assert trigsimp(cosh(x)**2 * cosh(y)**2 - cosh(x)**2 * sinh(y)**2 -
sinh(x)**2,
recursive=True) == 1
assert trigsimp(sinh(x)**2 * sinh(y)**2 - sinh(x)**2 * cosh(y)**2 +
cosh(x)**2,
recursive=True) == 1
assert abs(trigsimp(2.0 * cosh(x)**2 - 2.0 * sinh(x)**2) - 2.0) < 1e-10
assert trigsimp(sinh(x)**2 / cosh(x)**2) == tanh(x)**2
assert trigsimp(sinh(x)**3 / cosh(x)**3) == tanh(x)**3
assert trigsimp(sinh(x)**10 / cosh(x)**10) == tanh(x)**10
assert trigsimp(cosh(x)**3 / sinh(x)**3) == 1 / tanh(x)**3
assert trigsimp(cosh(x) / sinh(x)) == 1 / tanh(x)
assert trigsimp(cosh(x)**2 / sinh(x)**2) == 1 / tanh(x)**2
assert trigsimp(cosh(x)**10 / sinh(x)**10) == 1 / tanh(x)**10
assert trigsimp(x * cosh(x) * tanh(x)) == x * sinh(x)
assert trigsimp(-sinh(x) + cosh(x) * tanh(x)) == 0
assert tan(x) != 1 / cot(x) # cot doesn't auto-simplify
assert trigsimp(tan(x) - 1 / cot(x)) == 0
assert trigsimp(3 * tanh(x)**7 - 2 / coth(x)**7) == tanh(x)**7
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_coth_rewrite():
x = Symbol('x')
assert coth(x).rewrite(exp) == (exp(x) + exp(-x))/(exp(x) - exp(-x)) \
== coth(x).rewrite('tractable')
assert coth(x).rewrite(sinh) == -I * sinh(I * pi / 2 - x) / sinh(x)
assert coth(x).rewrite(cosh) == -I * cosh(x) / cosh(I * pi / 2 - x)
assert coth(x).rewrite(tanh) == 1 / tanh(x)
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_gruntz_hyperbolic():
assert gruntz(cosh(x), x, oo) == oo
assert gruntz(cosh(x), x, -oo) == oo
assert gruntz(sinh(x), x, oo) == oo
assert gruntz(sinh(x), x, -oo) == -oo
assert gruntz(2*cosh(x)*exp(x), x, oo) == oo
assert gruntz(2*cosh(x)*exp(x), x, -oo) == 1
assert gruntz(2*sinh(x)*exp(x), x, oo) == oo
assert gruntz(2*sinh(x)*exp(x), x, -oo) == -1
assert gruntz(tanh(x), x, oo) == 1
assert gruntz(tanh(x), x, -oo) == -1
assert gruntz(coth(x), x, oo) == 1
assert gruntz(coth(x), x, -oo) == -1
def test_gruntz_hyperbolic():
assert gruntz(cosh(x), x, oo) is oo
assert gruntz(cosh(x), x, -oo) is oo
assert gruntz(sinh(x), x, oo) is oo
assert gruntz(sinh(x), x, -oo) is -oo
assert gruntz(2 * cosh(x) * exp(x), x, oo) is oo
assert gruntz(2 * cosh(x) * exp(x), x, -oo) == 1
assert gruntz(2 * sinh(x) * exp(x), x, oo) is oo
assert gruntz(2 * sinh(x) * exp(x), x, -oo) == -1
assert gruntz(tanh(x), x, oo) == 1
assert gruntz(tanh(x), x, -oo) == -1
assert gruntz(coth(x), x, oo) == 1
assert gruntz(coth(x), x, -oo) == -1
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 test_sech_rewrite():
x = Symbol('x')
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_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_sinh_rewrite():
x = Symbol('x')
assert sinh(x).rewrite(exp) == (exp(x) - exp(-x))/2 \
== sinh(x).rewrite('tractable')
assert sinh(x).rewrite(cosh) == -I * cosh(x + I * pi / 2)
tanh_half = tanh(S.Half * x)
assert sinh(x).rewrite(tanh) == 2 * tanh_half / (1 - tanh_half**2)
coth_half = coth(S.Half * x)
assert sinh(x).rewrite(coth) == 2 * coth_half / (coth_half**2 - 1)
