Hyperbolic functions

The full set of hyperbolic and inverse hyperbolic functions is available:

• hyperbolic sine: sinh()

• hyperbolic cosine: cosh()

• hyperbolic tangent: tanh()

• hyperbolic cotangent: coth()

• hyperbolic secant: sech()

• hyperbolic cosecant: csch()

• inverse hyperbolic sine: asinh()

• inverse hyperbolic cosine: acosh()

• inverse hyperbolic tangent: atanh()

• inverse hyperbolic cotangent: acoth()

• inverse hyperbolic secant: asech()

• inverse hyperbolic cosecant: acsch()

REFERENCES:

• Wikipedia article Hyperbolic function

• Wikipedia article Inverse hyperbolic functions

• 18. Roy, F. W. J. Olver, Elementary Functions, https://dlmf.nist.gov/4

EXAMPLES:

Inverse hyperbolic functions have logarithmic expressions, so expressions of the form exp(c*f(x)) simplify:

sage: exp(2*atanh(x))
-(x + 1)/(x - 1)
sage: exp(2*acoth(x))
(x + 1)/(x - 1)

sage: exp(2*asinh(x))
(x + sqrt(x^2 + 1))^2
sage: exp(2*acosh(x))
(x + sqrt(x^2 - 1))^2

sage: exp(2*asech(x))
(sqrt(-x^2 + 1)/x + 1/x)^2
sage: exp(2*acsch(x))
(sqrt(1/x^2 + 1) + 1/x)^2

class sage.functions.hyperbolic.Function_arccosh

Bases: GinacFunction

The inverse of the hyperbolic cosine function.

EXAMPLES:

sage: acosh(1/2)
arccosh(1/2)
sage: acosh(1 + I*1.0)
1.06127506190504 + 0.904556894302381*I
sage: float(acosh(2))
1.3169578969248168
sage: cosh(float(acosh(2)))
2.0
sage: acosh(complex(1, 2))  # abs tol 1e-15
(1.5285709194809982+1.1437177404024204j)

Warning

If the input is in the complex field or symbolic (which includes rational and integer input), the output will be complex. However, if the input is a real decimal, the output will be real or \(NaN\). See the examples for details.

sage: acosh(0.5)
NaN
sage: acosh(1/2)
arccosh(1/2)
sage: acosh(1/2).n()
NaN
sage: acosh(CC(0.5))
1.04719755119660*I
sage: acosh(0)
1/2*I*pi
sage: acosh(-1)
I*pi

To prevent automatic evaluation use the hold argument:

sage: acosh(-1,hold=True)
arccosh(-1)

To then evaluate again, use the unhold method:

sage: acosh(-1,hold=True).unhold()
I*pi

conjugate(arccosh(x))==arccosh(conjugate(x)) unless on the branch cut which runs along the real axis from +1 to -inf.:

sage: conjugate(acosh(x))
conjugate(arccosh(x))
sage: var('y', domain='positive')
y
sage: conjugate(acosh(y))
conjugate(arccosh(y))
sage: conjugate(acosh(y+I))
conjugate(arccosh(y + I))
sage: conjugate(acosh(1/16))
conjugate(arccosh(1/16))
sage: conjugate(acosh(2))
arccosh(2)
sage: conjugate(acosh(I/2))
arccosh(-1/2*I)

class sage.functions.hyperbolic.Function_arccoth

Bases: GinacFunction

The inverse of the hyperbolic cotangent function.

EXAMPLES:

sage: acoth(2.0)
0.549306144334055
sage: acoth(2)
1/2*log(3)
sage: acoth(1 + I*1.0)
0.402359478108525 - 0.553574358897045*I
sage: acoth(2).n(200)
0.54930614433405484569762261846126285232374527891137472586735

sage: bool(diff(acoth(x), x) == diff(atanh(x), x))
True
sage: diff(acoth(x), x)
-1/(x^2 - 1)

sage: float(acoth(2))
0.5493061443340549
sage: float(acoth(2).n(53))   # Correct result to 53 bits
0.5493061443340549
sage: float(acoth(2).n(100))  # Compute 100 bits and then round to 53
0.5493061443340549

class sage.functions.hyperbolic.Function_arccsch

Bases: GinacFunction

The inverse of the hyperbolic cosecant function.

