The Art of Mathematics

# arsech or arsch — arc-hyperbolic secant function

## 1. Definition

Arc-hyperbolic secant is inverse of hyperbolic secant function.

arsechxsechinvx

With the help of natural logarithm it can be represented as:

arsechx ≡ ln{[1 + √(1 − x2)] /x}

## 2. Graph

Arc-hyperbolic secant is monotone function defined in the range (0, 1], point x = 0 is singular one. Its graph is depicted below — fig. 1. Fig. 1. Graph of the arc-hyperbolic secant function y = arsechx.

Function codomain is non-negative part of real axis: [0, +∞).

## 3. Identities

Reciprocal argument:

arsech(1/x) = arcoshx

Sum and difference:

arsechx + arsechy = arsech(xy /{1 + √[(1 − x2)(1 − y2)]})
arsechx − arsechy = arsech(xy /{1 − √[(1 − x2)(1 − y2)]})

## 4. Derivative and indefinite integral

Arc-hyperbolic secant derivative:

arsech′x = −1 /[x √(1 − x2)]

Indefinite integral of the arc-hyperbolic secant:

∫ arsechx dx = x arsechx + arcsinx + C

where C is an arbitrary constant.

## 5. How to use

To calculate arc-hyperbolic secant of the number:

``arsech(1);``

To get arc-hyperbolic secant of the complex number:

``arsech(1+i);``

To get arc-hyperbolic secant of the current result:

``arsech(rslt);``

To get arc-hyperbolic secant of the number z in calculator memory:

``arsech(mem[z]);``

## 6. Support

Arc-hyperbolic secant of the real argument is supported in free version of the Librow calculator.

Arc-hyperbolic secant of the complex argument is supported in professional version of the Librow calculator.