def test_cosh_rewrite():
x = Symbol('x')
assert cosh(x).rewrite(exp) == (exp(x) + exp(-x))/2 \
== cosh(x).rewrite('tractable')
assert cosh(x).rewrite(sinh) == -I * sinh(x + I * pi / 2)
tanh_half = tanh(S.Half * x)**2
assert cosh(x).rewrite(tanh) == (1 + tanh_half) / (1 - tanh_half)
coth_half = coth(S.Half * x)**2
assert cosh(x).rewrite(coth) == (coth_half + 1) / (coth_half - 1)
def test_hyperbolic():
assert sinh(x).nseries(x, n=6) == x + x**3 / 6 + x**5 / 120 + O(x**6)
assert cosh(x).nseries(x, n=5) == 1 + x**2 / 2 + x**4 / 24 + O(x**5)
assert tanh(x).nseries(x, n=6) == x - x**3 / 3 + 2 * x**5 / 15 + O(x**6)
assert coth(x).nseries(x, n=6) == \
1/x - x**3/45 + x/3 + 2*x**5/945 + O(x**6)
assert asinh(x).nseries(x, n=6) == x - x**3 / 6 + 3 * x**5 / 40 + O(x**6)
assert acosh(x).nseries(x, n=6) == \
pi*I/2 - I*x - 3*I*x**5/40 - I*x**3/6 + O(x**6)
assert atanh(x).nseries(x, n=6) == x + x**3 / 3 + x**5 / 5 + O(x**6)
assert acoth(x).nseries(
x, n=6) == x + x**3 / 3 + x**5 / 5 + pi * I / 2 + O(x**6)
def test_inverses():
x = Symbol('x')
assert sinh(x).inverse() == asinh
raises(AttributeError, lambda: cosh(x).inverse())
assert tanh(x).inverse() == atanh
assert coth(x).inverse() == acoth
assert asinh(x).inverse() == sinh
assert acosh(x).inverse() == cosh
assert atanh(x).inverse() == tanh
assert acoth(x).inverse() == coth
assert asech(x).inverse() == sech
assert acsch(x).inverse() == csch
def test_conjugate():
a = Symbol("a", real=True)
b = Symbol("b", real=True)
c = Symbol("c", imaginary=True)
d = Symbol("d", imaginary=True)
x = Symbol('x')
z = a + I * b + c + I * d
zc = a - I * b - c + I * d
assert conjugate(z) == zc
assert conjugate(exp(z)) == exp(zc)
assert conjugate(exp(I * x)) == exp(-I * conjugate(x))
assert conjugate(z**5) == zc**5
assert conjugate(abs(x)) == abs(x)
assert conjugate(sign(z)) == sign(zc)
assert conjugate(sin(z)) == sin(zc)
assert conjugate(cos(z)) == cos(zc)
assert conjugate(tan(z)) == tan(zc)
assert conjugate(cot(z)) == cot(zc)
assert conjugate(sinh(z)) == sinh(zc)
assert conjugate(cosh(z)) == cosh(zc)
assert conjugate(tanh(z)) == tanh(zc)
assert conjugate(coth(z)) == coth(zc)
def test_trigsimp1a():
assert trigsimp(sin(2)**2 * cos(3) * exp(2) /
cos(2)**2) == tan(2)**2 * cos(3) * exp(2)
assert trigsimp(tan(2)**2 * cos(3) * exp(2) *
cos(2)**2) == sin(2)**2 * cos(3) * exp(2)
assert trigsimp(cot(2) * cos(3) * exp(2) *
sin(2)) == cos(3) * exp(2) * cos(2)
assert trigsimp(tan(2) * cos(3) * exp(2) /
sin(2)) == cos(3) * exp(2) / cos(2)
assert trigsimp(cot(2) * cos(3) * exp(2) /
cos(2)) == cos(3) * exp(2) / sin(2)
assert trigsimp(cot(2) * cos(3) * exp(2) * tan(2)) == cos(3) * exp(2)
assert trigsimp(sinh(2) * cos(3) * exp(2) /
cosh(2)) == tanh(2) * cos(3) * exp(2)
assert trigsimp(tanh(2) * cos(3) * exp(2) *
cosh(2)) == sinh(2) * cos(3) * exp(2)
assert trigsimp(coth(2) * cos(3) * exp(2) *
sinh(2)) == cosh(2) * cos(3) * exp(2)
