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from sympy.calculus.accumulationbounds import AccumBoundsfrom sympy.core.function import (expand_mul, expand_trig)from sympy.core.numbers import (E, I, Integer, Rational, nan, oo, pi, zoo)from sympy.core.singleton import Sfrom sympy.core.symbol import (Symbol, symbols)from sympy.functions.elementary.complexes import (im, re)from sympy.functions.elementary.exponential import (exp, log)from sympy.functions.elementary.hyperbolic import (acosh, acoth, acsch, asech, asinh, atanh, cosh, coth, csch, sech, sinh, tanh)from sympy.functions.elementary.miscellaneous import sqrtfrom sympy.functions.elementary.trigonometric import (acos, asin, cos, cot, sec, sin, tan)from sympy.series.order import O
from sympy.core.expr import unchangedfrom sympy.core.function import ArgumentIndexErrorfrom sympy.testing.pytest import raises
def test_sinh(): x, y = symbols('x,y')
k = Symbol('k', integer=True)
assert sinh(nan) is nan assert sinh(zoo) is nan
assert sinh(oo) is oo assert sinh(-oo) is -oo
assert sinh(0) == 0
assert unchanged(sinh, 1) assert sinh(-1) == -sinh(1)
assert unchanged(sinh, x) assert sinh(-x) == -sinh(x)
assert unchanged(sinh, pi) assert sinh(-pi) == -sinh(pi)
assert unchanged(sinh, 2**1024 * E) assert sinh(-2**1024 * E) == -sinh(2**1024 * E)
assert sinh(pi*I) == 0 assert sinh(-pi*I) == 0 assert sinh(2*pi*I) == 0 assert sinh(-2*pi*I) == 0 assert sinh(-3*10**73*pi*I) == 0 assert sinh(7*10**103*pi*I) == 0
assert sinh(pi*I/2) == I assert sinh(-pi*I/2) == -I assert sinh(pi*I*Rational(5, 2)) == I assert sinh(pi*I*Rational(7, 2)) == -I
assert sinh(pi*I/3) == S.Half*sqrt(3)*I assert sinh(pi*I*Rational(-2, 3)) == Rational(-1, 2)*sqrt(3)*I
assert sinh(pi*I/4) == S.Half*sqrt(2)*I assert sinh(-pi*I/4) == Rational(-1, 2)*sqrt(2)*I assert sinh(pi*I*Rational(17, 4)) == S.Half*sqrt(2)*I assert sinh(pi*I*Rational(-3, 4)) == Rational(-1, 2)*sqrt(2)*I
assert sinh(pi*I/6) == S.Half*I assert sinh(-pi*I/6) == Rational(-1, 2)*I assert sinh(pi*I*Rational(7, 6)) == Rational(-1, 2)*I assert sinh(pi*I*Rational(-5, 6)) == Rational(-1, 2)*I
assert sinh(pi*I/105) == sin(pi/105)*I assert sinh(-pi*I/105) == -sin(pi/105)*I
assert unchanged(sinh, 2 + 3*I)
assert sinh(x*I) == sin(x)*I
assert sinh(k*pi*I) == 0 assert sinh(17*k*pi*I) == 0
assert sinh(k*pi*I/2) == sin(k*pi/2)*I
assert sinh(x).as_real_imag(deep=False) == (cos(im(x))*sinh(re(x)), sin(im(x))*cosh(re(x))) x = Symbol('x', extended_real=True) assert sinh(x).as_real_imag(deep=False) == (sinh(x), 0)
x = Symbol('x', real=True) assert sinh(I*x).is_finite is True assert sinh(x).is_real is True assert sinh(I).is_real is False p = Symbol('p', positive=True) assert sinh(p).is_zero is False assert sinh(0, evaluate=False).is_zero is True assert sinh(2*pi*I, evaluate=False).is_zero is True
def test_sinh_series(): x = Symbol('x') assert sinh(x).series(x, 0, 10) == \ x + x**3/6 + x**5/120 + x**7/5040 + x**9/362880 + O(x**10)
def test_sinh_fdiff(): x = Symbol('x') raises(ArgumentIndexError, lambda: sinh(x).fdiff(2))
def test_cosh(): x, y = symbols('x,y')
k = Symbol('k', integer=True)
assert cosh(nan) is nan assert cosh(zoo) is nan
assert cosh(oo) is oo assert cosh(-oo) is oo