EXAMPLES:

sage: acsch(2.0)
0.481211825059603
sage: acsch(2)
arccsch(2)
sage: acsch(1 + I*1.0)
0.530637530952518 - 0.452278447151191*I
sage: acsch(1).n(200)
0.88137358701954302523260932497979230902816032826163541075330
sage: float(acsch(1))
0.881373587019543

sage: diff(acsch(x), x)
-1/(sqrt(x^2 + 1)*x)
sage: latex(acsch(x))
\operatorname{arcsch}\left(x\right)

class sage.functions.hyperbolic.Function_arcsech

Bases: GinacFunction

The inverse of the hyperbolic secant function.

EXAMPLES:

sage: asech(0.5)
1.31695789692482
sage: asech(1/2)
arcsech(1/2)
sage: asech(1 + I*1.0)
0.530637530952518 - 1.11851787964371*I
sage: asech(1/2).n(200)
1.3169578969248167086250463473079684440269819714675164797685
sage: float(asech(1/2))
1.3169578969248168

sage: diff(asech(x), x)
-1/(sqrt(-x^2 + 1)*x)
sage: latex(asech(x))
\operatorname{arsech}\left(x\right)
sage: asech(x)._sympy_()
asech(x)

class sage.functions.hyperbolic.Function_arcsinh

Bases: GinacFunction

The inverse of the hyperbolic sine function.

EXAMPLES:

sage: asinh
arcsinh
sage: asinh(0.5)
0.481211825059603
sage: asinh(1/2)
arcsinh(1/2)
sage: asinh(1 + I*1.0)
1.06127506190504 + 0.666239432492515*I

To prevent automatic evaluation use the hold argument:

sage: asinh(-2,hold=True)
arcsinh(-2)

To then evaluate again, use the unhold method:

sage: asinh(-2,hold=True).unhold()
-arcsinh(2)

conjugate(asinh(x))==asinh(conjugate(x)) unless on the branch cuts which run along the imaginary axis outside the interval [-I, +I].:

sage: conjugate(asinh(x))
conjugate(arcsinh(x))
sage: var('y', domain='positive')
y
sage: conjugate(asinh(y))
arcsinh(y)
sage: conjugate(asinh(y+I))
conjugate(arcsinh(y + I))
sage: conjugate(asinh(1/16))
arcsinh(1/16)
sage: conjugate(asinh(I/2))
arcsinh(-1/2*I)
sage: conjugate(asinh(2*I))
conjugate(arcsinh(2*I))

class sage.functions.hyperbolic.Function_arctanh

Bases: GinacFunction

The inverse of the hyperbolic tangent function.

EXAMPLES:

sage: atanh(0.5)
0.549306144334055
sage: atanh(1/2)
1/2*log(3)
sage: atanh(1 + I*1.0)
0.402359478108525 + 1.01722196789785*I

To prevent automatic evaluation use the hold argument:

sage: atanh(-1/2,hold=True)
arctanh(-1/2)

To then evaluate again, use the unhold method:

sage: atanh(-1/2,hold=True).unhold()
-1/2*log(3)

conjugate(arctanh(x)) == arctanh(conjugate(x)) unless on the branch cuts which run along the real axis outside the interval [-1, +1].

sage: conjugate(atanh(x))
conjugate(arctanh(x))
sage: var('y', domain='positive')
y
sage: conjugate(atanh(y))
conjugate(arctanh(y))
sage: conjugate(atanh(y+I))
conjugate(arctanh(y + I))
sage: conjugate(atanh(1/16))
1/2*log(17/15)
sage: conjugate(atanh(I/2))
arctanh(-1/2*I)
sage: conjugate(atanh(-2*I))
arctanh(2*I)

class sage.functions.hyperbolic.Function_cosh

Bases: GinacFunction

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

sage: latex(cosh(x))
\cosh\left(x\right)
sage: cosh(x)._sympy_()
cosh(x)

To prevent automatic evaluation, use the hold parameter:

sage: cosh(arcsinh(x),hold=True)
cosh(arcsinh(x))

To then evaluate again, use the unhold method:

sage: cosh(arcsinh(x),hold=True).unhold()
sqrt(x^2 + 1)

class sage.functions.hyperbolic.Function_coth

Bases: GinacFunction

The hyperbolic cotangent function.