assert trigsimp(tanh(2) * cos(3) * exp(2) /
sinh(2)) == cos(3) * exp(2) / cosh(2)
assert trigsimp(coth(2) * cos(3) * exp(2) /
cosh(2)) == cos(3) * exp(2) / sinh(2)
assert trigsimp(coth(2) * cos(3) * exp(2) * tanh(2)) == cos(3) * exp(2)
def test_leading_term():
x = Symbol('x')
assert cosh(x).as_leading_term(x) == 1
assert coth(x).as_leading_term(x) == 1 / x
assert acosh(x).as_leading_term(x) == I * pi / 2
assert acoth(x).as_leading_term(x) == I * pi / 2
for func in [sinh, tanh, asinh, atanh]:
assert func(x).as_leading_term(x) == x
for func in [sinh, cosh, tanh, coth, asinh, acosh, atanh, acoth]:
for arg in (1 / x, S.Half):
eq = func(arg)
assert eq.as_leading_term(x) == eq
for func in [csch, sech]:
eq = func(S.Half)
assert eq.as_leading_term(x) == eq
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_laplace_transform():
from sympy import lowergamma
from sympy.functions.special.delta_functions import DiracDelta
from sympy.functions.special.error_functions import (fresnelc, fresnels)
LT = laplace_transform
a, b, c, = symbols('a, b, c', positive=True)
t, w, x = symbols('t, w, x')
f = Function("f")
g = Function("g")
# Test rule-base evaluation according to
# http://eqworld.ipmnet.ru/en/auxiliary/inttrans/
# Power-law functions (laplace2.pdf)
assert LT(a*t+t**2+t**(S(5)/2), t, s) ==\
(a/s**2 + 2/s**3 + 15*sqrt(pi)/(8*s**(S(7)/2)), 0, True)
assert LT(b/(t+a), t, s) == (-b*exp(-a*s)*Ei(-a*s), 0, True)
assert LT(1/sqrt(t+a), t, s) ==\
(sqrt(pi)*sqrt(1/s)*exp(a*s)*erfc(sqrt(a)*sqrt(s)), 0, True)
assert LT(sqrt(t)/(t+a), t, s) ==\
(-pi*sqrt(a)*exp(a*s)*erfc(sqrt(a)*sqrt(s)) + sqrt(pi)*sqrt(1/s),
0, True)
assert LT((t+a)**(-S(3)/2), t, s) ==\
(-2*sqrt(pi)*sqrt(s)*exp(a*s)*erfc(sqrt(a)*sqrt(s)) + 2/sqrt(a),
0, True)
assert LT(t**(S(1)/2)*(t+a)**(-1), t, s) ==\
(-pi*sqrt(a)*exp(a*s)*erfc(sqrt(a)*sqrt(s)) + sqrt(pi)*sqrt(1/s),
0, True)
assert LT(1/(a*sqrt(t) + t**(3/2)), t, s) ==\
(pi*sqrt(a)*exp(a*s)*erfc(sqrt(a)*sqrt(s)), 0, True)
assert LT((t+a)**b, t, s) ==\
(s**(-b - 1)*exp(-a*s)*lowergamma(b + 1, a*s), 0, True)
assert LT(t**5/(t+a), t, s) == (120*a**5*lowergamma(-5, a*s), 0, True)
# Exponential functions (laplace3.pdf)
assert LT(exp(t), t, s) == (1/(s - 1), 1, True)
assert LT(exp(2*t), t, s) == (1/(s - 2), 2, True)
assert LT(exp(a*t), t, s) == (1/(s - a), a, True)
assert LT(exp(a*(t-b)), t, s) == (exp(-a*b)/(-a + s), a, True)
assert LT(t*exp(-a*(t)), t, s) == ((a + s)**(-2), -a, True)
assert LT(t*exp(-a*(t-b)), t, s) == (exp(a*b)/(a + s)**2, -a, True)
assert LT(b*t*exp(-a*t), t, s) == (b/(a + s)**2, -a, True)
assert LT(t**(S(7)/4)*exp(-8*t)/gamma(S(11)/4), t, s) ==\
((s + 8)**(-S(11)/4), -8, True)
assert LT(t**(S(3)/2)*exp(-8*t), t, s) ==\
(3*sqrt(pi)/(4*(s + 8)**(S(5)/2)), -8, True)
assert LT(t**a*exp(-a*t), t, s) == ((a+s)**(-a-1)*gamma(a+1), -a, True)
assert LT(b*exp(-a*t**2), t, s) ==\
(sqrt(pi)*b*exp(s**2/(4*a))*erfc(s/(2*sqrt(a)))/(2*sqrt(a)), 0, True)
assert LT(exp(-2*t**2), t, s) ==\
