Areasinus hyperbolicus (abbreviated or ) and Areakosinus hyperbolicus (abbreviated or ) belong to the area functions and are the inverse functions of the hyperbolic sine and hyperbolic cosine .
arsinh
{\ displaystyle \ operatorname {arsinh}}
asinh
{\ displaystyle \ operatorname {asinh}}
arcosh
{\ displaystyle \ operatorname {arcosh}}
acosh
{\ displaystyle \ operatorname {acosh}}
designation
"Areasinus hyperbolicus" and "Areakosinus hyperbolicus" are the names accepted by experts. The short forms "Areasinus" and "Areakosinus", which appear occasionally on the Internet, are unusual. And is an abbreviation for formulas . The representations or are considered out of date and the representations that are rarely found and are to be avoided, as they conflict with the meaning or .
arsh
{\ displaystyle \ operatorname {arsh}}
arch
{\ displaystyle \ operatorname {arch}}
A.
r
S.
i
n
{\ displaystyle {\ mathfrak {ArSin}}}
A.
r
C.
O
s
{\ displaystyle {\ mathfrak {ArCos}}}
sinh
-
1
{\ displaystyle \ sinh ^ {- 1}}
cosh
-
1
{\ displaystyle \ cosh ^ {- 1}}
1
sinh
{\ displaystyle {\ tfrac {1} {\ sinh}}}
1
cosh
{\ displaystyle {\ tfrac {1} {\ cosh}}}
Definitions
The functions can be expressed by the following formulas:
Areasinus hyperbolicus:
arsinh
(
x
)
=
ln
(
x
+
x
2
+
1
)
{\ displaystyle \ operatorname {arsinh} (x) = \ ln \ left (x + {\ sqrt {x ^ {2} +1}} \ right)}
With
x
∈
R.
{\ displaystyle \, x \ in \ mathbb {R}}
Areakosine hyperbolicus:
arcosh
(
x
)
=
ln
(
x
+
x
2
-
1
)
{\ displaystyle \ operatorname {arcosh} (x) = \ ln \ left (x + {\ sqrt {x ^ {2} -1}} \ right)}
For
x
≥
1
{\ displaystyle x \ geq 1}
Here stands for the natural logarithm .
ln
{\ displaystyle \ ln}
conversion
Together with the signum function , the following applies:
so-called
{\ displaystyle \ operatorname {sgn}}
arsinh
(
x
)
=
so-called
(
x
)
⋅
arcosh
(
x
2
+
1
)
{\ displaystyle \ operatorname {arsinh} (x) = \ operatorname {sgn} (x) \ cdot \ operatorname {arcosh} \ left ({\ sqrt {x ^ {2} +1}} \ right)}
The following applies to:
x
≥
1
{\ displaystyle x \ geq 1}
arcosh
(
x
)
=
arsinh
(
x
2
-
1
)
{\ displaystyle \ operatorname {arcosh} (x) = \ operatorname {arsinh} \ left ({\ sqrt {x ^ {2} -1}} \ right)}
properties
Graph of function arsinh (x)
Graph of function arcosh (x)
Hyperbolic area
Hyperbolic areakosine
Domain of definition
-
∞
<
x
<
+
∞
{\ displaystyle - \ infty <x <+ \ infty}
1
≤
x
<
+
∞
{\ displaystyle 1 \ leq x <+ \ infty}
Range of values
-
∞
<
f
(
x
)
<
+
∞
{\ displaystyle - \ infty <f (x) <+ \ infty}
0
≤
f
(
x
)
<
+
∞
{\ displaystyle 0 \ leq f (x) <+ \ infty}
periodicity
no
no
monotony
strictly monotonously increasing
strictly monotonously increasing
Symmetries
Point symmetry to the origin, odd function
no
asymptote
f
(
x
)
→
±
ln
(
2
|
x
|
)
{\ displaystyle f (x) \ to \ pm \ ln (2 | x |)}
For
x
→
±
∞
{\ displaystyle x \ to \ pm \ infty}
f
(
x
)
→
ln
(
2
x
)
{\ displaystyle f (x) \ to \ ln (2x)}
For
x
→
+
∞
{\ displaystyle x \ to + \ infty}
zeropoint
x
=
0
{\ displaystyle x = 0}
x
=
1
{\ displaystyle x = 1}
Jump points
no
no
Poles
no
no
Extremes
no
no
Turning points
x
=
0
{\ displaystyle x = 0}
no
Series developments
As with all trigonometric and hyperbolic functions, there are also series expansion. The double faculty and the generalization of the binomial coefficient occur.
