def test_hyper_as_trig():
from sympy.simplify.fu import _osborne as o, _osbornei as i, TR12
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 o(sinh(x), d) == I*sin(x*d)
assert o(tanh(x), d) == I*tan(x*d)
assert o(coth(x), d) == cot(x*d)/I
assert o(cosh(x), d) == cos(x*d)
assert o(sech(x), d) == sec(x*d)
assert o(csch(x), d) == csc(x*d)/I
for func in (sinh, cosh, tanh, coth, sech, csch):
h = func(pi)
assert i(o(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 i(cos(x*y + z), y) == cosh(x + z*I)
assert i(sin(x*y + z), y) == sinh(x + z*I)/I
assert i(tan(x*y + z), y) == tanh(x + z*I)/I
assert i(cot(x*y + z), y) == coth(x + z*I)*I
assert i(sec(x*y + z), y) == sech(x + z*I)
assert i(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_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_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)
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_rewrite_trigh():
# if this import passes then the test below should also pass
from sympy import sech
assert solveset_real(sinh(x) + sech(x), x) == FiniteSet(
2*atanh(-S.Half + sqrt(5)/2 - sqrt(-2*sqrt(5) + 2)/2),
2*atanh(-S.Half + sqrt(5)/2 + sqrt(-2*sqrt(5) + 2)/2),
2*atanh(-sqrt(5)/2 - S.Half + sqrt(2 + 2*sqrt(5))/2),
2*atanh(-sqrt(2 + 2*sqrt(5))/2 - sqrt(5)/2 - S.Half))
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_rewrite_trigh():
# if this import passes then the test below should also pass
from sympy import sech
assert solveset_real(sinh(x) + sech(x), x) == FiniteSet(
2 * atanh(-S.Half + sqrt(5) / 2 - sqrt(-2 * sqrt(5) + 2) / 2),
2 * atanh(-S.Half + sqrt(5) / 2 + sqrt(-2 * sqrt(5) + 2) / 2),
2 * atanh(-sqrt(5) / 2 - S.Half + sqrt(2 + 2 * sqrt(5)) / 2),
2 * atanh(-sqrt(2 + 2 * sqrt(5)) / 2 - sqrt(5) / 2 - S.Half))
def test_asech():
x = Symbol('x')
assert asech(-x) == asech(-x)
# values at fixed points
assert asech(1) == 0
assert asech(-1) == pi*I
assert asech(0) == oo
assert asech(2) == I*pi/3
assert asech(-2) == 2*I*pi / 3
# at infinites
assert asech(oo) == I*pi/2
assert asech(-oo) == I*pi/2
assert asech(zoo) == nan
assert asech(I) == log(1 + sqrt(2)) - I*pi/2
assert asech(-I) == log(1 + sqrt(2)) + I*pi/2
assert asech(sqrt(2) - sqrt(6)) == 11*I*pi / 12
assert asech(sqrt(2 - 2/sqrt(5))) == I*pi / 10
assert asech(-sqrt(2 - 2/sqrt(5))) == 9*I*pi / 10
assert asech(2 / sqrt(2 + sqrt(2))) == I*pi / 8
assert asech(-2 / sqrt(2 + sqrt(2))) == 7*I*pi / 8
assert asech(sqrt(5) - 1) == I*pi / 5
assert asech(1 - sqrt(5)) == 4*I*pi / 5
assert asech(-sqrt(2*(2 + sqrt(2)))) == 5*I*pi / 8