(sqrt(2)*sqrt(pi)*exp(s**2/8)*erfc(sqrt(2)*s/4)/4, 0, True)
assert LT(b*exp(2*t**2), t, s) == b*LaplaceTransform(exp(2*t**2), t, s)
assert LT(t*exp(-a*t**2), t, s) ==\
(1/(2*a) - s*erfc(s/(2*sqrt(a)))/(4*sqrt(pi)*a**(S(3)/2)), 0, True)
assert LT(exp(-a/t), t, s) ==\
(2*sqrt(a)*sqrt(1/s)*besselk(1, 2*sqrt(a)*sqrt(s)), 0, True)
assert LT(sqrt(t)*exp(-a/t), t, s) ==\
(sqrt(pi)*(2*sqrt(a)*sqrt(s) + 1)*sqrt(s**(-3))*exp(-2*sqrt(a)*\
sqrt(s))/2, 0, True)
assert LT(exp(-a/t)/sqrt(t), t, s) ==\
(sqrt(pi)*sqrt(1/s)*exp(-2*sqrt(a)*sqrt(s)), 0, True)
assert LT( exp(-a/t)/(t*sqrt(t)), t, s) ==\
(sqrt(pi)*sqrt(1/a)*exp(-2*sqrt(a)*sqrt(s)), 0, True)
assert LT(exp(-2*sqrt(a*t)), t, s) ==\
( 1/s -sqrt(pi)*sqrt(a) * exp(a/s)*erfc(sqrt(a)*sqrt(1/s))/\
s**(S(3)/2), 0, True)
assert LT(exp(-2*sqrt(a*t))/sqrt(t), t, s) == (exp(a/s)*erfc(sqrt(a)*\
sqrt(1/s))*(sqrt(pi)*sqrt(1/s)), 0, True)
assert LT(t**4*exp(-2/t), t, s) ==\
(8*sqrt(2)*(1/s)**(S(5)/2)*besselk(5, 2*sqrt(2)*sqrt(s)), 0, True)
# Hyperbolic functions (laplace4.pdf)
assert LT(sinh(a*t), t, s) == (a/(-a**2 + s**2), a, True)
assert LT(b*sinh(a*t)**2, t, s) == (2*a**2*b/(-4*a**2*s**2 + s**3),
2*a, True)
# The following line confirms that issue #21202 is solved
assert LT(cosh(2*t), t, s) == (s/(-4 + s**2), 2, True)
assert LT(cosh(a*t), t, s) == (s/(-a**2 + s**2), a, True)
assert LT(cosh(a*t)**2, t, s) == ((-2*a**2 + s**2)/(-4*a**2*s**2 + s**3),
2*a, True)
assert LT(sinh(x + 3), x, s) == (
(-s + (s + 1)*exp(6) + 1)*exp(-3)/(s - 1)/(s + 1)/2, 0, Abs(s) > 1)
# The following line replaces the old test test_issue_7173()
assert LT(sinh(a*t)*cosh(a*t), t, s) == (a/(-4*a**2 + s**2), 2*a, True)
assert LT(sinh(a*t)/t, t, s) == (log((a + s)/(-a + s))/2, a, True)
assert LT(t**(-S(3)/2)*sinh(a*t), t, s) ==\
(-sqrt(pi)*(sqrt(-a + s) - sqrt(a + s)), a, True)
assert LT(sinh(2*sqrt(a*t)), t, s) ==\
(sqrt(pi)*sqrt(a)*exp(a/s)/s**(S(3)/2), 0, True)
assert LT(sqrt(t)*sinh(2*sqrt(a*t)), t, s) ==\
(-sqrt(a)/s**2 + sqrt(pi)*(a + s/2)*exp(a/s)*erf(sqrt(a)*\
sqrt(1/s))/s**(S(5)/2), 0, True)
assert LT(sinh(2*sqrt(a*t))/sqrt(t), t, s) ==\
(sqrt(pi)*exp(a/s)*erf(sqrt(a)*sqrt(1/s))/sqrt(s), 0, True)
assert LT(sinh(sqrt(a*t))**2/sqrt(t), t, s) ==\
(sqrt(pi)*(exp(a/s) - 1)/(2*sqrt(s)), 0, True)
assert LT(t**(S(3)/7)*cosh(a*t), t, s) ==\
(((a + s)**(-S(10)/7) + (-a+s)**(-S(10)/7))*gamma(S(10)/7)/2, a, True)
assert LT(cosh(2*sqrt(a*t)), t, s) ==\
(sqrt(pi)*sqrt(a)*exp(a/s)*erf(sqrt(a)*sqrt(1/s))/s**(S(3)/2) + 1/s,
0, True)
assert LT(sqrt(t)*cosh(2*sqrt(a*t)), t, s) ==\
(sqrt(pi)*(a + s/2)*exp(a/s)/s**(S(5)/2), 0, True)
assert LT(cosh(2*sqrt(a*t))/sqrt(t), t, s) ==\
(sqrt(pi)*exp(a/s)/sqrt(s), 0, True)
assert LT(cosh(sqrt(a*t))**2/sqrt(t), t, s) ==\
(sqrt(pi)*(exp(a/s) + 1)/(2*sqrt(s)), 0, True)
# logarithmic functions (laplace5.pdf)
assert LT(log(t), t, s) == (-log(s+S.EulerGamma)/s, 0, True)
assert LT(log(t/a), t, s) == (-log(a*s + S.EulerGamma)/s, 0, True)
