sinh, cosh, tanh, asinh, acosh, atanh [Function]
sinh
number => result
cosh
number => result
tanh
number => result
asinh
number => result
acosh
number => result
atanh
number => result
Arguments and Values::
number--a number.
result--a number.
Description::
These functions compute the hyperbolic sine, cosine, tangent,
arc sine, arc cosine, and arc tangent functions,
which are mathematically defined for an argument x
as given in Figure 12-15.
Function Definition
Hyperbolic sine (e^x-e^-x)/2
Hyperbolic cosine (e^x+e^-x)/2
Hyperbolic tangent (e^x-e^-x)/(e^x+e^-x)
Hyperbolic arc sine log (x+\sqrt1+x^2)
Hyperbolic arc cosine 2 log (\sqrt(x+1)/2 + \sqrt(x-1)/2)
Hyperbolic arc tangent (log (1+x) - log (1-x))/2
Figure 12--15: Mathematical definitions for hyperbolic functions
The following definition for the inverse hyperbolic cosine
determines the range and branch cuts:
arccosh z = 2 log (\sqrt(z+1)/2 + \sqrt(z-1)/2\Bigr).
The branch cut for the inverse hyperbolic cosine function
lies along the real axis to the left of~1 (inclusive), extending
indefinitely along the negative real axis, continuous with quadrant II
and (between 0 and~1) with quadrant I.
The range is that half-strip of the complex plane containing numbers whose
real part is non-negative and whose imaginary
part is between -\pi (exclusive) and~\pi (inclusive).
A number with real part zero is in the range
if its imaginary part is between zero (inclusive) and~\pi (inclusive).
The following definition for the inverse hyperbolic sine determines
the range and branch cuts:
arcsinh z = log (z+\sqrt1+z^2\Bigr).
The branch cut for the inverse hyperbolic sine function is in two pieces:
one along the positive imaginary axis above i
(inclusive), continuous with quadrant I, and one along the negative imaginary
axis below -i (inclusive), continuous with quadrant III.
The range is that strip of the complex plane containing numbers whose imaginary
part is between -\pi/2 and~\pi/2. A number with imaginary
part equal to -\pi/2 is in the range if and only if its real
part is non-positive; a number with imaginary part equal to \pi/2 is in
the range if and only if its imaginary part is non-negative.
The following definition for the inverse hyperbolic tangent
determines the range and branch cuts:
arctanh z = log (1+z) - log (1-z)\over2.
Note that:
i arctan z = arctanh iz.
The branch cut for the inverse hyperbolic tangent function
is in two pieces: one along the negative real axis to the left of
-1 (inclusive), continuous with quadrant III, and one along
the positive real axis to the right of~1 (inclusive), continuous with
quadrant I. The points -1 and~1 are excluded from the
domain.
The range is that strip of the complex plane containing
numbers whose imaginary part is between -\pi/2 and
\pi/2. A number with imaginary part equal to -\pi/2
is in the range if and only if its real part is strictly negative; a number with
imaginary part equal to \pi/2 is in the range if and only if its imaginary
part is strictly positive.
Thus the range of the inverse hyperbolic tangent function is identical to
that of the inverse hyperbolic sine function with the points
-\pi i/2 and~\pi i/2 excluded.
Examples::
(sinh 0) => 0.0
(cosh (complex 0 -1)) => #C(0.540302 -0.0)
Exceptional Situations::
Should signal an error of type type-error
if number is not a number.
Might signal arithmetic-error.
The result of acosh may be a complex even if number
is not a complex; this occurs when number is less than one.
Also, the result of atanh may be a complex even if number
is not a complex; this occurs when the absolute value of number
is greater than one.
The branch cut formulae are mathematically correct, assuming
completely accurate computation.
Implementors should consult a good text on
numerical analysis. The formulae given above are not necessarily
the simplest ones for real-valued computations; they are chosen
to define the branch cuts in desirable ways for the complex case.