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 Arithmetic Hypercomplex components Polynomials, Rationals, Exponentials, and Logarithms Trigonometric functions and inverses Hyperbolic functions and inverses

Functions of a hypercomplex variable

Hyperbolic functions and inverses

Hyperbolic tangent

• Complex hyperbolic tangent (complex map / 3D plot)
• is elementary
• is analytic on its domain
• is meromorphic
• has simple poles at all
• is conformal
• Dual hyperbolic tangent (dual map / 3D plot)
• Split-complex hyperbolic tangent (split-complex map / 3D plot)
• Quaternionic hyperbolic tangent ([ vectors] / [ sliders] / [ 3D rotations])
• Octonionic hyperbolic tangent ([ vectors] / [ sliders])

Hyperbolic cosecant

• Complex hyperbolic cosecant (complex map / 3D plot)
• is elementary
• is analytic on any open set on its domain
• is conformal
• Dual hyperbolic cosecant (dual map / 3D plot)
• Split-complex hyperbolic cosecant (split-complex map / 3D plot)
• Quaternionic hyperbolic cosecant ([ vectors] / [ sliders] / [ 3D rotations])
• Octonionic hyperbolic cosecant ([ vectors] / [ sliders])

Hyperbolic secant

• Complex hyperbolic secant (complex map / 3D plot / iterates)
• is elementary
• is analytic on any open set on its domain
• is conformal
• Dual hyperbolic secant (dual map / 3D plot)
• Split-complex hyperbolic secant (split-complex map / 3D plot)
• Quaternionic hyperbolic secant ([ vectors] / [ sliders] / [ 3D rotations])
• Octonionic hyperbolic secant ([ vectors] / [ sliders])

Hyperbolic cotangent

• Complex hyperbolic cotangent (complex map / 3D plot)
• is elementary
• is analytic on any open set on its domain
• is conformal
• Dual hyperbolic cotangent ([ dual map] / [ 3D plot])
• Split-complex hyperbolic cotangent ([ split-complex map] / [ 3D plot])
• Quaternionic hyperbolic cotangent ([ vectors] / [ sliders] / [ 3D rotations])
• Octonionic hyperbolic cotangent ([ vectors] / [ sliders])

Inverse hyperbolic sine

• Complex inverse hyperbolic sine (complex map of principal branch / 3D plot of principal branch / Riemann surface)
• is elementary
• is complex analytic
• is conformal except at and
• is multivalued
• Dual inverse hyperbolic sine ([ dual map] / [ 3D plot])
• Split-complex inverse hyperbolic sine ([ split-complex map] / [ 3D plot])
• Quaternionic inverse hyperbolic sine ([ vectors] / [ sliders] / [ 3D rotations])
• is extremely multivalued for elements of
• Octonionic inverse hyperbolic sine ([ vectors] / [ sliders])
• is extremely multivalued for elements of

Inverse hyperbolic cosine

• Complex inverse hyperbolic cosine (complex map of principal branch / 3D plot of principal branch / Riemann surface)
• is elementary
• is complex analytic
• is conformal except at and
• is multivalued
• Dual inverse hyperbolic cosine ([ dual map of principal branch] / [ 3D plot of principal branch])
• Split-complex inverse hyperbolic cosine ([ split-complex map of principal branch] / [ 3D plot of principal branch])
• Quaternionic inverse hyperbolic cosine ([ vectors] / [ sliders] / [ 3D rotations])
• is extremely multivalued for elements of
• Octonionic inverse hyperbolic cosine ([ vectors] / [ sliders])
• is extremely multivalued for elements of

Inverse hyperbolic tangent

• Complex inverse hyperbolic tangent (complex map of principal branch / 3D plot of principal branch / Riemann surface)
• is elementary
• is analytic on any open set on its domain
• is conformal
• is multivalued
• Dual inverse hyperbolic tangent ([ dual map] / [ 3D plot])
• Split-complex inverse hyperbolic tangent ([ split-complex map] / [ 3D plot])
• Quaternionic inverse hyperbolic tangent ([ vectors] / [ sliders] / [ 3D rotations])
• is extremely multivalued for elements of
• Octonionic inverse hyperbolic tangent ([ vectors] / [ sliders])
• is extremely multivalued for elements of

Inverse hyperbolic cosecant

• Complex inverse hyperbolic cosecant (complex map of principal branch / 3D plot of principal branch / Riemann surface)
• is elementary
• is analytic on any open set on its domain
• is conformal
• is multivalued
• Dual inverse hyperbolic cosecant ([ dual map] / [ 3D plot])
• Split-complex inverse hyperbolic cosecant ([ split-complex map] / [ 3D plot])
• Quaternionic inverse hyperbolic cosecant ([ vectors] / [ sliders] / [ 3D rotations])
• Octonionic inverse hyperbolic cosecant ([ vectors] / [ sliders])

Inverse hyperbolic secant

• Complex inverse hyperbolic secant (complex map of principal branch / 3D plot of principal branch / Riemann surface)
• is elementary
• is analytic on any open set on its domain
• is conformal
• is multivalued
• Dual inverse hyperbolic secant ([ dual map] / [ 3D plot])
• Split-complex inverse hyperbolic secant ([ split-complex map] / [ 3D plot])
• Quaternionic inverse hyperbolic secant ([ vectors] / [ sliders] / [ 3D rotations])
• Octonionic inverse hyperbolic secant ([ vectors] / [ sliders])

Inverse hyperbolic cotangent

• Complex inverse hyperbolic cotangent (complex map of principal branch / 3D plot of principal branch / Riemann surface)
• is elementary
• is analytic on any open set on its domain
• is conformal
• is multivalued
• Dual inverse hyperbolic cotangent ([ dual map] / [ 3D plot])
• Split-complex inverse hyperbolic cotangent ([ split-complex map] / [ 3D plot])
• Quaternionic inverse hyperbolic cotangent ([ vectors] / [ sliders] / [ 3D rotations])
• Octonionic inverse hyperbolic cotangent ([ vectors] / [ sliders])

Hyperbolic excosecant

• Complex hyperbolic excosecant (complex map / 3D plot)
• is elementary
• is analytic on any open set on its domain
• is conformal
• Dual hyperbolic excosecant (dual map / 3D plot)
• Split-complex hyperbolic excosecant (split-complex map / 3D plot)
• Quaternionic inverse hyperbolic excosecant ([ vectors] / [ sliders] / [ 3D rotations])
• Octonionic inverse hyperbolic excosecant ([ vectors] / [ sliders])

Hyperbolic exsecant

• Complex hyperbolic exsecant (complex map / 3D plot)
• is elementary
• is analytic on any open set on its domain
• is conformal
• Dual hyperbolic exsecant (dual map / 3D plot)
• Split-complex hyperbolic exsecant (split-complex map / 3D plot)
• Quaternionic inverse hyperbolic exsecant ([ vectors] / [ sliders] / [ 3D rotations])
• Octonionic inverse hyperbolic exsecant ([ vectors] / [ sliders])