Areatangens hyperbolicus and areakotangens hyperbolicus are the inverse functions of the tangent hyperbolicus and cotangent hyperbolicus and thus area functions .
Spellings:
y
=
artanh
(
x
)
=
tanh
-
1
(
x
)
{\ displaystyle y = \ operatorname {artanh} (x) = \ tanh ^ {- 1} (x)}
y
=
arcoth
(
x
)
=
coth
-
1
(
x
)
{\ displaystyle y = \ operatorname {arcoth} (x) = \ coth ^ {- 1} (x)}
The latter is used less often to avoid confusion with the reciprocal of the hyperbolic (co-) tangent. It is .
artanh
(
x
)
=
tanh
-
1
(
x
)
≠
tanh
(
x
)
-
1
=
1
tanh
(
x
)
{\ displaystyle \ operatorname {artanh} (x) = \ tanh ^ {- 1} (x) \ not = \ tanh (x) ^ {- 1} = {\ tfrac {1} {\ tanh (x)}} }
Definitions Hyperbolic areatangens:
artanh
(
x
)
: =
1
2
ln
(
1
+
x
1
-
x
)
f
u
¨
r
|
x
|
<
1
{\ displaystyle \ operatorname {artanh} (x): = {\ frac {1} {2}} \ ln \ left ({\ frac {1 + x} {1-x}} \ right) \ quad \ mathrm { f {\ ddot {u}} r} \ quad | x | <1}
Hyperbolic areakotangent:
arcoth
(
x
)
: =
1
2
ln
(
x
+
1
x
-
1
)
f
u
¨
r
|
x
|
>
1
{\ displaystyle \ operatorname {arcoth} (x): = {\ frac {1} {2}} \ ln \ left ({\ frac {x + 1} {x-1}} \ right) \ quad \ mathrm { f {\ ddot {u}} r} \ quad | x |> 1}
Geometric definitions Geometric can be the Area hyperbolic tangent represented by the area in the plane containing the connection line between the coordinate origin and the hyperbola sweeps: Let and start and end point on the hyperbola, the surface of the link overlined.
(
x
,
y
)
=
(
0
,
0
)
{\ displaystyle (x, y) = (0,0)}
x
2
-
y
2
=
1
{\ displaystyle x ^ {2} -y ^ {2} = 1}
(
x
,
-
y
)
=
(
x
,
-
x
2
-
1
)
{\ displaystyle (x, -y) = \ left (x, - {\ sqrt {x ^ {2} -1}} \ right)}
(
x
,
y
)
=
(
x
,
+
x
2
-
1
)
{\ displaystyle (x, y) = \ left (x, + {\ sqrt {x ^ {2} -1}} \ right)}
A.
=
artanh
(
y
x
)
{\ displaystyle A = \ operatorname {artanh} \ left ({\ frac {y} {x}} \ right)}
properties
Graph of the function artanh (x)
Graph of the function arcoth (x)
Hyperbolic areatangens
Hyperbolic areakotangent
Domain of definition
-
1
<
x
<
1
{\ displaystyle -1 <x <1}
-
∞
<
x
<
-
1
{\ displaystyle - \ infty <x <-1}
1
<
x
<
∞
{\ displaystyle 1 <x <\ infty}
Range of values
-
∞
<
f
(
x
)
<
∞
{\ displaystyle - \ infty <f (x) <\ infty}
-
∞
<
f
(
x
)
<
∞
;
f
(
x
)
≠
0
{\ displaystyle - \ infty <f (x) <\ infty; \; f (x) \ neq 0}
periodicity
no
no
monotony
strictly monotonously increasing
no
Symmetries
odd function:
f
(
-
x
)
=
-
f
(
x
)
{\ displaystyle f (-x) = - f (x)}
odd function:
f
(
-
x
)
=
-
f
(
x
)
{\ displaystyle f (-x) = - f (x)}
Asymptotes
x
=
1
:
f
(
x
)
→
∞
For
x
→
1
{\ displaystyle x = 1 \ colon \, f (x) \ to \ infty {\ text {for}} x \ to 1}
x
=
-
1
:
f
(
x
)
→
-
∞
For
x
→
-
1
{\ displaystyle x = -1 \ colon \, f (x) \ to - \ infty {\ text {for}} x \ to -1}
y
=
0
:
f
(
x
)
→
0
For
x
→
±
∞
{\ displaystyle y = 0 \ colon \, f (x) \ to 0 {\ text {for}} x \ to \ pm \ infty}
zeropoint
x
=
0
{\ displaystyle x = 0}
no
Jump points
no
no
Poles
x
=
±
1
{\ displaystyle x = \ pm 1}
x
=
±
1
{\ displaystyle x = \ pm 1}
Extremes
no
no
Turning points
x
=
0
{\ displaystyle x = 0}
no
Series developments Taylor and Laurent series of the two functions are
artanh
(
x
)
=
∑
k
=
0
∞
x
2
k
+
1
2
k
+
1
=
x
+
1
3
x
3
+
1
5
x
5
+
1
7th
x
7th
+
...
