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§4.37 Inverse Hyperbolic Functions ‣ Hyperbolic Functions ‣ Chapter 4 Elementary Functions

§4.37 Inverse Hyperbolic Functions Contents
  1. §4.37(i) General Definitions
  2. §4.37(ii) Principal Values
  3. §4.37(iii) Reflection Formulas
  4. §4.37(iv) Logarithmic Forms
  5. §4.37(v) Fundamental Property
  6. §4.37(vi) Interrelations
§4.37(i) General Definitions

The general values of the inverse hyperbolic functions are defined by

4.37.1 Arcsinh ⁡ z = ∫ 0 z d t ( 1 + t2 ) 1 / 2 , 4.37.2 Arccosh ⁡ z = ∫ 1 z d t ( t2 − 1 ) 1 / 2 , 4.37.3 Arctanh ⁡ z = ∫ 0 z d t 1 − t2 , z ≠ ± 1 , 4.37.4 Arccsch ⁡ z = Arcsinh ⁡ ( 1 / z ) , 4.37.5 Arcsech ⁡ z = Arccosh ⁡ ( 1 / z ) , 4.37.6 Arccoth ⁡ z = Arctanh ⁡ ( 1 / z ) .

In (4.37.1) the integration path may not pass through either of the points t = ± i , and the function ( 1 + t2 ) 1 / 2 assumes its principal value when t is real. In (4.37.2) the integration path may not pass through either of the points ± 1 , and the function ( t2 − 1 ) 1 / 2 assumes its principal value when t ∈ ( 1 , ∞ ) . Elsewhere on the integration paths in (4.37.1) and (4.37.2) the branches are determined by continuity. In (4.37.3) the integration path may not intersect ± 1 . Each of the six functions is a multivalued function of z . Arcsinh ⁡ z and Arccsch ⁡ z have branch points at z = ± i ; the other four functions have branch points at z = ± 1 .

§4.37(ii) Principal Values

The principal values (or principal branches) of the inverse sinh , cosh , and tanh are obtained by introducing cuts in the z -plane as indicated in Figure 4.37.1(i)-(iii), and requiring the integration paths in (4.37.1)–(4.37.3) not to cross these cuts. Compare the principal value of the logarithm (§4.2(i)). The principal branches are denoted by arcsinh , arccosh , arctanh respectively. Each is two-valued on the corresponding cut(s), and each is real on the part of the real axis that remains after deleting the intersections with the corresponding cuts.

The principal values of the inverse hyperbolic cosecant, hyperbolic secant, and hyperbolic tangent are given by

4.37.7 arccsch ⁡ z = arcsinh ⁡ ( 1 / z ) , 4.37.8 arcsech ⁡ z = arccosh ⁡ ( 1 / z ) . 4.37.9 arccoth ⁡ z = arctanh ⁡ ( 1 / z ) , z ≠ ± 1 .

These functions are analytic in the cut plane depicted in Figure 4.37.1(iv), (v), (vi), respectively.

Except where indicated otherwise, it is assumed throughout the DLMF that the inverse hyperbolic functions assume their principal values.

(i) arcsinh ⁡ z (ii) arccosh ⁡ z (iii) arctanh ⁡ z (iv) arccsch ⁡ z (v) arcsech ⁡ z (vi) arccoth ⁡ z Figure 4.37.1: z -plane. Branch cuts for the inverse hyperbolic functions. Magnify

Graphs of the principal values for real arguments are given in §4.29. This section also indicates conformal mappings, and surface plots for complex arguments.

§4.37(iii) Reflection Formulas 4.37.10 arcsinh ⁡ ( − z ) = − arcsinh ⁡ z . 4.37.11 arccosh ⁡ ( − z ) = ± π ⁢ i + arccosh ⁡ z , ℑ ⁡ z ≷ 0 . 4.37.12 arctanh ⁡ ( − z ) = − arctanh ⁡ z , z ≠ ± 1 . 4.37.13 arccsch ⁡ ( − z ) = − arccsch ⁡ z . 4.37.14 arcsech ⁡ ( − z ) = ∓ π ⁢ i + arcsech ⁡ z , ℑ ⁡ z ≷ 0 . 4.37.15 arccoth ⁡ ( − z ) = − arccoth ⁡ z , z ≠ ± 1 . §4.37(iv) Logarithmic Forms

Throughout this subsection all quantities assume their principal values. For more concrete principal branch expressions for inverse hyperbolic functions see Dempsey (2025).