assert LT(log(1+a*t), t, s) == (-exp(s/a)*Ei(-s/a)/s, 0, True)
assert LT(log(t+a), t, s) == ((log(a) - exp(s/a)*Ei(-s/a)/s)/s, 0, True)
assert LT(log(t)/sqrt(t), t, s) ==\
(sqrt(pi)*(-log(s) - 2*log(2) - S.EulerGamma)/sqrt(s), 0, True)
assert LT(t**(S(5)/2)*log(t), t, s) ==\
(15*sqrt(pi)*(-log(s)-2*log(2)-S.EulerGamma+S(46)/15)/(8*s**(S(7)/2)),
0, True)
assert (LT(t**3*log(t), t, s, noconds=True)-6*(-log(s) - S.EulerGamma\
+ S(11)/6)/s**4).simplify() == S.Zero
assert LT(log(t)**2, t, s) ==\
(((log(s) + EulerGamma)**2 + pi**2/6)/s, 0, True)
assert LT(exp(-a*t)*log(t), t, s) ==\
((-log(a + s) - S.EulerGamma)/(a + s), -a, True)
# Trigonometric functions (laplace6.pdf)
assert LT(sin(a*t), t, s) == (a/(a**2 + s**2), 0, True)
assert LT(Abs(sin(a*t)), t, s) ==\
(a*coth(pi*s/(2*a))/(a**2 + s**2), 0, True)
assert LT(sin(a*t)/t, t, s) == (atan(a/s), 0, True)
assert LT(sin(a*t)**2/t, t, s) == (log(4*a**2/s**2 + 1)/4, 0, True)
assert LT(sin(a*t)**2/t**2, t, s) ==\
(a*atan(2*a/s) - s*log(4*a**2/s**2 + 1)/4, 0, True)
assert LT(sin(2*sqrt(a*t)), t, s) ==\
(sqrt(pi)*sqrt(a)*exp(-a/s)/s**(S(3)/2), 0, True)
assert LT(sin(2*sqrt(a*t))/t, t, s) == (pi*erf(sqrt(a)*sqrt(1/s)), 0, True)
assert LT(cos(a*t), t, s) == (s/(a**2 + s**2), 0, True)
assert LT(cos(a*t)**2, t, s) ==\
((2*a**2 + s**2)/(s*(4*a**2 + s**2)), 0, True)
assert LT(sqrt(t)*cos(2*sqrt(a*t)), t, s) ==\
(sqrt(pi)*(-2*a + s)*exp(-a/s)/(2*s**(S(5)/2)), 0, True)
assert LT(cos(2*sqrt(a*t))/sqrt(t), t, s) ==\
(sqrt(pi)*sqrt(1/s)*exp(-a/s), 0, True)
assert LT(sin(a*t)*sin(b*t), t, s) ==\
(2*a*b*s/((s**2 + (a - b)**2)*(s**2 + (a + b)**2)), 0, True)
assert LT(cos(a*t)*sin(b*t), t, s) ==\
(b*(-a**2 + b**2 + s**2)/((s**2 + (a - b)**2)*(s**2 + (a + b)**2)),
0, True)
assert LT(cos(a*t)*cos(b*t), t, s) ==\
(s*(a**2 + b**2 + s**2)/((s**2 + (a - b)**2)*(s**2 + (a + b)**2)),
0, True)
assert LT(c*exp(-b*t)*sin(a*t), t, s) == (a*c/(a**2 + (b + s)**2),
-b, True)
assert LT(c*exp(-b*t)*cos(a*t), t, s) == ((b + s)*c/(a**2 + (b + s)**2),
-b, True)
assert LT(cos(x + 3), x, s) == ((s*cos(3) - sin(3))/(s**2 + 1), 0, True)
# Error functions (laplace7.pdf)
assert LT(erf(a*t), t, s) == (exp(s**2/(4*a**2))*erfc(s/(2*a))/s, 0, True)
assert LT(erf(sqrt(a*t)), t, s) == (sqrt(a)/(s*sqrt(a + s)), 0, True)
assert LT(exp(a*t)*erf(sqrt(a*t)), t, s) ==\
(sqrt(a)/(sqrt(s)*(-a + s)), a, True)
assert LT(erf(sqrt(a/t)/2), t, s) == ((1-exp(-sqrt(a)*sqrt(s)))/s, 0, True)
assert LT(erfc(sqrt(a*t)), t, s) ==\
((-sqrt(a) + sqrt(a + s))/(s*sqrt(a + s)), 0, True)
assert LT(exp(a*t)*erfc(sqrt(a*t)), t, s) ==\
(1/(sqrt(a)*sqrt(s) + s), 0, True)
assert LT(erfc(sqrt(a/t)/2), t, s) == (exp(-sqrt(a)*sqrt(s))/s, 0, True)
# Bessel functions (laplace8.pdf)
assert LT(besselj(0, a*t), t, s) == (1/sqrt(a**2 + s**2), 0, True)
assert LT(besselj(1, a*t), t, s) ==\
(a/(sqrt(a**2 + s**2)*(s + sqrt(a**2 + s**2))), 0, True)
assert LT(besselj(2, a*t), t, s) ==\
