Hyperbolic functions

Hyperbolic functions

cosh()

mpmath.cosh(x, **kwargs)

Computes the hyperbolic cosine of \(x\), \(\cosh(x) = (e^x + e^{-x})/2\). Values and limits include:

>>> from sympy.mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> cosh(0)
1.0
>>> cosh(1)
1.543080634815243778477906
>>> cosh(-inf), cosh(+inf)
(+inf, +inf)

The hyperbolic cosine is an even, convex function with a global minimum at \(x = 0\), having a Maclaurin series that starts:

>>> nprint(chop(taylor(cosh, 0, 5)))
[1.0, 0.0, 0.5, 0.0, 0.0416667, 0.0]

Generalized to complex numbers, the hyperbolic cosine is equivalent to a cosine with the argument rotated in the imaginary direction, or \(\cosh x = \cos ix\):

>>> cosh(2+3j)
(-3.724545504915322565473971 + 0.5118225699873846088344638j)
>>> cos(3-2j)
(-3.724545504915322565473971 + 0.5118225699873846088344638j)

sinh()

mpmath.sinh(x, **kwargs)

Computes the hyperbolic sine of \(x\), \(\sinh(x) = (e^x - e^{-x})/2\). Values and limits include:

>>> from sympy.mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> sinh(0)
0.0
>>> sinh(1)
1.175201193643801456882382
>>> sinh(-inf), sinh(+inf)
(-inf, +inf)

The hyperbolic sine is an odd function, with a Maclaurin series that starts:

>>> nprint(chop(taylor(sinh, 0, 5)))
[0.0, 1.0, 0.0, 0.166667, 0.0, 0.00833333]

Generalized to complex numbers, the hyperbolic sine is essentially a sine with a rotation \(i\) applied to the argument; more precisely, \(\sinh x = -i \sin ix\):

>>> sinh(2+3j)
(-3.590564589985779952012565 + 0.5309210862485198052670401j)
>>> j*sin(3-2j)
(-3.590564589985779952012565 + 0.5309210862485198052670401j)

tanh()

mpmath.tanh(x, **kwargs)

Computes the hyperbolic tangent of \(x\), \(\tanh(x) = \sinh(x)/\cosh(x)\). Values and limits include:

>>> from sympy.mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> tanh(0)
0.0
>>> tanh(1)
0.7615941559557648881194583
>>> tanh(-inf), tanh(inf)
(-1.0, 1.0)

The hyperbolic tangent is an odd, sigmoidal function, similar to the inverse tangent and error function. Its Maclaurin series is:

>>> nprint(chop(taylor(tanh, 0, 5)))
[0.0, 1.0, 0.0, -0.333333, 0.0, 0.133333]

Generalized to complex numbers, the hyperbolic tangent is essentially a tangent with a rotation \(i\) applied to the argument; more precisely, \(\tanh x = -i \tan ix\):

>>> tanh(2+3j)
(0.9653858790221331242784803 - 0.009884375038322493720314034j)
>>> j*tan(3-2j)
(0.9653858790221331242784803 - 0.009884375038322493720314034j)

sech()

mpmath.sech(x)

Computes the hyperbolic secant of \(x\), \(\mathrm{sech}(x) = \frac{1}{\cosh(x)}\).

csch()

mpmath.csch(x)

Computes the hyperbolic cosecant of \(x\), \(\mathrm{csch}(x) = \frac{1}{\sinh(x)}\).

coth()

mpmath.coth(x)

Computes the hyperbolic cotangent of \(x\), \(\mathrm{coth}(x) = \frac{\cosh(x)}{\sinh(x)}\).

Inverse hyperbolic functions

acosh()

mpmath.acosh(x, **kwargs)

Computes the inverse hyperbolic cosine of \(x\), \(\mathrm{cosh}^{-1}(x) = \log(x+\sqrt{x+1}\sqrt{x-1})\).

asinh()

mpmath.asinh(x, **kwargs)

Computes the inverse hyperbolic sine of \(x\), \(\mathrm{sinh}^{-1}(x) = \log(x+\sqrt{1+x^2})\).

atanh()

mpmath.atanh(x, **kwargs)

Computes the inverse hyperbolic tangent of \(x\), \(\mathrm{tanh}^{-1}(x) = \frac{1}{2}\left(\log(1+x)-\log(1-x)\right)\).

asech()

mpmath.asech(x)

Computes the inverse hyperbolic secant of \(x\), \(\mathrm{sech}^{-1}(x) = \cosh^{-1}(1/x)\).

acsch()

mpmath.acsch(x)

Computes the inverse hyperbolic cosecant of \(x\), \(\mathrm{csch}^{-1}(x) = \sinh^{-1}(1/x)\).

acoth()

mpmath.acoth(x)

Computes the inverse hyperbolic cotangent of \(x\), \(\mathrm{coth}^{-1}(x) = \tanh^{-1}(1/x)\).