arcoth
(
x
)
=
∑
k
=
1
∞
x
-
(
2
k
-
1
)
2
k
-
1
=
x
-
1
+
1
3
x
-
3
+
1
5
x
-
5
+
1
7th
x
-
7th
+
...
=
∑
k
=
0
∞
1
(
2
k
+
1
)
⋅
x
2
k
+
1
=
1
x
+
1
3
x
3
+
1
5
x
5
+
1
7th
x
7th
+
...
{\ displaystyle {\ begin {alignedat} {2} \ operatorname {artanh} (x) & = \ sum _ {k = 0} ^ {\ infty} {\ frac {x ^ {2k + 1}} {2k + 1}} & = x + {\ frac {1} {3}} x ^ {3} + {\ frac {1} {5}} x ^ {5} + {\ frac {1} {7}} x ^ {7} + \ ldots & {} \\\ operatorname {arcoth} (x) & = \ sum _ {k = 1} ^ {\ infty} {\ frac {x ^ {- (2k-1)}} { 2k-1}} & = x ^ {- 1} + {\ frac {1} {3}} x ^ {- 3} + {\ frac {1} {5}} x ^ {- 5} + {\ frac {1} {7}} x ^ {- 7} + \ ldots & {} \\ & = \ sum _ {k = 0} ^ {\ infty} {\ frac {1} {(2k + 1) \ cdot x ^ {2k + 1}}} & = {\ frac {1} {x}} + {\ frac {1} {3x ^ {3}}} + {\ frac {1} {5x ^ {5} }} + {\ frac {1} {7x ^ {7}}} + \ ldots & {} \ end {alignedat}}}
Derivatives
d
d
x
artanh
(
x
)
=
1
1
-
x
2
;
|
x
|
<
1
{\ displaystyle {\ frac {\ mathrm {d}} {\ mathrm {d} x}} \ operatorname {artanh} (x) = {\ frac {1} {1-x ^ {2}}} \ ,; \ quad | x | <1}
d
d
x
arcoth
(
x
)
=
1
1
-
x
2
;
|
x
|
>
1
{\ displaystyle {\ frac {\ mathrm {d}} {\ mathrm {d} x}} \ operatorname {arcoth} (x) = {\ frac {1} {1-x ^ {2}}} \ ,; \ quad | x |> 1}
Integrals The antiderivatives are:
∫
artanh
(
x
)
d
x
=
x
⋅
artanh
(
x
)
+
1
2
ln
(
1
-
x
2
)
+
C.
{\ displaystyle \ int \ operatorname {artanh} (x) \, \ mathrm {d} x = x \ cdot \ operatorname {artanh} (x) + {\ frac {1} {2}} \ ln \ left (1 -x ^ {2} \ right) + C}
∫
arcoth
(
x
)
d
x
=
x
⋅
arcoth
(
x
)
+
1
2
ln
(
x
2
-
1
)
+
C.
{\ displaystyle \ int \ operatorname {arcoth} (x) \, \ mathrm {d} x = x \ cdot \ operatorname {arcoth} (x) + {\ frac {1} {2}} \ ln \ left (x ^ {2} -1 \ right) + C}
Addition theorems
artanh
(
x
)
±
artanh
(
y
)
=
artanh
(
x
±
y
1
±
x
y
)
{\ displaystyle \ operatorname {artanh} (x) \ pm \ operatorname {artanh} (y) = \ operatorname {artanh} \ left ({\ frac {x \ pm y} {1 \ pm xy}} \ \ right) }
arcoth
(
x
)
±
arcoth
(
y
)
=
arcoth
(
1
±
x
y
x
±
y
)
{\ displaystyle \ operatorname {arcoth} (x) \ pm \ operatorname {arcoth} (y) = \ operatorname {arcoth} \ left ({\ frac {1 \ pm xy} {x \ pm y}} \ \ right) }
See also Web links
<img src="https://de.wikipedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" title="" width="1" height="1" style="border: none; position: absolute;">
