Hyperbolic functions¶
Hyperbolic functions¶
cosh()
¶
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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()
¶
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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()
¶
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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()
¶
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mpmath.
sech
(x)¶ Computes the hyperbolic secant of \(x\), \(\mathrm{sech}(x) = \frac{1}{\cosh(x)}\).
Inverse hyperbolic functions¶
acosh()
¶
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mpmath.
acosh
(x, **kwargs)¶ Computes the inverse hyperbolic cosine of \(x\), \(\mathrm{cosh}^{-1}(x) = \log(x+\sqrt{x+1}\sqrt{x-1})\).
asinh()
¶
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mpmath.
asinh
(x, **kwargs)¶ Computes the inverse hyperbolic sine of \(x\), \(\mathrm{sinh}^{-1}(x) = \log(x+\sqrt{1+x^2})\).
atanh()
¶
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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()
¶
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mpmath.
asech
(x)¶ Computes the inverse hyperbolic secant of \(x\), \(\mathrm{sech}^{-1}(x) = \cosh^{-1}(1/x)\).