assert cosh(0) == 1
assert unchanged(cosh, 1) assert cosh(-1) == cosh(1)
assert unchanged(cosh, x) assert cosh(-x) == cosh(x)
assert cosh(pi*I) == cos(pi) assert cosh(-pi*I) == cos(pi)
assert unchanged(cosh, 2**1024 * E) assert cosh(-2**1024 * E) == cosh(2**1024 * E)
assert cosh(pi*I/2) == 0 assert cosh(-pi*I/2) == 0 assert cosh((-3*10**73 + 1)*pi*I/2) == 0 assert cosh((7*10**103 + 1)*pi*I/2) == 0
assert cosh(pi*I) == -1 assert cosh(-pi*I) == -1 assert cosh(5*pi*I) == -1 assert cosh(8*pi*I) == 1
assert cosh(pi*I/3) == S.Half assert cosh(pi*I*Rational(-2, 3)) == Rational(-1, 2)
assert cosh(pi*I/4) == S.Half*sqrt(2) assert cosh(-pi*I/4) == S.Half*sqrt(2) assert cosh(pi*I*Rational(11, 4)) == Rational(-1, 2)*sqrt(2) assert cosh(pi*I*Rational(-3, 4)) == Rational(-1, 2)*sqrt(2)
assert cosh(pi*I/6) == S.Half*sqrt(3) assert cosh(-pi*I/6) == S.Half*sqrt(3) assert cosh(pi*I*Rational(7, 6)) == Rational(-1, 2)*sqrt(3) assert cosh(pi*I*Rational(-5, 6)) == Rational(-1, 2)*sqrt(3)
assert cosh(pi*I/105) == cos(pi/105) assert cosh(-pi*I/105) == cos(pi/105)
assert unchanged(cosh, 2 + 3*I)
assert cosh(x*I) == cos(x)
assert cosh(k*pi*I) == cos(k*pi) assert cosh(17*k*pi*I) == cos(17*k*pi)
assert unchanged(cosh, k*pi)
assert cosh(x).as_real_imag(deep=False) == (cos(im(x))*cosh(re(x)), sin(im(x))*sinh(re(x))) x = Symbol('x', extended_real=True) assert cosh(x).as_real_imag(deep=False) == (cosh(x), 0)
x = Symbol('x', real=True) assert cosh(I*x).is_finite is True assert cosh(I*x).is_real is True assert cosh(I*2 + 1).is_real is False assert cosh(5*I*S.Pi/2, evaluate=False).is_zero is True assert cosh(x).is_zero is False
def test_cosh_series(): x = Symbol('x') assert cosh(x).series(x, 0, 10) == \ 1 + x**2/2 + x**4/24 + x**6/720 + x**8/40320 + O(x**10)
def test_cosh_fdiff(): x = Symbol('x') raises(ArgumentIndexError, lambda: cosh(x).fdiff(2))
def test_tanh(): x, y = symbols('x,y')
k = Symbol('k', integer=True)
assert tanh(nan) is nan assert tanh(zoo) is nan
assert tanh(oo) == 1 assert tanh(-oo) == -1
assert tanh(0) == 0
assert unchanged(tanh, 1) assert tanh(-1) == -tanh(1)
assert unchanged(tanh, x) assert tanh(-x) == -tanh(x)
assert unchanged(tanh, pi) assert tanh(-pi) == -tanh(pi)
assert unchanged(tanh, 2**1024 * E) assert tanh(-2**1024 * E) == -tanh(2**1024 * E)
assert tanh(pi*I) == 0 assert tanh(-pi*I) == 0 assert tanh(2*pi*I) == 0 assert tanh(-2*pi*I) == 0 assert tanh(-3*10**73*pi*I) == 0 assert tanh(7*10**103*pi*I) == 0
assert tanh(pi*I/2) is zoo assert tanh(-pi*I/2) is zoo assert tanh(pi*I*Rational(5, 2)) is zoo assert tanh(pi*I*Rational(7, 2)) is zoo
assert tanh(pi*I/3) == sqrt(3)*I assert tanh(pi*I*Rational(-2, 3)) == sqrt(3)*I
assert tanh(pi*I/4) == I assert tanh(-pi*I/4) == -I assert tanh(pi*I*Rational(17, 4)) == I assert tanh(pi*I*Rational(-3, 4)) == I
assert tanh(pi*I/6) == I/sqrt(3) assert tanh(-pi*I/6) == -I/sqrt(3) assert tanh(pi*I*Rational(7, 6)) == I/sqrt(3) assert tanh(pi*I*Rational(-5, 6)) == I/sqrt(3)
assert tanh(pi*I/105) == tan(pi/105)*I assert tanh(-pi*I/105) == -tan(pi/105)*I
assert unchanged(tanh, 2 + 3*I)
assert tanh(x*I) == tan(x)*I
assert tanh(k*pi*I) == 0 assert tanh(17*k*pi*I) == 0
assert tanh(k*pi*I/2) == tan(k*pi/2)*I
