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 the next figure.
Mathematical definitions for hyperbolic functions
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+sqrt(1+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 |
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)).
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+sqrt(1+z^{2})).
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))/2.
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.