The inverse hyperbolic functions, sometimes also called the area hyperbolic functions (Spanier and Oldham 1987, p. 263) are the multivalued function that are the inverse functions of the hyperbolic functions. They are denoted , , , , , and . Variants of these notations beginning with a capital letter are commonly used to denote their principal values (e.g., Harris and Stocker 1998, p. 263).
These functions are multivalued, and hence require branch cuts in the complex plane. Differing branch cut conventions are possible, but those adopted in this work follow those used by the Wolfram Language, summarized below.
function name | function | the Wolfram Language | branch cut(s) |
inverse hyperbolic cosecant | ArcCsch[z] | ||
inverse hyperbolic cosine | ArcCosh[z] | ||
inverse hyperbolic cotangent | ArcCoth[z] | ||
inverse hyperbolic secant | ArcSech[z] | and | |
inverse hyperbolic sine | ArcSinh[z] | and | |
inverse hyperbolic tangent | ArcTanh[z] | and |
The inverse hyperbolic functions as defined in this work have the following ranges for domains on the real line , again following the convention of the Wolfram Language.
function name | function | domain | range |
inverse hyperbolic cosecant | |||
inverse hyperbolic cosine | |||
inverse hyperbolic cotangent | or | ||
inverse hyperbolic secant | |||
inverse hyperbolic sine | |||
inverse hyperbolic tangent |
They are defined in the complex plane by
(1)
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(2)
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(3)
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(4)
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(5)
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(6)
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