# properties
# asech(x) == acosh(1/x)
assert asech(sqrt(2)) == acosh(1/sqrt(2))
assert asech(2/sqrt(3)) == acosh(sqrt(3)/2)
assert asech(2/sqrt(2 + sqrt(2))) == acosh(sqrt(2 + sqrt(2))/2)
assert asech(S(2)) == acosh(1/S(2))
# asech(x) == I*acos(1/x)
# (Note: the exact formula is asech(x) == +/- I*acos(1/x))
assert asech(-sqrt(2)) == I*acos(-1/sqrt(2))
assert asech(-2/sqrt(3)) == I*acos(-sqrt(3)/2)
assert asech(-S(2)) == I*acos(-S.Half)
assert asech(-2/sqrt(2)) == I*acos(-sqrt(2)/2)
# sech(asech(x)) / x == 1
assert expand_mul(sech(asech(sqrt(6) - sqrt(2))) / (sqrt(6) - sqrt(2))) == 1
assert expand_mul(sech(asech(sqrt(6) + sqrt(2))) / (sqrt(6) + sqrt(2))) == 1
assert (sech(asech(sqrt(2 + 2/sqrt(5)))) / (sqrt(2 + 2/sqrt(5)))).simplify() == 1
assert (sech(asech(-sqrt(2 + 2/sqrt(5)))) / (-sqrt(2 + 2/sqrt(5)))).simplify() == 1
assert (sech(asech(sqrt(2*(2 + sqrt(2))))) / (sqrt(2*(2 + sqrt(2))))).simplify() == 1
assert expand_mul(sech(asech((1 + sqrt(5)))) / ((1 + sqrt(5)))) == 1
assert expand_mul(sech(asech((-1 - sqrt(5)))) / ((-1 - sqrt(5)))) == 1
assert expand_mul(sech(asech((-sqrt(6) - sqrt(2)))) / ((-sqrt(6) - sqrt(2)))) == 1
# numerical evaluation
assert str(asech(5*I).n(6)) == '0.19869 - 1.5708*I'
assert str(asech(-5*I).n(6)) == '0.19869 + 1.5708*I'
def perpendicular(Perpendicular):
if Perpendicular != 'EckartPure':
return '''\n\ndouble ''' + Perpendicular + '''_::DerivativeY(double const X,double const Y)const&noexcept
{return ''' + sympy.ccode(
eckart(Perpendicular).diff(Y).subs(
sympy.sech(BarrierWidth * X)**2,
1 - sympy.tanh(BarrierWidth * X)**2)) + ''';}
double ''' + Perpendicular + '''_::DerivativeY2(double const X,double const Y)const&noexcept
{return ''' + sympy.ccode(
eckart(Perpendicular).diff(Y, 2).subs(
sympy.sech(BarrierWidth * X)**2,
1 - sympy.tanh(BarrierWidth * X)**2)) + ''';}
double ''' + Perpendicular + '''_::DerivativeYX(double const X,double const Y)const&noexcept
{return ''' + sympy.ccode(
eckart(Perpendicular).diff(X).diff(Y).subs(
sympy.sech(BarrierWidth * X)**2, 1 -
sympy.tanh(BarrierWidth * X)**2)) + ''';}'''
return ''
def MultiDimensionalSecant(self, *args):
assert len(args) >= 2
assert len(args) % 2 == 0
variables = int(len(args) / 2)
offset = 2 + 3 * variables
variable_objects = []
equation_objects = []
for i, e in enumerate(args):
if i % 2 == 0:
variable_objects.append(e)
else:
equation_objects.append(e)
for x_string in equation_objects:
if isinstance(x_string, dict):
raise NotImplementedError(
"At this time multi-variable functions do not support composition."