(a**2/(sqrt(a**2 + s**2)*(s + sqrt(a**2 + s**2))**2), 0, True)
assert LT(t*besselj(0, a*t), t, s) ==\
(s/(a**2 + s**2)**(S(3)/2), 0, True)
assert LT(t*besselj(1, a*t), t, s) ==\
(a/(a**2 + s**2)**(S(3)/2), 0, True)
assert LT(t**2*besselj(2, a*t), t, s) ==\
(3*a**2/(a**2 + s**2)**(S(5)/2), 0, True)
assert LT(besselj(0, 2*sqrt(a*t)), t, s) == (exp(-a/s)/s, 0, True)
assert LT(t**(S(3)/2)*besselj(3, 2*sqrt(a*t)), t, s) ==\
(a**(S(3)/2)*exp(-a/s)/s**4, 0, True)
assert LT(besselj(0, a*sqrt(t**2+b*t)), t, s) ==\
(exp(b*s - b*sqrt(a**2 + s**2))/sqrt(a**2 + s**2), 0, True)
assert LT(besseli(0, a*t), t, s) == (1/sqrt(-a**2 + s**2), a, True)
assert LT(besseli(1, a*t), t, s) ==\
(a/(sqrt(-a**2 + s**2)*(s + sqrt(-a**2 + s**2))), a, True)
assert LT(besseli(2, a*t), t, s) ==\
(a**2/(sqrt(-a**2 + s**2)*(s + sqrt(-a**2 + s**2))**2), a, True)
assert LT(t*besseli(0, a*t), t, s) == (s/(-a**2 + s**2)**(S(3)/2), a, True)
assert LT(t*besseli(1, a*t), t, s) == (a/(-a**2 + s**2)**(S(3)/2), a, True)
assert LT(t**2*besseli(2, a*t), t, s) ==\
(3*a**2/(-a**2 + s**2)**(S(5)/2), a, True)
assert LT(t**(S(3)/2)*besseli(3, 2*sqrt(a*t)), t, s) ==\
(a**(S(3)/2)*exp(a/s)/s**4, 0, True)
assert LT(bessely(0, a*t), t, s) ==\
(-2*asinh(s/a)/(pi*sqrt(a**2 + s**2)), 0, True)
assert LT(besselk(0, a*t), t, s) ==\
(log(s + sqrt(-a**2 + s**2))/sqrt(-a**2 + s**2), a, True)
assert LT(sin(a*t)**8, t, s) ==\
(40320*a**8/(s*(147456*a**8 + 52480*a**6*s**2 + 4368*a**4*s**4 +\
120*a**2*s**6 + s**8)), 0, True)
# Test general rules and unevaluated forms
# These all also test whether issue #7219 is solved.
assert LT(Heaviside(t-1)*cos(t-1), t, s) == (s*exp(-s)/(s**2 + 1), 0, True)
assert LT(a*f(t), t, w) == a*LaplaceTransform(f(t), t, w)
assert LT(a*Heaviside(t+1)*f(t+1), t, s) ==\
a*LaplaceTransform(f(t + 1)*Heaviside(t + 1), t, s)
assert LT(a*Heaviside(t-1)*f(t-1), t, s) ==\
a*LaplaceTransform(f(t), t, s)*exp(-s)
assert LT(b*f(t/a), t, s) == a*b*LaplaceTransform(f(t), t, a*s)
assert LT(exp(-f(x)*t), t, s) == (1/(s + f(x)), -f(x), True)
assert LT(exp(-a*t)*f(t), t, s) == LaplaceTransform(f(t), t, a + s)
assert LT(exp(-a*t)*erfc(sqrt(b/t)/2), t, s) ==\
(exp(-sqrt(b)*sqrt(a + s))/(a + s), -a, True)
assert LT(sinh(a*t)*f(t), t, s) ==\
LaplaceTransform(f(t), t, -a+s)/2 - LaplaceTransform(f(t), t, a+s)/2
assert LT(sinh(a*t)*t, t, s) ==\
(-1/(2*(a + s)**2) + 1/(2*(-a + s)**2), a, True)
assert LT(cosh(a*t)*f(t), t, s) ==\
LaplaceTransform(f(t), t, -a+s)/2 + LaplaceTransform(f(t), t, a+s)/2
assert LT(cosh(a*t)*t, t, s) ==\
(1/(2*(a + s)**2) + 1/(2*(-a + s)**2), a, True)
assert LT(sin(a*t)*f(t), t, s) ==\
I*(-LaplaceTransform(f(t), t, -I*a + s) +\
LaplaceTransform(f(t), t, I*a + s))/2
assert LT(sin(a*t)*t, t, s) ==\
(2*a*s/(a**4 + 2*a**2*s**2 + s**4), 0, True)
assert LT(cos(a*t)*f(t), t, s) ==\
LaplaceTransform(f(t), t, -I*a + s)/2 +\
LaplaceTransform(f(t), t, I*a + s)/2
assert LT(cos(a*t)*t, t, s) ==\
((-a**2 + s**2)/(a**4 + 2*a**2*s**2 + s**4), 0, True)
