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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

### Trigonometric functions and inverses

#### Cosecant

• Complex cosecant (complex map / 3D plot)
• is elementary
• is analytic on any open set on its domain
• has simple poles at all
• has an essential singularity at
• is conformal
• Dual cosecant (dual map / 3D plot)
• Split-complex cosecant (split-complex map / 3D plot)
• Quaternionic cosecant ([ vectors] / [ sliders] / [ 3D rotations])
• Octonionic cosecant ([ vectors] / [ sliders])

#### Secant

• Complex secant (complex map / 3D plot / iterates)
• is elementary
• is analytic on any open set on its domain
• has simple poles at all
• has an essential singularity at
• is conformal
• Dual secant (dual map / 3D plot)
• Split-complex secant (split-complex map / 3D plot)
• Quaternionic secant ([ vectors] / [ sliders] / [ 3D rotations])
• Octonionic secant ([ vectors] / [ sliders])

#### Cotangent

• Complex cotangent (complex map / 3D plot)
• is elementary
• is analytic on any open set on its domain
• has simple poles at all
• has an essential singularity at
• is conformal
• Dual cotangent (dual map / 3D plot)
• Split-complex cotangent (split-complex map / 3D plot)
• Quaternionic cotangent ([ vectors] / [ sliders] / [ 3D rotations])
• Octonionic cotangent ([ vectors] / [ sliders])

#### Cis

• Complex cis (complex map / 3D plot)
• is elementary
• is complex analytic
• is conformal
• is entire

#### Cocis

• Complex cocis (complex map / 3D plot)
• is elementary
• is complex analytic
• is conformal
• is entire

#### Inverse cosecant

• Complex inverse cosecant (complex map of principal branch / 3D plot of principal branch / Riemann surface)
• is elementary
• is analytic everywhere in its domain except at and
• is conformal except at and
• is multivalued
• has branch points at and
• Dual inverse cosecant ([ dual map of principal branch] / [ 3D plot of principal branch])
• Split-complex inverse cosecant ([ split-complex map of principal branch] / [ 3D plot of principal branch])
• Quaternionic inverse cosecant ([ vectors] / [ sliders] / [ 3D rotations])
• Octonionic inverse cosecant ([ vectors] / [ sliders])

#### Inverse secant

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

#### Inverse cotangent

• Complex inverse 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 cotangent ([ dual map of principal branch] / [ 3D plot of principal branch])
• Split-complex inverse cotangent ([ split-complex map of principal branch] / [ 3D plot of principal branch])
• Quaternionic inverse cotangent ([ vectors] / [ sliders] / [ 3D rotations])
• Octonionic inverse cotangent ([ vectors] / [ sliders])

#### Versine

• Complex versine (complex map / 3D plot)
• is elementary
• is complex analytic
• is conformal except at elements of
• is entire
• Dual versine (dual map / 3D plot)
• Split-complex versine (split-complex map / 3D plot)
• Quaternionic versine ([ vectors] / [ sliders] / [ 3D rotations])
• Octonionic versine ([ vectors] / [ sliders])

#### Coversine

• Complex coversine (complex map / 3D plot)
• is elementary
• is complex analytic
• is conformal except at elements of
• is entire
• Dual coversine (dual map / 3D plot)
• Split-complex coversine (split-complex map / 3D plot)
• Quaternionic coversine ([ vectors] / [ sliders] / [ 3D rotations])
• Octonionic coversine ([ vectors] / [ sliders])

#### Haversine

• Complex haversine (complex map / 3D plot)
• is elementary
• is complex analytic
• is conformal except at elements of
• is entire
• Dual haversine (dual map / 3D plot)
• Split-complex haversine (split-complex map / 3D plot)
• Quaternionic haversine ([ vectors] / [ sliders] / [ 3D rotations])
• Octonionic haversine ([ vectors] / [ sliders])

#### Excosecant

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

#### Exsecant

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

#### Chord

• Complex chord ([ complex map] / [ 3D plot])
• is elementary
• is complex analytic
• is conformal
• Dual chord ([ dual map] / [ 3D plot])
• Split-complex chord ([ split-complex map] / [ 3D plot])
• Quaternionic chord ([ vectors] / [ sliders] / [ 3D rotations])
• Octonionic chord ([ vectors] / [ sliders])

#### Inverse versine

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

#### Inverse coversine

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

#### Inverse haversine

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

#### Inverse excosecant

• Complex inverse excosecant (complex map of principal branch / 3D plot of principal branch / Riemann surface)
• is elementary
• is analytic everywhere in its domain except at and
• is conformal except at and
• is multivalued
• has branch points at and
• Dual inverse excosecant ([ dual map of principal branch] / [ 3D plot of principal branch])
• Split-complex inverse excosecant ([ split-complex map of principal branch] / [ 3D plot of principal branch])
• Quaternionic inverse excosecant ([ vectors] / [ sliders] / [ 3D rotations])
• Octonionic inverse excosecant ([ vectors] / [ sliders])

#### Inverse exsecant

• Complex inverse exsecant (complex map of principal branch / 3D plot of principal branch / Riemann surface)
• is elementary
• is analytic everywhere in its domain except at and
• is conformal except at and
• is multivalued
• has branch points at and
• Dual inverse exsecant ([ dual map of principal branch] / [ 3D plot of principal branch])
• Split-complex inverse exsecant ([ split-complex map of principal branch] / [ 3D plot of principal branch])
• Quaternionic inverse exsecant ([ vectors] / [ sliders] / [ 3D rotations])
• Octonionic inverse exsecant ([ vectors] / [ sliders])

#### Inverse chord

• Complex inverse chord ([ complex map of principal branch] / [ 3D plot of principal branch] / [ Riemann surface])
• is elementary
• is analytic everywhere in its domain except at and
• is conformal except at and
• is multivalued
• has branch points at and
• Dual inverse chord ([ dual map of principal branch] / [ 3D plot of principal branch])
• Split-complex inverse chord ([ split-complex map of principal branch] / [ 3D plot of principal branch])
• Quaternionic inverse chord ([ vectors] / [ sliders] / [ 3D rotations])
• Octonionic inverse chord ([ vectors] / [ sliders])

#### Schläflian function

• Complex polygonal Schläflian function (complex map / 3D plot)
• is elementary
• is analytic everywhere in its domain
• is conformal except at elements of
• is multivalued
• Dual polygonal Schläflian function ([ dual map] / [ 3D plot])
• Split-complex polygonal Schläflian function ([ split-complex map] / [ 3D plot])
• Quaternionic polygonal Schläflian function ([ vectors] / [ sliders] / [ 3D rotations])
• Octonionic polygonal Schläflian function ([ vectors] / [ sliders])