Areasinus hyperbolicus and Areakosinus hyperbolicus

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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 .

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 The representations or are considered out of date and the representations that are rarely found and are to be avoided, as they conflict with the meaning or .

Definitions

The functions can be expressed by the following formulas:

Areasinus hyperbolicus:

With

Areakosine hyperbolicus:

For

Here stands for the natural logarithm .

conversion

Together with the signum function , the following applies:

The following applies to:

properties

Graph of function arsinh (x)
Graph of function arcosh (x)
  Hyperbolic area Hyperbolic areakosine
Domain of definition
Range of values
periodicity no no
monotony strictly monotonously increasing strictly monotonously increasing
Symmetries Point symmetry to the origin,
odd function
no
asymptote For For
zeropoint
Jump points no no
Poles no no
Extremes no no
Turning points 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:

Derivatives

The derivation of the hyperbolic area is:

The derivation of the hyperbolic areakosine is:

for x> 1

Antiderivatives

The antiderivatives of the hyperbolic area and the hyperbolic area are:

Other identities


Numerical calculation

Basically, the hyperbolic area sine can be calculated using the well-known formula

can be calculated if the natural logarithm function is available. However, there are the following problems:

  • 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:

for applied.

The following cases can then be distinguished for:

Case 1: is a large, positive number with :

where the number of significant decimal digits is the number type used, which is double 16 for the 64-bit floating point type .
This formula results from the following consideration:
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 :

Case 2: is close to 0, e.g. B. for :

Using the Taylor series:

Case 3: All others :

The hyperbolic areacosine can be calculated using the well-known formula

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 :

where is the number of significant decimal digits of the number type used.

Case 2 ::

The result is not defined.

Case 3: All others , i.e. for :

See also

Web links

Individual evidence

  1. ^ Franz Brzoska, Walter Bartsch: Mathematical formula collection . 2nd improved edition. Fachbuchverlag Leipzig, 1956.