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 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 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 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 $x$, $\coth (x)=\frac{\cosh (x)}{\sinh (x)}$.

Inverse hyperbolic functions

acosh()

mpmath.acosh(x, **kwargs)

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

asinh()

mpmath.asinh(x, **kwargs)

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

atanh()

mpmath.atanh(x, **kwargs)

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

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 $x$, ${\coth }^{-1}(x)={\tanh }^{-1}(1/x)$.