assert tanh(x).as_real_imag(deep=False) == (sinh(re(x))*cosh(re(x))/(cos(im(x))**2 + sinh(re(x))**2), sin(im(x))*cos(im(x))/(cos(im(x))**2 + sinh(re(x))**2)) x = Symbol('x', extended_real=True) assert tanh(x).as_real_imag(deep=False) == (tanh(x), 0) assert tanh(I*pi/3 + 1).is_real is False assert tanh(x).is_real is True assert tanh(I*pi*x/2).is_real is None
def test_tanh_series(): x = Symbol('x') assert tanh(x).series(x, 0, 10) == \ x - x**3/3 + 2*x**5/15 - 17*x**7/315 + 62*x**9/2835 + O(x**10)
def test_tanh_fdiff(): x = Symbol('x') raises(ArgumentIndexError, lambda: tanh(x).fdiff(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_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)
def test_coth_fdiff(): x = Symbol('x') raises(ArgumentIndexError, lambda: coth(x).fdiff(2))
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_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_fdiff(): x = Symbol('x') raises(ArgumentIndexError, lambda: csch(x).fdiff(2))
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))
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_fdiff(): x = Symbol('x') raises(ArgumentIndexError, lambda: sech(x).fdiff(2))
def test_asinh(): x, y = symbols('x,y') assert unchanged(asinh, x) assert asinh(-x) == -asinh(x)
#at specific points assert asinh(nan) is nan assert asinh( 0) == 0 assert asinh(+1) == log(sqrt(2) + 1)
assert asinh(-1) == log(sqrt(2) - 1) assert asinh(I) == pi*I/2 assert asinh(-I) == -pi*I/2 assert asinh(I/2) == pi*I/6 assert asinh(-I/2) == -pi*I/6
# at infinites assert asinh(oo) is oo assert asinh(-oo) is -oo
assert asinh(I*oo) is oo assert asinh(-I *oo) is -oo
assert asinh(zoo) is zoo
#properties assert asinh(I *(sqrt(3) - 1)/(2**Rational(3, 2))) == pi*I/12 assert asinh(-I *(sqrt(3) - 1)/(2**Rational(3, 2))) == -pi*I/12
assert asinh(I*(sqrt(5) - 1)/4) == pi*I/10 assert asinh(-I*(sqrt(5) - 1)/4) == -pi*I/10
assert asinh(I*(sqrt(5) + 1)/4) == pi*I*Rational(3, 10) assert asinh(-I*(sqrt(5) + 1)/4) == pi*I*Rational(-3, 10)
# Symmetry assert asinh(Rational(-1, 2)) == -asinh(S.Half)
# inverse composition assert unchanged(asinh, sinh(Symbol('v1')))
assert asinh(sinh(0, evaluate=False)) == 0 assert asinh(sinh(-3, evaluate=False)) == -3 assert asinh(sinh(2, evaluate=False)) == 2 assert asinh(sinh(I, evaluate=False)) == I assert asinh(sinh(-I, evaluate=False)) == -I assert asinh(sinh(5*I, evaluate=False)) == -2*I*pi + 5*I assert asinh(sinh(15 + 11*I)) == 15 - 4*I*pi + 11*I assert asinh(sinh(-73 + 97*I)) == 73 - 97*I + 31*I*pi assert asinh(sinh(-7 - 23*I)) == 7 - 7*I*pi + 23*I assert asinh(sinh(13 - 3*I)) == -13 - I*pi + 3*I p = Symbol('p', positive=True) assert asinh(p).is_zero is False assert asinh(sinh(0, evaluate=False), evaluate=False).is_zero is True
def test_asinh_rewrite(): x = Symbol('x') assert asinh(x).rewrite(log) == log(x + sqrt(x**2 + 1))
def test_asinh_series(): x = Symbol('x') assert asinh(x).series(x, 0, 8) == \ x - x**3/6 + 3*x**5/40 - 5*x**7/112 + O(x**8) t5 = asinh(x).taylor_term(5, x) assert t5 == 3*x**5/40 assert asinh(x).taylor_term(7, x, t5, 0) == -5*x**7/112
def test_asinh_fdiff(): x = Symbol('x') raises(ArgumentIndexError, lambda: asinh(x).fdiff(2))
