Hyperbolic Areakosekans and hyperbolic Areakosekans belong to the area functions . They are the inverse of the hyperbolic secant and hyperbolic cosecant . They are written as functions or less often or and less often .
arsech
{\ displaystyle \ operatorname {arsech}}
six
-
1
{\ displaystyle \ operatorname {six} ^ {- 1}}
arcsch
(
x
)
{\ displaystyle \ operatorname {arcsch} (x)}
csch
-
1
(
x
)
{\ displaystyle \ operatorname {csch} ^ {- 1} (x)}
Definitions
The hyperbolic and hyperbolic areakoscans are usually defined as:
arsech
(
x
)
=
ln
(
1
+
1
-
x
2
x
)
{\ displaystyle \ operatorname {arsech} (x) = \ ln \ left ({\ frac {1 + {\ sqrt {1-x ^ {2}}}} {x}} \ right)}
arcsch
(
x
)
=
ln
(
1
+
1
+
x
2
x
)
{\ displaystyle \ operatorname {arcsch} (x) = \ ln \ left ({\ frac {1 + {\ sqrt {1 + x ^ {2}}}} {x}} \ right)}
Here stands for the natural logarithm .
ln
{\ displaystyle \ ln}
properties
Graph of the hyperbolic Areasekans function
Graph of the hyperbolic areakosecan function
Areasecans hyperbolicus
Hyperbolic areakosekans
Domain of definition
0
<
x
≤
1
{\ displaystyle 0 <x \ leq 1}
-
∞
<
x
<
+
∞
;
x
≠
0
{\ displaystyle - \ infty <x <+ \ infty \,; \, x \ neq 0}
Range of values
0
≤
f
(
x
)
<
+
∞
{\ displaystyle 0 \ leq f (x) <+ \ infty}
-
∞
<
f
(
x
)
<
+
∞
;
f
(
x
)
≠
0
{\ displaystyle - \ infty <f (x) <+ \ infty \,; \, f (x) \ neq 0}
periodicity
no
no
monotony
strictly falling monotonously
x
≠
0
{\ displaystyle x \ neq 0}
strictly falling monotonously
Symmetries
no
Odd function
f
(
x
)
=
-
f
(
-
x
)
{\ displaystyle f (x) = - f (-x)}
asymptote
f
(
x
)
→
0
{\ displaystyle f (x) \ to 0}
;
x
→
+
1
{\ displaystyle x \ to +1}
f
(
x
)
→
0
{\ displaystyle f (x) \ to 0}
;
x
→
±
∞
{\ displaystyle x \ to \ pm \ infty}
zeropoint
x
=
1
{\ displaystyle x = 1}
no
Jump points
no
no
Poles
x
=
0
{\ displaystyle x = 0}
x
=
0
{\ displaystyle x = 0}
Extremes
no
no
Turning points
x
=
1
2
2
{\ displaystyle x = {\ frac {1} {2}} {\ sqrt {2}}}
no
Special values
The following applies:
arcsch
2
=
ln
Φ
{\ displaystyle \ operatorname {arcsch} \, 2 = \ ln \ Phi}
where denotes the golden ratio .
Φ
{\ displaystyle \! \ \ Phi}
Series developments
arsech
(
x
)
=
ln
(
2
x
)
-
∑
k
=
1
∞
(
2
k
-
1
)
!
!
x
2
k
(
2
k
)
!
!
2
k
f
u
¨
r
0
<
x
≤
1
arcsch
(
x
)
=
∑
k
=
1
∞
P
k
-
1
(
0
)
k
x
k
=
∑
k
=
1
∞
(
-
1
)
k
⋅
(
1
2
)
k
-
1
(
2
k
-
1
)
(
k
-
1
)
!
x
1
-
2
k
{\ displaystyle {\ begin {alignedat} {2} \ operatorname {arsech} (x) & = \ ln \ left ({\ frac {2} {x}} \ right) - \ sum _ {k = 1} ^ {\ infty} {\ frac {(2k-1) !! x ^ {2k}} {(2k) !! 2k}} & \ qquad \ mathrm {f {\ ddot {u}} r} \, 0 < x \ leq 1 \\\ operatorname {arcsch} (x) & = \ sum _ {k = 1} ^ {\ infty} {\ frac {P_ {k-1} (0)} {k}} x ^ { k} \\ & = \ sum _ {k = 1} ^ {\ infty} {\ frac {(-1) ^ {k} \ cdot ({\ tfrac {1} {2}}) _ {k-1 }} {(2k-1) (k-1)!}} \, X ^ {1-2k} \ end {alignedat}}}
It is the th Legendre polynomial and represents the Pochhammer symbol .
P
k
{\ displaystyle P_ {k}}
k
{\ displaystyle k}
(
1
2
)
n
{\ displaystyle ({\ tfrac {1} {2}}) _ {n}}
Derivatives
d
d
x
a
r
s
e
c
H
(
x
)
=
-
1
x
1
-
x
2
{\ displaystyle {\ frac {\ mathrm {d}} {\ mathrm {d} x}} {\ rm {arsech}} (x) = - {\ frac {1} {x {\ sqrt {1-x ^ {2}}}}}}
.
d
d
x
arcsch
(
x
)
=
-
1
|
x
|
1
+
x
2
{\ displaystyle {\ frac {\ mathrm {d}} {\ mathrm {d} x}} \ operatorname {arcsch} (x) = - {\ frac {1} {| x | {\ sqrt {1 + x ^ {2}}}}}}
.
Integrals
Primitives of Areasekans hyperbolic and hyperbolic Areakosekans are:
∫
arsech
(
x
)
d
x
=
x
⋅
arsech
(
x
)
-
arctan
(
1
x
2
-
1
)
+
C.
{\ displaystyle \ int \ operatorname {arsech} (x) \, \ mathrm {d} x = x \ cdot \ operatorname {arsech} (x) - \ arctan \ left ({\ sqrt {{\ frac {1} { x ^ {2}}} - 1}} \ right) + C}
∫
arcsch
(
x
)
d
x
=
x
⋅
arcsch
(
x
)
+
ln
(
x
+
x
1
+
x
-
2
)
+
C.
.
{\ displaystyle \ int \ operatorname {arcsch} (x) \, \ mathrm {d} x = x \ cdot \ operatorname {arcsch} (x) + \ ln \ left (x + x {\ sqrt {1+ {x } ^ {- 2}}} \ right) + C.}
Conversion and relationships to other trigonometric functions
arsech
(
x
)
=
arcosh
(
1
x
)
{\ displaystyle \ operatorname {arsech} \, (x) = \ operatorname {arcosh} \ left ({\ frac {1} {x}} \ right)}
arcsch
(
x
)
=
arsinh
(
1
x
)
{\ displaystyle \ operatorname {arcsch} \, (x) = \ operatorname {arsinh} \ left ({\ frac {1} {x}} \ right)}
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;">