# The following two lines test whether issues #5813 and #7176 are solved.
assert LT(diff(f(t), (t, 1)), t, s) == s*LaplaceTransform(f(t), t, s)\
- f(0)
assert LT(diff(f(t), (t, 3)), t, s) == s**3*LaplaceTransform(f(t), t, s)\
- s**2*f(0) - s*Subs(Derivative(f(t), t), t, 0)\
- Subs(Derivative(f(t), (t, 2)), t, 0)
assert LT(a*f(b*t)+g(c*t), t, s) == a*LaplaceTransform(f(t), t, s/b)/b +\
LaplaceTransform(g(t), t, s/c)/c
assert inverse_laplace_transform(
f(w), w, t, plane=0) == InverseLaplaceTransform(f(w), w, t, 0)
assert LT(f(t)*g(t), t, s) == LaplaceTransform(f(t)*g(t), t, s)
# additional basic tests from wikipedia
assert LT((t - a)**b*exp(-c*(t - a))*Heaviside(t - a), t, s) == \
((s + c)**(-b - 1)*exp(-a*s)*gamma(b + 1), -c, True)
assert LT((exp(2*t) - 1)*exp(-b - t)*Heaviside(t)/2, t, s, noconds=True) \
== exp(-b)/(s**2 - 1)
# DiracDelta function: standard cases
assert LT(DiracDelta(t), t, s) == (1, 0, True)
assert LT(DiracDelta(a*t), t, s) == (1/a, 0, True)
assert LT(DiracDelta(t/42), t, s) == (42, 0, True)
assert LT(DiracDelta(t+42), t, s) == (0, 0, True)
assert LT(DiracDelta(t)+DiracDelta(t-42), t, s) == \
(1 + exp(-42*s), 0, True)
assert LT(DiracDelta(t)-a*exp(-a*t), t, s) == (s/(a + s), 0, True)
assert LT(exp(-t)*(DiracDelta(t)+DiracDelta(t-42)), t, s) == \
(exp(-42*s - 42) + 1, -oo, True)
# Collection of cases that cannot be fully evaluated and/or would catch
# some common implementation errors
assert LT(DiracDelta(t**2), t, s) == LaplaceTransform(DiracDelta(t**2), t, s)
assert LT(DiracDelta(t**2 - 1), t, s) == (exp(-s)/2, -oo, True)
assert LT(DiracDelta(t*(1 - t)), t, s) == \
LaplaceTransform(DiracDelta(-t**2 + t), t, s)
assert LT((DiracDelta(t) + 1)*(DiracDelta(t - 1) + 1), t, s) == \
(LaplaceTransform(DiracDelta(t)*DiracDelta(t - 1), t, s) + \
1 + exp(-s) + 1/s, 0, True)
assert LT(DiracDelta(2*t-2*exp(a)), t, s) == (exp(-s*exp(a))/2, 0, True)
assert LT(DiracDelta(-2*t+2*exp(a)), t, s) == (exp(-s*exp(a))/2, 0, True)
# Heaviside tests
assert LT(Heaviside(t), t, s) == (1/s, 0, True)
assert LT(Heaviside(t - a), t, s) == (exp(-a*s)/s, 0, True)
assert LT(Heaviside(t-1), t, s) == (exp(-s)/s, 0, True)
assert LT(Heaviside(2*t-4), t, s) == (exp(-2*s)/s, 0, True)
assert LT(Heaviside(-2*t+4), t, s) == ((1 - exp(-2*s))/s, 0, True)
assert LT(Heaviside(2*t+4), t, s) == (1/s, 0, True)
assert LT(Heaviside(-2*t+4), t, s) == ((1 - exp(-2*s))/s, 0, True)
# Fresnel functions
assert laplace_transform(fresnels(t), t, s) == \
((-sin(s**2/(2*pi))*fresnels(s/pi) + sin(s**2/(2*pi))/2 -
cos(s**2/(2*pi))*fresnelc(s/pi) + cos(s**2/(2*pi))/2)/s, 0, True)
assert laplace_transform(fresnelc(t), t, s) == (
((2*sin(s**2/(2*pi))*fresnelc(s/pi) - 2*cos(s**2/(2*pi))*fresnels(s/pi)
+ sqrt(2)*cos(s**2/(2*pi) + pi/4))/(2*s), 0, True))
# Matrix tests
Mt = Matrix([[exp(t), t*exp(-t)], [t*exp(-t), exp(t)]])
Ms = Matrix([[ 1/(s - 1), (s + 1)**(-2)],
[(s + 1)**(-2), 1/(s - 1)]])
# The default behaviour for Laplace tranform of a Matrix returns a Matrix
# of Tuples and is deprecated:
with warns_deprecated_sympy():
Ms_conds = Matrix([[(1/(s - 1), 1, True), ((s + 1)**(-2),
-1, True)], [((s + 1)**(-2), -1, True), (1/(s - 1), 1, True)]])
with warns_deprecated_sympy():
assert LT(Mt, t, s) == Ms_conds
# The new behavior is to return a tuple of a Matrix and the convergence
# conditions for the matrix as a whole:
assert LT(Mt, t, s, legacy_matrix=False) == (Ms, 1, True)
