arcsinh, arccosh,
arctanh, arccsch, arcsech, arccoth -- the inverse hyperbolic
functions
Introductionarcsinh(x) represents the inverse of the sine
function.
arccosh(x) represents the inverse of the cosine
function.
arctanh(x) represents the inverse of the tangent
function.
arccsch(x) represents the inverse of the cosecant
function.
arcsech(x) represents the inverse of the secant
function.
arccoth(x) represents the inverse of the cotangent
function.
Call(s)arcsinh(x)
arccosh(x)
arctanh(x)
arccsch(x)
arcsech(x)
arccoth(x)
Parametersx |
- | an arithmetical expression |
Returnsan arithmetical expression.
x
Side
EffectsWhen called with a floating point argument, the functions are
sensitive to the environment variable DIGITS which determines the numerical
working precision.
Related
Functionssinh, cosh, tanh, csch, sech, coth
Detailsarcsinh(0)=0,
arccosh(0)=I*PI/2,
arccosh(1)=0,
arctanh(0)=0,
arccosh(0)=I*PI/2.
y:=arcsinh(x) satisfies -PI/2<=Im(y)<=PI/2,
y:=arccosh(x) satisfies -PI<Im(y)<=PI,
y:=arctanh(x) satisfies -PI/2<Im(y)<PI/2,
y:=arccoth(x) satisfies -PI/2<Im(y)<=PI/2.
arcsinh(x) = ln(x + sqrt(x^2 + 1)),
arccosh(x) = ln(x + sqrt(x^2 - 1)),
arctanh(x) = (ln(1 + x) - ln(1 - x))/2,
arccsch(x) = arcsinh(1/x),
arcsech(x) = arccosh(1/x),
arccoth(x) = arctanh(1/x).
Cf. example 2.
For arcsinh, the branch cuts are the intervals
(-I*infinity,-I) and (I,I*infinity) on the
imaginary axis.
For arccosh, the branch cuts are the real interval
(-infinity,1) and the imaginary axis.
For arctanh, the branch cuts are the real intervals
(-infinity,-1] and [1,infinity).
For arccsch, the branch cut is the interval
(-I,I) on the imaginary axis.
For arcsech, the branch cuts are the real intervals
(-infinity,0) and (1,infinity) together with the
imaginary axis.
For arccoth, the branch cut is the real interval
[-1,1].
The values jump when the argument crosses a branch cut. Cf. example 3.
arccsch and arcsech
immediately rewrite themselves, returning
arccsch(x)=arcsinh(1/x) and
arcsech(x)=arccosh(1/x),
respectively. MuPAD's arccoth is defined by
arccoth(x)=arctanh(1/x). However,
it does not rewrite itself automatically in terms of
arctanh.
Example
1We demonstrate some calls with exact and symbolic input data:
>> arcsinh(1), arccosh(1/sqrt(2)), arctanh(5 + I), arccsch(1/3), arcsech(I), arccoth(2)
/ 1/2 \
| 2 |
arcsinh(1), arccosh| ---- |, arctanh(5 + I), arcsinh(3),
\ 2 /
arccosh(- I), arccoth(2)
>> arcsinh(-x), arccosh(x + 1), arctanh(1/x)
/ 1 \
-arcsinh(x), arccosh(x + 1), arctanh| - |
\ x /
Floating point values are computed for floating point arguments:
>> arcsinh(0.1234), arccosh(5.6 + 7.8*I), arccoth(1.0/10^20)
0.1230889466, 2.956002937 + 0.9506879769 I, - 1.570796327 I
Example
2The inverse hyperbolic functions can be rewritten in terms of the logarithm function:
>> rewrite(arcsinh(x), ln), rewrite(arctanh(x), ln)
2 1/2 ln(x + 1) ln(1 - x)
ln(x + (x + 1) ), --------- - ---------
2 2
Example
3The values jump when crossing a branch cut:
>> arctanh(2.0 + I/10^10), arctanh(2.0 - I/10^10)
0.5493061443 + 1.570796327 I, 0.5493061443 - 1.570796327 I
On the branch cut, the values of arctanh
coincide with the limit ``from below'' for real arguments
x>1. The values coincide with the limit ``from above''
for real x<-1:
>> arctanh(1.2), arctanh(1.2 - I/10^10), arctanh(1.2 + I/10^10)
1.198947636 - 1.570796327 I, 1.198947636 - 1.570796327 I,
1.198947636 + 1.570796327 I
>> arctanh(-1.2), arctanh(-1.2 + I/10^10), arctanh(-1.2 - I/10^10)
- 1.198947636 + 1.570796327 I, - 1.198947636 + 1.570796327 I,
- 1.198947636 - 1.570796327 I
Example
4Various system functions such as diff, float, limit, or series handle expressions involving
the inverse hyperbolic functions:
>> diff(arcsinh(x^2), x), float(arccosh(3)*arctanh(5 + I))
2 x
-----------, 0.3427241326 + 2.698556745 I
4 1/2
(x + 1)
>> limit(arcsinh(x)/arctanh(x), x = 0)
1
>> series(arctanh(sinh(x)) - arcsinh(tanh(x)), x = 0, 10)
7 9
3 83 x 4 x 10
x + ----- - ---- + O(x )
120 189
series
attributes were improved. Floating point results are now consistent
with the branch cuts defined by the logarithmic representation.asinh,..,acoth in previous
MuPAD versions.