Inverse Hyperbolic Sine 4.37.16 arcsinh ⁡ z = ln ⁡ ( ( z2 + 1 ) 1 / 2 + z ) , z / i ∈ ℂ ∖ ( − ∞ , − 1 ) ∪ ( 1 , ∞ ) ;

compare Figure 4.37.1(i). On the cuts

4.37.17 arcsinh ⁡ ( i ⁢ y ) = 1 2 ⁢ π ⁢ i ± ln ⁡ ( ( y2 − 1 ) 1 / 2 + y ) , y ∈ [ 1 , ∞ ) , 4.37.18 arcsinh ⁡ ( i ⁢ y ) = − 1 2 ⁢ π ⁢ i ± ln ⁡ ( ( y2 − 1 ) 1 / 2 − y ) , y ∈ ( − ∞ , − 1 ] ,

the upper/lower signs corresponding to the right/left sides.

Inverse Hyperbolic Cosine 4.37.19 arccosh ⁡ z = ln ⁡ ( ± ( z2 − 1 ) 1 / 2 + z ) , z ∈ ℂ ∖ ( − ∞ , 1 ) ,

the upper or lower sign being taken according as ℜ ⁡ z ≷ 0 ; compare Figure 4.37.1(ii). Also,

It should be noted that the imaginary axis is not a cut; the function defined by (4.37.19) and (4.37.20) is analytic everywhere except on ( − ∞ , 1 ] . Compare Figure 4.37.1(ii).

An equivalent definition is

4.37.21 arccosh ⁡ z = 2 ⁢ ln ⁡ ( ( z + 1 2 ) 1 / 2 + ( z − 1 2 ) 1 / 2 ) , z ∈ ℂ ∖ ( − ∞ , 1 ) ;

see Kahan (1987).

On the part of the cuts from − 1 to 1

4.37.22 arccosh ⁡ x = ± ln ⁡ ( i ⁢ ( 1 − x2 ) 1 / 2 + x ) , x ∈ ( − 1 , 1 ] ,

the upper/lower sign corresponding to the upper/lower side.

On the part of the cut from − ∞ to − 1

4.37.23 arccosh ⁡ x = ± π ⁢ i + ln ⁡ ( ( x2 − 1 ) 1 / 2 − x ) , x ∈ ( − ∞ , − 1 ] ,

the upper/lower sign corresponding to the upper/lower side.

Inverse Hyperbolic Tangent 4.37.24 arctanh ⁡ z = 1 2 ⁢ ln ⁡ ( 1 + z 1 − z ) , z ∈ ℂ ∖ ( − ∞ , − 1 ] ∪ [ 1 , ∞ ) ;

compare Figure 4.37.1(iii). On the cuts

4.37.25 arctanh ⁡ x = ± 1 2 ⁢ π ⁢ i + 1 2 ⁢ ln ⁡ ( x + 1 x − 1 ) , x ∈ ( − ∞ , − 1 ) ∪ ( 1 , ∞ ) ,

the upper/lower sign corresponding to the upper/lower sides.

Other Inverse Functions

For the corresponding results for arccsch ⁡ z , arcsech ⁡ z , and arccoth ⁡ z , use (4.37.7)–(4.37.9); compare §4.23(iv).

§4.37(v) Fundamental Property

With k ∈ ℤ , the general solutions of the equations

are respectively given by

4.37.29 w = Arcsinh ⁡ z = ( − 1 ) k ⁢ arcsinh ⁡ z + k ⁢ π ⁢ i , 4.37.30 w = Arccosh ⁡ z = ± arccosh ⁡ z + 2 ⁢ k ⁢ π ⁢ i , 4.37.31 w = Arctanh ⁡ z = arctanh ⁡ z + k ⁢ π ⁢ i , z ≠ ± 1 . §4.37(vi) Interrelations

Table 4.30.1 can also be used to find interrelations between inverse hyperbolic functions. For example, arcsech ⁡ a = arccoth ⁡ ( ( 1 − a2 ) − 1 / 2 ) .

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