The series developments are:
arsinh
(
x
)
=
x
∑
k
=
0
∞
(
2
k
-
1
)
!
!
(
-
x
2
)
k
(
2
k
)
!
!
(
2
k
+
1
)
=
∑
k
=
0
∞
(
-
1
2
k
)
x
2
k
+
1
2
k
+
1
=
x
-
1
2
x
3
3
+
1
⋅
3
2
⋅
4th
x
5
5
-
1
⋅
3
⋅
5
2
⋅
4th
⋅
6th
x
7th
7th
+
⋯
For
|
x
|
<
1
arsinh
(
x
)
=
so-called
(
x
)
⋅
[
ln
(
2
|
x
|
)
-
∑
k
=
1
∞
(
2
k
-
1
)
!
!
2
k
(
2
k
)
!
!
(
-
x
2
)
k
]
For
|
x
|
>
1
arcosh
(
x
)
=
ln
(
2
x
)
-
∑
k
=
1
∞
(
2
k
-
1
)
!
!
2
k
⋅
(
2
k
)
!
!
x
-
2
k
{\ displaystyle {\ begin {alignedat} {2} \ operatorname {arsinh} (x) & = x \ sum _ {k = 0} ^ {\ infty} {\ frac {(2k-1) !! (- x ^ {2}) ^ {k}} {(2k) !! (2k + 1)}} = \ sum _ {k = 0} ^ {\ infty} {\ frac {{\ binom {- {\ frac { 1} {2}}} {k}} x ^ {2k + 1}} {2k + 1}} & {} \\ & = x - {\ frac {1} {2}} {\ frac {x ^ {3}} {3}} + {\ frac {1 \ cdot 3} {2 \ cdot 4}} {\ frac {x ^ {5}} {5}} - {\ frac {1 \ cdot 3 \ cdot 5} {2 \ cdot 4 \ cdot 6}} {\ frac {x ^ {7}} {7}} + \ cdots & {\ text {for}} | x | <1 \\\ operatorname {arsinh} ( x) & = \ operatorname {sgn} (x) \ cdot \ left [\ ln (2 | x |) - \ sum _ {k = 1} ^ {\ infty} {\ frac {(2k-1) !! } {2k (2k) !! (- x ^ {2}) ^ {k}}} \ right] & {\ text {for}} | x |> 1 \\\ operatorname {arcosh} (x) & = \ ln (2x) - \ sum _ {k = 1} ^ {\ infty} {\ frac {(2k-1) !!} {2k \ cdot (2k) !!}} x ^ {- 2k} & { } \ end {alignedat}}}
Derivatives
The derivation of the hyperbolic area is:
d
d
x
arsinh
(
x
)
=
1
x
2
+
1
{\ displaystyle {\ frac {\ mathrm {d}} {\ mathrm {d} x}} \ operatorname {arsinh} (x) = {\ frac {1} {\ sqrt {x ^ {2} +1}} }}
The derivation of the hyperbolic areakosine is:
d
d
x
arcosh
(
x
)
=
1
x
2
-
1
{\ displaystyle {\ frac {\ mathrm {d}} {\ mathrm {d} x}} \ operatorname {arcosh} (x) = {\ frac {1} {\ sqrt {x ^ {2} -1}} }}
for x> 1
Antiderivatives
The antiderivatives of the hyperbolic area and the hyperbolic area are:
∫
arsinh
(
x
)
d
x
=
x
⋅
arsinh
(
x
)
-
x
2
+
1
+
C.
{\ displaystyle \ int \ operatorname {arsinh} (x) \ \ mathrm {d} x = x \ cdot \ operatorname {arsinh} (x) - {\ sqrt {x ^ {2} +1}} + C}
∫
arcosh
(
x
)
d
x
=
x
⋅
arcosh
(
x
)
-
x
2
-
1
+
C.