EXAMPLES:

sage: coth(pi)
coth(pi)
sage: coth(0)
Infinity
sage: coth(pi*I)
Infinity
sage: coth(pi*I/2)
0
sage: coth(7*pi*I/2)
0
sage: coth(8*pi*I/2)
Infinity
sage: coth(7.*pi*I/2)
-I*cot(3.50000000000000*pi)
sage: coth(3.1415)
1.00374256795520
sage: float(coth(pi))
1.0037418731973213
sage: RR(coth(pi))
1.00374187319732
sage: coth(complex(1, 2))  # abs tol 1e-15
(0.8213297974938518+0.17138361290918508j)

sage: bool(diff(coth(x), x) == diff(1/tanh(x), x))
True
sage: diff(coth(x), x)
-1/sinh(x)^2
sage: latex(coth(x))
\coth\left(x\right)
sage: coth(x)._sympy_()
coth(x)

class sage.functions.hyperbolic.Function_csch

Bases: GinacFunction

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
sage: csch(0)
Infinity
sage: csch(pi*I)
Infinity
sage: csch(pi*I/2)
-I
sage: csch(7*pi*I/2)
I
sage: csch(7.*pi*I/2)
-I*csc(3.50000000000000*pi)

sage: bool(diff(csch(x), x) == diff(1/sinh(x), x))
True
sage: diff(csch(x), x)
-coth(x)*csch(x)
sage: latex(csch(x))
\operatorname{csch}\left(x\right)
sage: csch(x)._sympy_()
csch(x)

class sage.functions.hyperbolic.Function_sech

Bases: GinacFunction

The hyperbolic secant function.

EXAMPLES:

sage: sech(pi)
sech(pi)
sage: sech(3.1415)
0.0862747018248192
sage: float(sech(pi))
0.0862667383340544...
sage: RR(sech(pi))
0.0862667383340544
sage: sech(0)
1
sage: sech(pi*I)
-1
sage: sech(pi*I/2)
Infinity
sage: sech(7*pi*I/2)
Infinity
sage: sech(8*pi*I/2)
1
sage: sech(8.*pi*I/2)
sec(4.00000000000000*pi)

sage: bool(diff(sech(x), x) == diff(1/cosh(x), x))
True
sage: diff(sech(x), x)
-sech(x)*tanh(x)
sage: latex(sech(x))
\operatorname{sech}\left(x\right)
sage: sech(x)._sympy_()
sech(x)

class sage.functions.hyperbolic.Function_sinh

Bases: GinacFunction

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

sage: latex(sinh(x))
\sinh\left(x\right)
sage: sinh(x)._sympy_()
sinh(x)

To prevent automatic evaluation, use the hold parameter:

sage: sinh(arccosh(x),hold=True)
sinh(arccosh(x))

To then evaluate again, use the unhold method:

sage: sinh(arccosh(x),hold=True).unhold()
sqrt(x + 1)*sqrt(x - 1)

class sage.functions.hyperbolic.Function_tanh

Bases: GinacFunction

The hyperbolic tangent function.

EXAMPLES:

sage: tanh(pi)
tanh(pi)
sage: tanh(3.1415)
0.996271386633702
sage: float(tanh(pi))
0.99627207622075
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))  # rel tol 2e-15
0.9975247319761636 - 0.002790687681003147*I

To prevent automatic evaluation, use the hold parameter:

sage: tanh(arcsinh(x),hold=True)
tanh(arcsinh(x))

To then evaluate again, use the unhold method:

sage: tanh(arcsinh(x),hold=True).unhold()
x/sqrt(x^2 + 1)