Hyperbolic functions

Hyperbolic functions#

Cardinal hyperbolic sine, \(\mathrm{sinhc}(x) = \sinh(x)/x\)#

math53.sinhc(x)#

Returns \(\mathrm{sinhc}(x) = \sinh(x)/x\), accurate also for \(x\) near 0.

An example in Python

>>> from xlcalcnet import ereal
>>> ereal.Sinhc(0.5)
ereal('5.2359877559829887307E-1')
>>> ereal.Sinhc('0.51')
ereal('5.3518479027559984754E-1')

Auxiliary function, \(\mathrm{sinhmx}(x) = \sinh(x)-x\)#

math53.sinhmx(x)#

Returns sinh(x)-x, accurate also for \(x\) near 0.

An example in Python

>>> from xlcalcnet import ereal
>>> ereal.Sinhmx(0.5)
ereal('5.2359877559829887307E-1')
>>> ereal.Sinhmx('0.51')
ereal('5.3518479027559984754E-1')

Auxiliary function, \(\mathrm{coshm1}(x) = \cosh(x)-1\)#

math53.coshm1(x)#

Returns \(\cosh(x)-1\), accurate also for \(x\) near 0.

An example in Python

>>> from xlcalcnet import ereal
>>> ereal.Coshm1(0.5)
ereal('5.2359877559829887307E-1')
>>> ereal.Coshm1('0.51')
ereal('5.3518479027559984754E-1')

Auxiliary function \(\mathrm{acosh}(1+x)\)#

math53.acosh1p(z)#

Returns \(\mathrm{acosh}(1+x), x \ge 0\), accurate also for \(x\) near 0.

An example in Python

>>> from xlcalcnet import ereal
>>> ereal.Acosh1p(0.5)
ereal('5.2359877559829887307E-1')
>>> ereal.Acosh1p('0.51')
ereal('5.3518479027559984754E-1')

Auxiliary function \(\log(\cosh(x))\)#

math53.logcosh(x)#

Returns ln(cosh(x)), accurate for x ~ 0 and without overflow for large x

An example in Python

>>> from xlcalcnet import ereal
>>> ereal.Logcosh(0.5)
ereal('5.2359877559829887307E-1')
>>> ereal.Logcosh('0.51')
ereal('5.3518479027559984754E-1')

Auxiliary function \(\log(\sinh(x))\)#

math53.logsinh(x)#

Returns ln(sinh(x)), x > 0, accurate for x ~ 0 and without overflow for large x

An example in Python

>>> from xlcalcnet import ereal
>>> ereal.Logsinh(0.5)
ereal('5.2359877559829887307E-1')
>>> ereal.Logsinh('0.51')
ereal('5.3518479027559984754E-1')

Gudermannian function \(\mathrm{gd}(x) = \mathrm{asin}(\mathrm{tanh}(x))\)#

math53.gd(x)#

Returns the Gudermannian function \(\mathrm{gd}(x) = \mathrm{asin}(\mathrm{tanh}(x))\). See also Wikipedia [1345], MathWorld [983].

An example in Python

>>> from xlcalcnet import ereal
>>> ereal.Gudermann(0.5)
ereal('5.2359877559829887307E-1')
>>> ereal.Gudermann('0.51')
ereal('5.3518479027559984754E-1')

Inverse Gudermannian function \(\mathrm{arcgd}(x) = \mathrm{atanh}(\sin(x))\)#

math53.arcgd(z)#

Returns the inverse Gudermannian function \(\mathrm{arcgd}(x) = \mathrm{atanh}(\sin(x)), |x| < \pi/2.\)

See also Wikipedia [1347], MathWorld [987].

An example in Python

>>> from xlcalcnet import ereal
>>> ereal.ArcGd(0.5)
ereal('5.2359877559829887307E-1')
>>> ereal.ArcGd('0.51')
ereal('5.3518479027559984754E-1')

Langevin function, \(L(x)\)#

math53.langevin_l(x)#

Returns the Langevin function \(L(x)\), defined as \(L(x) = \coth(x) - 1/x\) for \(x \ne 0\), and \(L(0) = 0\) for \(x = 0\).

See also Ehrhardt [297] (3.10.16).

https://en.wikipedia.org/wiki/Brillouin_and_Langevin_functions

An example in Python

>>> from xlcalcnet import ereal
>>> ereal.LangevinL(0.2)
ereal('5.2359877559829887307E-1')
>>> ereal.LangevinL(0.21)
ereal('5.3518479027559984754E-1')

Inverse Langevin function, \(L^{-1}(x)\)#

math53.langevin_l_inv(x)#

Returns the functional inverse \(L^{-1}\) of the Langevin function, i.e. \(L(L^{-1}(x))= x\), \(|x| < 1\).

See also Ehrhardt [297] (3.10.17).

https://en.wikipedia.org/wiki/Brillouin_and_Langevin_functions

An example in Python

>>> from xlcalcnet import ereal
>>> ereal.LangevinLInv(0.2)
ereal('5.2359877559829887307E-1')
>>> ereal.LangevinLInv(0.21)
ereal('5.3518479027559984754E-1')

Solutions of Kepler’s equation, \(\mathrm{kepler}(M,e)\)#

math53.kepler(M, e)#

Returns the solutions (eccentric anomaly \(x\)) of Kepler’s equation from the mean anomaly \(M\) and the eccentricity \(e\), more precisely the solutions \(x\) of

\[\begin{split}M =\begin{cases} x - e \sin(x), & e<1,\\ x+x^3/3, & e=1 \text{ (Barker's equation)},\\ e \sinh(x)-x, & e>1. \end{cases}\end{split}\]

See also Ehrhardt [297] (3.10.23).

An example in Python

>>> from xlcalcnet import ereal
>>> ereal.Kepler(3, 0.44)
ereal('5.2359877559829887307E-1')
>>> ereal.Kepler(3, 0.14404)
ereal('5.3518479027559984754E-1')
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