{\ displaystyle \ int \ operatorname {arcosh} (x) \ \ mathrm {d} x = x \ cdot \ operatorname {arcosh} (x) - {\ sqrt {x ^ {2} -1}} + C}
Other identities
arcosh
(
2
x
2
-
1
)
=
2
arcosh
(
x
)
For
x
≥
1
arcosh
(
8th
x
4th
-
8th
x
2
+
1
)
=
4th
arcosh
(
x
)
For
x
≥
1
arcosh
(
2
x
2
+
1
)
=
2
arsinh
(
x
)
For
x
≥
0
arcosh
(
8th
x
4th
+
8th
x
2
+
1
)
=
4th
arsinh
(
x
)
For
x
≥
0
{\ displaystyle {\ begin {aligned} \ operatorname {arcosh} (2x ^ {2} -1) = 2 \ operatorname {arcosh} (x) \ qquad {\ text {for}} x \ geq 1 \\\ operatorname {arcosh} (8x ^ {4} -8x ^ {2} +1) = 4 \ operatorname {arcosh} (x) \ qquad {\ text {for}} x \ geq 1 \\\ operatorname {arcosh} (2x ^ {2} +1) = 2 \ operatorname {arsinh} (x) \ qquad {\ text {for}} x \ geq 0 \\\ operatorname {arcosh} (8x ^ {4} + 8x ^ {2} + 1) = 4 \ operatorname {arsinh} (x) \ qquad {\ text {for}} x \ geq 0 \ end {aligned}}}
arsinh
u
±
arsinh
v
=
arsinh
(
u
1
+
v
2
±
v
1
+
u
2
)
{\ displaystyle \ operatorname {arsinh} u \ pm \ operatorname {arsinh} v = \ operatorname {arsinh} \ left (u {\ sqrt {1 + v ^ {2}}} \ pm v {\ sqrt {1 + u ^ {2}}} \ right)}
arcosh
u
±
arcosh
v
=
arcosh
(
u
v
±
(
u
2
-
1
)
(
v
2
-
1
)
)
{\ displaystyle \ operatorname {arcosh} u \ pm \ operatorname {arcosh} v = \ operatorname {arcosh} \ left (uv \ pm {\ sqrt {(u ^ {2} -1) (v ^ {2} -1 )}} \ right)}
arsinh
u
+
arcosh
v
=
arsinh
(
u
v
+
(
1
+
u
2
)
(
v
2
-
1
)
)
=
arcosh
(
v
1
+
u
2
+
u
v
2
-
1
)
{\ displaystyle {\ begin {aligned} \ operatorname {arsinh} u + \ operatorname {arcosh} v & = \ operatorname {arsinh} \ left (uv + {\ sqrt {(1 + u ^ {2}) (v ^ {2} -1)}} \ right) \\ & = \ operatorname {arcosh} \ left (v {\ sqrt {1 + u ^ {2}}} + u {\ sqrt {v ^ {2} -1}} \ right) \ end {aligned}}}
Numerical calculation
Basically, the hyperbolic area sine can be calculated using the well-known formula
arsinh
(
x
)
=
ln
(
x
+
x
2
+
1
)
{\ displaystyle \ operatorname {arsinh} (x) = \ ln \ left (x + {\ sqrt {x ^ {2} +1}} \ right)}
can be calculated if the natural logarithm function is available. However, there are the following problems:
ln
x
{\ displaystyle \ ln x}
Large, positive operands trigger an overflow, although the end result can always be represented.
For operands close to 0 there is a numerical cancellation, which makes the result imprecise.
First of all, the operand should be made positive:
x
{\ displaystyle x}
arsinh
x
=
-
arsinh
(
-
x
)
{\ displaystyle \ operatorname {arsinh} x = - \ operatorname {arsinh} (-x)}
for applied.
x
<
0
{\ displaystyle x <0}
The following cases can then be distinguished for:
x
≥
0
{\ displaystyle x \ geq 0}
Case 1: is a large, positive number with :
x
{\ displaystyle x}
x
≥
10
k
2
{\ displaystyle x \ geq {10} ^ {\ frac {k} {2}}}
arsinh
x
=
ln
2
+
ln
x
,
{\ displaystyle \ operatorname {arsinh} x = \ ln {2} + \ ln {x},}
where the number of significant decimal digits is the number type used, which is double 16 for the 64-bit floating point type .