# With noconds=True the transformed matrix is returned without conditions
# either way:
assert LT(Mt, t, s, noconds=True) == Ms
assert LT(Mt, t, s, legacy_matrix=False, noconds=True) == Ms
from sympy import Derivative, diff, sqrt, exp, symbols, Matrix, diff, init_printing, latex, simplify
from sympy.abc import a
from sympy.functions.elementary.hyperbolic import coth
init_printing()
Hx, Hy, Hz, normH = symbols("Hx Hy Hz |H|")
Mx, My, Mz = symbols("Mx My Mz")
alpha, Ms = symbols("alpha Ms")
H = Matrix([Hx, Hy, Hz])
M = Matrix([Mx, My, Mz])
He = H + alpha * M
Mah = Ms * (coth(He.norm()) - a / He.norm())
B = Matrix([mu * Hx, mu * Hy, mu * Hz])
H = Matrix([Hx, Hy, Hz])
dBdHcommon = mu0 / ((m0 + normH * (mur0 - 1))**2)
def test_sympy__functions__elementary__hyperbolic__coth():
from sympy.functions.elementary.hyperbolic import coth
assert _test_args(coth(2))
def test_sympy__functions__elementary__hyperbolic__coth():
from sympy.functions.elementary.hyperbolic import coth
assert _test_args(coth(2))
def test_coth():
x, y = symbols('x,y')
k = Symbol('k', integer=True)
assert coth(nan) is nan
assert coth(zoo) is nan
assert coth(oo) == 1
assert coth(-oo) == -1
assert coth(0) is zoo
assert unchanged(coth, 1)
assert coth(-1) == -coth(1)
assert unchanged(coth, x)
assert coth(-x) == -coth(x)
assert coth(pi * I) == -I * cot(pi)
assert coth(-pi * I) == cot(pi) * I
assert unchanged(coth, 2**1024 * E)
assert coth(-2**1024 * E) == -coth(2**1024 * E)
assert coth(pi * I) == -I * cot(pi)
assert coth(-pi * I) == I * cot(pi)
assert coth(2 * pi * I) == -I * cot(2 * pi)
assert coth(-2 * pi * I) == I * cot(2 * pi)
assert coth(-3 * 10**73 * pi * I) == I * cot(3 * 10**73 * pi)
assert coth(7 * 10**103 * pi * I) == -I * cot(7 * 10**103 * pi)
assert coth(pi * I / 2) == 0
assert coth(-pi * I / 2) == 0
assert coth(pi * I * Rational(5, 2)) == 0
assert coth(pi * I * Rational(7, 2)) == 0
assert coth(pi * I / 3) == -I / sqrt(3)
assert coth(pi * I * Rational(-2, 3)) == -I / sqrt(3)
assert coth(pi * I / 4) == -I
assert coth(-pi * I / 4) == I
assert coth(pi * I * Rational(17, 4)) == -I
assert coth(pi * I * Rational(-3, 4)) == -I
assert coth(pi * I / 6) == -sqrt(3) * I
assert coth(-pi * I / 6) == sqrt(3) * I
assert coth(pi * I * Rational(7, 6)) == -sqrt(3) * I
assert coth(pi * I * Rational(-5, 6)) == -sqrt(3) * I
assert coth(pi * I / 105) == -cot(pi / 105) * I
assert coth(-pi * I / 105) == cot(pi / 105) * I
assert unchanged(coth, 2 + 3 * I)
assert coth(x * I) == -cot(x) * I
assert coth(k * pi * I) == -cot(k * pi) * I
assert coth(17 * k * pi * I) == -cot(17 * k * pi) * I
assert coth(k * pi * I) == -cot(k * pi) * I
assert coth(log(tan(2))) == coth(log(-tan(2)))
assert coth(1 + I * pi / 2) == tanh(1)
assert coth(x).as_real_imag(
deep=False) == (sinh(re(x)) * cosh(re(x)) /
(sin(im(x))**2 + sinh(re(x))**2), -sin(im(x)) *
cos(im(x)) / (sin(im(x))**2 + sinh(re(x))**2))
x = Symbol('x', extended_real=True)
assert coth(x).as_real_imag(deep=False) == (coth(x), 0)
assert expand_trig(coth(2 * x)) == (coth(x)**2 + 1) / (2 * coth(x))
assert expand_trig(coth(
3 * x)) == (coth(x)**3 + 3 * coth(x)) / (1 + 3 * coth(x)**2)
assert expand_trig(
coth(x + y)) == (1 + coth(x) * coth(y)) / (coth(x) + coth(y))
def test_coth_fdiff():
x = Symbol('x')
raises(ArgumentIndexError, lambda: coth(x).fdiff(2))
def test_coth_series():
x = Symbol('x')
assert coth(x).series(x, 0, 8) == \
1/x + x/3 - x**3/45 + 2*x**5/945 - x**7/4725 + O(x**8)