)
sub_eq = 1
j = 2
for x in variable_objects:
sub_eq *= Equation.tf_secant_h(self.w[self._wc + j] * x +
self.w[self._wc + j + 1])
j += 2
eq = self.w[self._wc] * (sub_eq + self.w[self._wc + 1])
syms = get_symbols(1, offset + variables + 1)
sub_equation_string = 1
j = 2
for i in range(len(equation_objects)):
sub_equation_string *= sech(syms[j] * syms[-1 * (i + 1)] + syms[j + 1])
j += 2
equation_string = syms[0] * (sub_equation_string + syms[1])
parameters_list = []
for i, j in enumerate(equation_objects):
parameters_list.append([j, syms[-1 * (i + 1)]])
parameters_list += [[self.w[self._wc + i], syms[i]] for i in range(offset)]
return {
"equation": eq,
"parameters": parameters_list,
"symbolic": equation_string,
"offset": offset
}
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_sech_series():
x = Symbol('x')
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():
x, y = symbols('x, y')
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_real is True
{return ''' + sympy.ccode(
eckart(Perpendicular).diff(X).diff(Y).subs(
sympy.sech(BarrierWidth * X)**2, 1 -
sympy.tanh(BarrierWidth * X)**2)) + ''';}'''
return ''
for Perpendicular in Perpendiculars.keys():
print('''#include<cmath>
#include"''' + Perpendicular[0].lower() + Perpendicular[1:] + '''.h"
double ''' + Perpendicular +
'''_::Potential(double const X,double const Y)const&noexcept
{return ''' + sympy.ccode(
eckart(Perpendicular).subs(
sympy.sech(BarrierWidth * X)**2,
1 - sympy.tanh(BarrierWidth * X)**2)) + ''';}
double ''' + Perpendicular +
'''_::Derivative(double const X,double const Y)const&noexcept
{return ''' + sympy.ccode(
eckart(Perpendicular).diff(X).subs(
sympy.sech(BarrierWidth * X)**2,
1 - sympy.tanh(BarrierWidth * X)**2)) + ''';}
double ''' + Perpendicular +
'''_::Derivative2(double const X,double const Y)const&noexcept
{return ''' + sympy.ccode(
eckart(Perpendicular).diff(X, 2).subs(
sympy.sech(BarrierWidth * X)**2,
1 - sympy.tanh(BarrierWidth * X)**2)) + ''';}''' +
def test_sech_series():
x = Symbol('x')
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():
x, y = symbols("x, y")
k = Symbol("k", integer=True)
n = Symbol("n", positive=True)
assert sech(nan) is nan
assert sech(zoo) is 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) is zoo
assert sech(-pi * I / 2) is zoo
assert sech((-3 * 10 ** 73 + 1) * pi * I / 2) is zoo
assert sech((7 * 10 ** 103 + 1) * pi * I / 2) is 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(pi * I * Rational(-2, 3)) == -2
assert sech(pi * I / 4) == sqrt(2)
assert sech(-pi * I / 4) == sqrt(2)
assert sech(pi * I * Rational(5, 4)) == -sqrt(2)
assert sech(pi * I * Rational(-5, 4)) == -sqrt(2)
assert sech(pi * I / 6) == 2 / sqrt(3)
assert sech(-pi * I / 6) == 2 / sqrt(3)
assert sech(pi * I * Rational(7, 6)) == -2 / sqrt(3)
assert sech(pi * I * Rational(-5, 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_real is True
def test_issue_11254a():
assert not integrate(sech(x), (x, 0, 1)).has(Integral)
def eckart(Perpendicular):
return BarrierHeight * sympy.sech(
BarrierWidth * X)**2 + (1 + Couple * sympy.sech(BarrierWidth * X)**2
) * Perpendiculars[Perpendicular]
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
# Combinatoria
sp.factorial(num) # Factorial
sp.functions.combinatorial.numbers.nP(num1, num2) # Permutación
sp.functions.combinatorial.numbers.nC(num1, num2) # Combinación
def test_sech_fdiff():
x = Symbol('x')
raises(ArgumentIndexError, lambda: sech(x).fdiff(2))
def test_sech():
x, y = symbols('x, y')
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_real is True
def test_sech():
x, y = symbols('x, y')
k = Symbol('k', integer=True)
n = Symbol('n', positive=True)
assert sech(nan) is nan
assert sech(zoo) is 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) is zoo
assert sech(-pi * I / 2) is zoo
assert sech((-3 * 10**73 + 1) * pi * I / 2) is zoo
assert sech((7 * 10**103 + 1) * pi * I / 2) is 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(pi * I * Rational(-2, 3)) == -2
assert sech(pi * I / 4) == sqrt(2)
assert sech(-pi * I / 4) == sqrt(2)
assert sech(pi * I * Rational(5, 4)) == -sqrt(2)
assert sech(pi * I * Rational(-5, 4)) == -sqrt(2)
assert sech(pi * I / 6) == 2 / sqrt(3)
assert sech(-pi * I / 6) == 2 / sqrt(3)
assert sech(pi * I * Rational(7, 6)) == -2 / sqrt(3)
assert sech(pi * I * Rational(-5, 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_real is True
assert expand_trig(sech(x +
y)) == 1 / (cosh(x) * cosh(y) + sinh(x) * sinh(y))