k
{\ displaystyle k}
This formula results from the following consideration:
10
k
{\ displaystyle {10} ^ {k}}
is the smallest positive number from which the last digit before the decimal point is no longer stored, which is why . Now the one is to be calculated from the following applies: . This is true of what follows. So you can replace the hyperbolic area in the well-known formula with :
10
k
+
1
≈
10
k
{\ displaystyle {10} ^ {k} + {1} \ approx {10} ^ {k}}
x
{\ displaystyle x}
x
2
+
1
≈
x
2
{\ displaystyle x ^ {2} +1 \ approx {x} ^ {2}}
x
2
≥
10
k
{\ displaystyle {x} ^ {2} \ geq {10} ^ {k}}
x
≥
10
k
2
{\ displaystyle {x} \ geq {10} ^ {\ frac {k} {2}}}
x
2
+
1
{\ displaystyle x ^ {2} +1}
x
2
{\ displaystyle x ^ {2}}
arsinh
(
x
)
=
ln
(
x
+
x
2
+
1
)
{\ displaystyle \ operatorname {arsinh} (x) = \ ln \ left (x + {\ sqrt {x ^ {2} +1}} \ right)}
≈
ln
(
x
+
x
2
)
=
ln
(
2
x
)
=
ln
2
+
ln
x
{\ displaystyle \ ln (x + {\ sqrt {x ^ {2}}}) = \ ln ({2x}) = \ ln {2} + \ ln {x}}
Case 2: is close to 0, e.g. B. for :
x
{\ displaystyle x}
x
<
0.125
{\ displaystyle x <0 {,} 125}
Using the Taylor series:
arsinh
x
=
x
-
1
2
x
3
3
+
1
⋅
3
2
⋅
4th
x
5
5
-
1
⋅
3
⋅
5
2
⋅
4th
⋅
6th
x
7th
7th
+
⋯
{\ displaystyle \ operatorname {arsinh} x = x - {\ frac {1} {2}} {\ frac {x ^ {3}} {3}} + {\ frac {1 \ cdot 3} {2 \ cdot 4}} {\ frac {x ^ {5}} {5}} - {\ frac {1 \ cdot 3 \ cdot 5} {2 \ cdot 4 \ cdot 6}} {\ frac {x ^ {7}} {7}} + \ dotsb}
Case 3: All others :
x
{\ displaystyle x}
arsinh
(
x
)
=
ln
(
x
+
x
2
+
1
)
{\ displaystyle \ operatorname {arsinh} (x) = \ ln \ left (x + {\ sqrt {x ^ {2} +1}} \ right)}
The hyperbolic areacosine can be calculated using the well-known formula
arcosh
(
x
)
=
ln
(
x
+
x
2
-
1
)
{\ displaystyle \ operatorname {arcosh} (x) = \ ln \ left (x + {\ sqrt {x ^ {2} -1}} \ right)}
be calculated. Here, too, the numerical problem arises that large positive operands trigger an overflow, although the end result can always be represented.
Case 1: is a large positive number with :
x
{\ displaystyle x}
x
≥
10
k
2
{\ displaystyle x \ geq {10} ^ {\ frac {k} {2}}}
arcosh
x
=
ln
2
+
ln
x
,
{\ displaystyle \ operatorname {arcosh} x = \ ln {2} + \ ln {x},}
where is the number of significant decimal digits of the number type used.
k
{\ displaystyle k}
Case 2 ::
x
<
1
{\ displaystyle x <1}
The result is not defined.
Case 3: All others , i.e. for :
x
{\ displaystyle x}
1
≤
x
<
10
k
2
{\ displaystyle 1 \ leq x <{10} ^ {\ frac {k} {2}}}
arcosh
(
x
)
=
ln
(
x
+
x
2
-
1
)
{\ displaystyle \ operatorname {arcosh} (x) = \ ln \ left (x + {\ sqrt {x ^ {2} -1}} \ right)}
See also
Web links
Individual evidence
^ Franz Brzoska, Walter Bartsch: Mathematical formula collection . 2nd improved edition. Fachbuchverlag Leipzig, 1956.
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