area functions

area functions

The most usual

area functions

:

•

The

inverse function

of the

hyperbolic sine

(in Latin

sinus hyperbolicus

) is

arsinh

(

area sini hyperbolici

):

arsinh

⁡

x

:=

ln

⁡

(

x

+

x

2

+

1

)

•

The inverse function of the

hyperbolic cosine

(in Latin

cosinus hyperbolicus

) is

arcosh

(

area cosini hyperbolici

):

arcosh

⁡

x

:=

ln

⁡

(

x

+

x

2

-

1

)

It is defined for

x

≧

1

.

•

The inverse function of the

hyperbolic tangent

(in Latin

tangens hyperbolica

) is

artanh

(

area tangentis hyperbolicae

):

artanh

⁡

x

:=

1

2

⁢

ln

⁡

1

+

x

1

-

x

It is defined for

-

1

<

x

<

1

.

•

The inverse function of the

hyperbolic cotangent

(in Latin

cotangens hyperbolica

) is

arcoth

(

area cotangentis hyperbolicae

):

arcoth

⁡

x

:=

1

2

⁢

ln

⁡

x

+

1

x

-

1

It is defined for

|

x

|

>

1

.

These four

functions

are denoted also by

sinh

-

1

⁡

x

,

cosh

-

1

⁡

x

,

tanh

-

1

⁡

x

and

coth

-

1

⁡

x

.

Derivatives

:

d

d

⁢

x

⁢

arsinh

⁡

x

=

1

x

2

+

1

d

d

⁢

x

⁢

arcosh

⁡

x

=

1

x

2

-

1

d

d

⁢

x

⁢

artanh

⁡

x

=

1

1

-

x

2

d

d

⁢

x

⁢

arcoth

⁡

x

=

1

1

-

x

2

The functions

arsinh

and

artanh

have the

Taylor series

arsinh

x

=

x

-

1

2

⋅

x

3

3

+

1

⋅

3

2

⋅

4

⋅

x

5

5

-

1

⋅

3

⋅

5

2

⋅

4

⋅

6

⋅

x

7

7

+

-

⋯

(

|

x

|

≦

1

)

,

artanh

x

=

x

+

x

3

3

+

x

5

5

+

x

7

7

+

⋯

(

|

x

|

<

1

)

.

Because the

inverse tangent

function (see the cyclometric functions) has the

arctan

x

=

x

-

x

3

3

+

x

5

5

-

x

7

7

+

-

⋯

(

|

x

|

≦

1

)

,
we see that

artanh

⁡

x

=

1

i

⁢

arctan

⁡

i

⁢

x

;

similarly we get

arsinh

⁡

x

=

1

i

⁢

arcsin

⁡

i

⁢

x

.

Some other formulae which may be obtained by means of the

addition formulae

of the

hyperbolic functions

:

arsinh

⁡

x

±

arsinh

⁡

y

=

arsinh

⁡

(

x

⁢

y

2

+

1

±

y

⁢

x

2

+

1

)

arcosh

⁡

x

±

arcosh

⁡

y

=

arcosh

⁡

(

x

⁢

y

±

x

2

-

1

⁢

y

2

-

1

)

artanh

⁡

x

±

artanh

⁡

y

=

artanh

⁡

x

±

y

1

±

x

⁢

y

The classic abbreviations “

arsinh

” and “

arcosh

” are explained as follows:  The unit hyperbola

x

2

-

y

2

=

1

(its right half) has the parametric

{

x

=

cosh

⁡

A

,

y

=

sinh

⁡

A

;

here

A

means the area by the hyperbola and the straight line

segments

O

⁢

P

and

O

⁢

Q

, where

O

is the origin,

P

is the point

(

x

,

y

)

of the hyperbola and

Q

is the point

(

x

,

-

y

)

of the hyperbola.  Thus, conversely,

A

is the area having hyperbolic cosine equal to

x

(

area cosini hyperbolici x

), similarly

A

is the area having hyperbolic sine equal to

y

(

area sini hyperbolici y

).

Note.

In some countries the abbreviation “ar” in the symbols arsinh etc. is replaced by  “a”, “Ar”, “arc” or “arg”.

Title

area functions

Canonical

name

AreaFunctions

Date of creation

2013-03-22 14:21:18

Last modified on

2013-03-22 14:21:18

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

38

Author

pahio (2872)

Entry type

Definition

Classification

msc 26A09

Synonym

inverse hyperbolic functions

Related topic

UnitHyperbola

Related topic

CyclometricFunctions

Related topic

HyperbolicAngle

Related topic

IntegralTables

Related topic

IntegrationOfSqrtx21

Related topic

IntegralRelatedToArcSine

Related topic

ArcLengthOfParabola

Related topic

ListOfImproperIntegrals

Related topic

InverseGudermannianFunction

Related topic

EulersSubstitutionsForIntegration

Related topic

ArcoshCurve

Related topic

EqualArcLength

Defines

arsinh

Defines

arcosh

Defines

artanh

Defines

arcoth

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