def test_acosh(): x = Symbol('x')
assert unchanged(acosh, -x)
#at specific points assert acosh(1) == 0 assert acosh(-1) == pi*I assert acosh(0) == I*pi/2 assert acosh(S.Half) == I*pi/3 assert acosh(Rational(-1, 2)) == pi*I*Rational(2, 3) assert acosh(nan) is nan
# at infinites assert acosh(oo) is oo assert acosh(-oo) is oo
assert acosh(I*oo) == oo + I*pi/2 assert acosh(-I*oo) == oo - I*pi/2
assert acosh(zoo) is zoo
assert acosh(I) == log(I*(1 + sqrt(2))) assert acosh(-I) == log(-I*(1 + sqrt(2))) assert acosh((sqrt(3) - 1)/(2*sqrt(2))) == pi*I*Rational(5, 12) assert acosh(-(sqrt(3) - 1)/(2*sqrt(2))) == pi*I*Rational(7, 12) assert acosh(sqrt(2)/2) == I*pi/4 assert acosh(-sqrt(2)/2) == I*pi*Rational(3, 4) assert acosh(sqrt(3)/2) == I*pi/6 assert acosh(-sqrt(3)/2) == I*pi*Rational(5, 6) assert acosh(sqrt(2 + sqrt(2))/2) == I*pi/8 assert acosh(-sqrt(2 + sqrt(2))/2) == I*pi*Rational(7, 8) assert acosh(sqrt(2 - sqrt(2))/2) == I*pi*Rational(3, 8) assert acosh(-sqrt(2 - sqrt(2))/2) == I*pi*Rational(5, 8) assert acosh((1 + sqrt(3))/(2*sqrt(2))) == I*pi/12 assert acosh(-(1 + sqrt(3))/(2*sqrt(2))) == I*pi*Rational(11, 12) assert acosh((sqrt(5) + 1)/4) == I*pi/5 assert acosh(-(sqrt(5) + 1)/4) == I*pi*Rational(4, 5)
assert str(acosh(5*I).n(6)) == '2.31244 + 1.5708*I' assert str(acosh(-5*I).n(6)) == '2.31244 - 1.5708*I'
# inverse composition assert unchanged(acosh, Symbol('v1'))
assert acosh(cosh(-3, evaluate=False)) == 3 assert acosh(cosh(3, evaluate=False)) == 3 assert acosh(cosh(0, evaluate=False)) == 0 assert acosh(cosh(I, evaluate=False)) == I assert acosh(cosh(-I, evaluate=False)) == I assert acosh(cosh(7*I, evaluate=False)) == -2*I*pi + 7*I assert acosh(cosh(1 + I)) == 1 + I assert acosh(cosh(3 - 3*I)) == 3 - 3*I assert acosh(cosh(-3 + 2*I)) == 3 - 2*I assert acosh(cosh(-5 - 17*I)) == 5 - 6*I*pi + 17*I assert acosh(cosh(-21 + 11*I)) == 21 - 11*I + 4*I*pi assert acosh(cosh(cosh(1) + I)) == cosh(1) + I assert acosh(1, evaluate=False).is_zero is True
def test_acosh_rewrite(): x = Symbol('x') assert acosh(x).rewrite(log) == log(x + sqrt(x - 1)*sqrt(x + 1))
def test_acosh_series(): x = Symbol('x') assert acosh(x).series(x, 0, 8) == \ -I*x + pi*I/2 - I*x**3/6 - 3*I*x**5/40 - 5*I*x**7/112 + O(x**8) t5 = acosh(x).taylor_term(5, x) assert t5 == - 3*I*x**5/40 assert acosh(x).taylor_term(7, x, t5, 0) == - 5*I*x**7/112
def test_acosh_fdiff(): x = Symbol('x') raises(ArgumentIndexError, lambda: acosh(x).fdiff(2))
def test_asech(): x = Symbol('x')
assert unchanged(asech, -x)
# values at fixed points assert asech(1) == 0 assert asech(-1) == pi*I assert asech(0) is oo assert asech(2) == I*pi/3 assert asech(-2) == 2*I*pi / 3 assert asech(nan) is nan
# at infinites assert asech(oo) == I*pi/2 assert asech(-oo) == I*pi/2 assert asech(zoo) == I*AccumBounds(-pi/2, pi/2)
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(2) == acosh(S.Half)
# 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(Rational(-1, 2)) 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 test_asech_series(): x = Symbol('x') t6 = asech(x).expansion_term(6, x) assert t6 == -5*x**6/96 assert asech(x).expansion_term(8, x, t6, 0) == -35*x**8/1024
def test_asech_rewrite(): x = Symbol('x') assert asech(x).rewrite(log) == log(1/x + sqrt(1/x - 1) * sqrt(1/x + 1))
