The inverse of the hyperbolic cosine function.
EXAMPLES:
sage: arccosh(1/2)
arccosh(1/2)
sage: arccosh(1 + I*1.0)
1.06127506190504 + 0.904556894302381*I
sage: float(arccosh(2))
1.3169578969248168
sage: cosh(float(arccosh(2)))
2.0
Warning: If the input is real the output will be real or NaN:
sage: arccosh(0.5)
NaN
But evaluation where the input is in the complex field yields a complex output:
sage: arccosh(CC(0.5))
1.04719755119660*I
TESTS:
sage: arccosh(x).operator()
arccosh
The inverse of the hyperbolic cotangent function.
EXAMPLES:
sage: arccoth(2.0)
0.549306144334055
sage: arccoth(2)
arccoth(2)
sage: arccoth(1 + I*1.0)
0.402359478108525 - 0.553574358897045*I
EXAMPLES:
sage: arccoth(2).n(200)
0.54930614433405484569762261846126285232374527891137472586735
The inverse of the hyperbolic cosecant function.
EXAMPLES:
sage: arccsch(2.0)
0.481211825059603
sage: arccsch(2)
arccsch(2)
sage: arccsch(1 + I*1.0)
0.530637530952518 - 0.452278447151191*I
EXAMPLES:
sage: arccsch(1).n(200)
0.88137358701954302523260932497979230902816032826163541075330
The inverse of the hyperbolic secant function.
EXAMPLES:
sage: arcsech(0.5)
1.31695789692482
sage: arcsech(1/2)
arcsech(1/2)
sage: arcsech(1 + I*1.0)
-0.530637530952518 + 1.11851787964371*I
EXAMPLES:
sage: arcsech(1/2).n(200)
1.3169578969248167086250463473079684440269819714675164797685
The inverse of the hyperbolic sine function.
EXAMPLES:
sage: arcsinh
arcsinh
sage: arcsinh(0.5)
0.481211825059603
sage: arcsinh(1/2)
arcsinh(1/2)
sage: arcsinh(1 + I*1.0)
1.06127506190504 + 0.666239432492515*I
sage: arcsinh._approx_(0.5)
0.48121182505960347
TESTS:
sage: arcsinh(x).operator()
arcsinh
The inverse of the hyperbolic tangent function.
EXAMPLES:
sage: arctanh(0.5)
0.549306144334055
sage: arctanh(1/2)
arctanh(1/2)
sage: arctanh(1 + I*1.0)
0.402359478108525 + 1.01722196789785*I
TESTS:
sage: arctanh(x).operator()
arctanh
The hyperbolic cosine function.
EXAMPLES:
sage: cosh(pi)
cosh(pi)
sage: cosh(3.1415)
11.5908832931176
sage: float(cosh(pi))
11.591953275521519
sage: RR(cosh(1/2))
1.12762596520638
The hyperbolic cotangent function.
EXAMPLES:
sage: coth(pi)
coth(pi)
sage: coth(3.1415)
1.00374256795520
sage: float(coth(pi))
1.0037418731973213
sage: RR(coth(pi))
1.00374187319732
The hyperbolic cosecant function.
EXAMPLES:
sage: csch(pi)
csch(pi)
sage: csch(3.1415)
0.0865975907592133
sage: float(csch(pi))
0.0865895375300469...
sage: RR(csch(pi))
0.0865895375300470
The hyperbolic secant function.
EXAMPLES:
sage: sech(pi)
sech(pi)
sage: sech(3.1415)
0.0862747018248192
sage: float(sech(pi))
0.086266738334054419
sage: RR(sech(pi))
0.0862667383340544
The hyperbolic sine function.
EXAMPLES:
sage: sinh(pi)
sinh(pi)
sage: sinh(3.1415)
11.5476653707437
sage: float(sinh(pi))
11.54873935725774...
sage: RR(sinh(pi))
11.5487393572577
The hyperbolic tangent function.
EXAMPLES:
sage: tanh(pi)
tanh(pi)
sage: tanh(3.1415)
0.996271386633702
sage: float(tanh(pi))
0.996272076220749...
sage: tan(3.1415/4)
0.999953674278156
sage: tanh(pi/4)
tanh(1/4*pi)
sage: RR(tanh(1/2))
0.462117157260010
sage: CC(tanh(pi + I*e))
0.997524731976164 - 0.00279068768100315*I
sage: ComplexField(100)(tanh(pi + I*e))
0.99752473197616361034204366446 - 0.0027906876810031453884245163923*I
sage: CDF(tanh(pi + I*e))
0.997524731976 - 0.002790687681*I