def test_asech_fdiff(): x = Symbol('x') raises(ArgumentIndexError, lambda: asech(x).fdiff(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_acsch_infinities(): assert acsch(oo) == 0 assert acsch(-oo) == 0 assert acsch(zoo) == 0
def test_acsch_rewrite(): x = Symbol('x') assert acsch(x).rewrite(log) == log(1/x + sqrt(1/x**2 + 1))
def test_acsch_fdiff(): x = Symbol('x') raises(ArgumentIndexError, lambda: acsch(x).fdiff(2))
def test_atanh(): x = Symbol('x')
#at specific points assert atanh(0) == 0 assert atanh(I) == I*pi/4 assert atanh(-I) == -I*pi/4 assert atanh(1) is oo assert atanh(-1) is -oo assert atanh(nan) is nan
# at infinites assert atanh(oo) == -I*pi/2 assert atanh(-oo) == I*pi/2
assert atanh(I*oo) == I*pi/2 assert atanh(-I*oo) == -I*pi/2
assert atanh(zoo) == I*AccumBounds(-pi/2, pi/2)
#properties assert atanh(-x) == -atanh(x)
assert atanh(I/sqrt(3)) == I*pi/6 assert atanh(-I/sqrt(3)) == -I*pi/6 assert atanh(I*sqrt(3)) == I*pi/3 assert atanh(-I*sqrt(3)) == -I*pi/3 assert atanh(I*(1 + sqrt(2))) == pi*I*Rational(3, 8) assert atanh(I*(sqrt(2) - 1)) == pi*I/8 assert atanh(I*(1 - sqrt(2))) == -pi*I/8 assert atanh(-I*(1 + sqrt(2))) == pi*I*Rational(-3, 8) assert atanh(I*sqrt(5 + 2*sqrt(5))) == I*pi*Rational(2, 5) assert atanh(-I*sqrt(5 + 2*sqrt(5))) == I*pi*Rational(-2, 5) assert atanh(I*(2 - sqrt(3))) == pi*I/12 assert atanh(I*(sqrt(3) - 2)) == -pi*I/12 assert atanh(oo) == -I*pi/2
# Symmetry assert atanh(Rational(-1, 2)) == -atanh(S.Half)
# inverse composition assert unchanged(atanh, tanh(Symbol('v1')))
assert atanh(tanh(-5, evaluate=False)) == -5 assert atanh(tanh(0, evaluate=False)) == 0 assert atanh(tanh(7, evaluate=False)) == 7 assert atanh(tanh(I, evaluate=False)) == I assert atanh(tanh(-I, evaluate=False)) == -I assert atanh(tanh(-11*I, evaluate=False)) == -11*I + 4*I*pi assert atanh(tanh(3 + I)) == 3 + I assert atanh(tanh(4 + 5*I)) == 4 - 2*I*pi + 5*I assert atanh(tanh(pi/2)) == pi/2 assert atanh(tanh(pi)) == pi assert atanh(tanh(-3 + 7*I)) == -3 - 2*I*pi + 7*I assert atanh(tanh(9 - I*2/3)) == 9 - I*2/3 assert atanh(tanh(-32 - 123*I)) == -32 - 123*I + 39*I*pi
def test_atanh_rewrite(): x = Symbol('x') assert atanh(x).rewrite(log) == (log(1 + x) - log(1 - x)) / 2
def test_atanh_series(): x = Symbol('x') assert atanh(x).series(x, 0, 10) == \ x + x**3/3 + x**5/5 + x**7/7 + x**9/9 + O(x**10)
def test_atanh_fdiff(): x = Symbol('x') raises(ArgumentIndexError, lambda: atanh(x).fdiff(2))
def test_acoth(): x = Symbol('x')
#at specific points assert acoth(0) == I*pi/2 assert acoth(I) == -I*pi/4 assert acoth(-I) == I*pi/4 assert acoth(1) is oo assert acoth(-1) is -oo assert acoth(nan) is nan
# at infinites assert acoth(oo) == 0 assert acoth(-oo) == 0 assert acoth(I*oo) == 0 assert acoth(-I*oo) == 0 assert acoth(zoo) == 0
#properties assert acoth(-x) == -acoth(x)
assert acoth(I/sqrt(3)) == -I*pi/3 assert acoth(-I/sqrt(3)) == I*pi/3 assert acoth(I*sqrt(3)) == -I*pi/6 assert acoth(-I*sqrt(3)) == I*pi/6 assert acoth(I*(1 + sqrt(2))) == -pi*I/8 assert acoth(-I*(sqrt(2) + 1)) == pi*I/8 assert acoth(I*(1 - sqrt(2))) == pi*I*Rational(3, 8) assert acoth(I*(sqrt(2) - 1)) == pi*I*Rational(-3, 8) assert acoth(I*sqrt(5 + 2*sqrt(5))) == -I*pi/10 assert acoth(-I*sqrt(5 + 2*sqrt(5))) == I*pi/10 assert acoth(I*(2 + sqrt(3))) == -pi*I/12 assert acoth(-I*(2 + sqrt(3))) == pi*I/12 assert acoth(I*(2 - sqrt(3))) == pi*I*Rational(-5, 12) assert acoth(I*(sqrt(3) - 2)) == pi*I*Rational(5, 12)
# Symmetry assert acoth(Rational(-1, 2)) == -acoth(S.Half)
def test_acoth_rewrite(): x = Symbol('x') assert acoth(x).rewrite(log) == (log(1 + 1/x) - log(1 - 1/x)) / 2
def test_acoth_series(): x = Symbol('x') assert acoth(x).series(x, 0, 10) == \ I*pi/2 + x + x**3/3 + x**5/5 + x**7/7 + x**9/9 + O(x**10)
def test_acoth_fdiff(): x = Symbol('x') raises(ArgumentIndexError, lambda: acoth(x).fdiff(2))
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_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 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_complex_2899(): a, b = symbols('a,b', real=True) for deep in [True, False]: for func in [sinh, cosh, tanh, coth]: assert func(a).expand(complex=True, deep=deep) == func(a)
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_issue_4136(): assert cosh(asinh(Integer(3)/2)) == sqrt(Integer(13)/4)
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_tanh_rewrite(): x = Symbol('x') assert tanh(x).rewrite(exp) == (exp(x) - exp(-x))/(exp(x) + exp(-x)) \ == tanh(x).rewrite('tractable') assert tanh(x).rewrite(sinh) == I*sinh(x)/sinh(I*pi/2 - x) assert tanh(x).rewrite(cosh) == I*cosh(I*pi/2 - x)/cosh(x) assert tanh(x).rewrite(coth) == 1/coth(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_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_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) 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_sinh_expansion(): x, y = symbols('x,y') assert sinh(x+y).expand(trig=True) == sinh(x)*cosh(y) + cosh(x)*sinh(y) assert sinh(2*x).expand(trig=True) == 2*sinh(x)*cosh(x) assert sinh(3*x).expand(trig=True).expand() == \ sinh(x)**3 + 3*sinh(x)*cosh(x)**2
def test_cosh_expansion(): x, y = symbols('x,y') assert cosh(x+y).expand(trig=True) == cosh(x)*cosh(y) + sinh(x)*sinh(y) assert cosh(2*x).expand(trig=True) == cosh(x)**2 + sinh(x)**2 assert cosh(3*x).expand(trig=True).expand() == \ 3*sinh(x)**2*cosh(x) + cosh(x)**3
def test_cosh_positive(): # See issue 11721 # cosh(x) is positive for real values of x k = symbols('k', real=True) n = symbols('n', integer=True)
assert cosh(k, evaluate=False).is_positive is True assert cosh(k + 2*n*pi*I, evaluate=False).is_positive is True assert cosh(I*pi/4, evaluate=False).is_positive is True assert cosh(3*I*pi/4, evaluate=False).is_positive is False
def test_cosh_nonnegative(): k = symbols('k', real=True) n = symbols('n', integer=True)
assert cosh(k, evaluate=False).is_nonnegative is True assert cosh(k + 2*n*pi*I, evaluate=False).is_nonnegative is True assert cosh(I*pi/4, evaluate=False).is_nonnegative is True assert cosh(3*I*pi/4, evaluate=False).is_nonnegative is False assert cosh(S.Zero, evaluate=False).is_nonnegative is True
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_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
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