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

Sinc analogues |
Logistic functions, Gudermannian, and inverses |
Algebraic curves |
Bessel-type functions |
Integer functions |

Factorials and binomial coefficient |
Productlogarithms, hyperoperators, and relatives |
Physical functions |
L^{p} trigonometric functions |
Exponential integral and relatives |

Elliptic integrals and relatives |
Gamma functions and relatives |
Error functions and relatives |
Zeta functions, Polylogarithms, and relatives |
Modular forms and relatives |

## Functions of a hypercomplex variable

### Trigonometric functions and inverses

#### Sine

- Complex sine (complex map / 3D plot / Mandelbrot / iterates)
- is elementary
- is complex analytic
- has simple zeros at all
- has an essential singularity at
- is conformal except at elements of
- is entire
- Complex reciprocal of sine ()
- Complex sine of reciprocal () (complex map / 3D plot)
- Complex natural logarithm of sine () (complex map of principal branch / 3D plot of principal branch / Riemann surface)
- Complex sine of natural logarithm () (complex map of principal branch / 3D plot of principal branch / Riemann surface)
- Complex inverse of sine ()
- Complex derivative of sine ()
- Complex integral of sine () (complex map / 3D plot)
- Complex sine of sine () (complex map / 3D plot)

- Dual sine (dual map / 3D plot)
- Split-complex sine (split-complex map / 3D plot)
- Quaternionic sine (vectors / sliders / 3D rotations)
- Octonionic sine (vectors / sliders)

#### Cosine

- Complex cosine (complex map / 3D plot / Mandelbrot / iterates)
- is elementary
- is complex analytic
- has simple zeros at all
- has an essential singularity at
- is conformal except at elements of
- is entire
- Complex reciprocal of cosine ()
- Complex cosine of reciprocal () (complex map / 3D plot)
- Complex natural logarithm of cosine () (complex map of principal branch / 3D plot of principal branch / Riemann surface)
- Complex cosine of natural logarithm () (complex map of principal branch / 3D plot of principal branch / Riemann surface)
- Complex inverse of cosine ()
- Complex derivative of cosine () (complex map / 3D plot)
- Complex integral of cosine () (complex map / 3D plot)
- Complex cosine of cosine () (complex map / 3D plot)

- Dual cosine (dual map / 3D plot)
- Split-complex cosine (split-complex map / 3D plot)
- Quaternionic cosine (vectors / sliders / 3D rotations)
- Octonionic cosine (vectors / sliders)

#### Tangent

- Complex tangent (complex map / 3D plot / iterates)
- is elementary
- is analytic on its domain
- is meromorphic
- has simple zeros at all
- has simple poles at all
- has an essential singularity at
- is conformal
- Complex reciprocal of tangent ()
- Complex tangent of reciprocal () (complex map / 3D plot)
- Complex inverse of tangent ()
- Complex derivative of tangent () (complex map / 3D plot)
- Complex integral of tangent () (complex map of principal branch / 3D plot of principal branch / Riemann surface)
- Complex tangent of tangent () (complex map / 3D plot)

- Dual tangent (dual map / 3D plot)
- Split-complex tangent (split-complex map / 3D plot)
- Quaternionic tangent (vectors / sliders / 3D rotations)
- Octonionic tangent (vectors / sliders)

#### 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 sine

- Complex inverse sine (complex map of principal branch / complex maps of branches / 3D plot of principal branch / Riemann surface)
- is elementary
- is analytic except at and
- has a simple zero at
- is conformal except at and
- is multivalued
- has branch points at and

- Dual inverse sine ([ dual map of principal branch] / [ 3D plot of principal branch])
- Split-complex inverse sine ([ split-complex map of principal branch] / [ 3D plot of principal branch])
- Quaternionic inverse sine (vectors of principal value / sliders of principal value / 3D rotations of principal value)
- is extremely multivalued for elements of

- Octonionic inverse sine (vectors of principal value / sliders of principal value)
- is extremely multivalued for elements of

#### Inverse cosine

- Complex inverse cosine (complex map of principal branch / complex maps of branches / 3D plot of principal branch / Riemann surface)
- is elementary
- is analytic except at and
- has a zero of order at
- is conformal except at and
- is multivalued
- has branch points at and

- Dual inverse cosine ([ dual map of principal branch] / [ 3D plot of principal branch])
- Split-complex inverse cosine ([ split-complex map of principal branch] / [ 3D plot of principal branch])
- Quaternionic inverse cosine (vectors of principal value / sliders of principal value / 3D rotations of principal value)
- is extremely multivalued for elements of

- Octonionic inverse cosine (vectors of principal value / sliders of principal value)
- is extremely multivalued for elements of

#### Inverse tangent

- Complex inverse tangent (complex map of principal branch / complex maps of branches / 3D plot of principal branch / Riemann surface)
- is elementary
- is analytic on any open set on its domain
- has a simple zero at
- is conformal
- is multivalued
- has branch points at and

- Complex two-argument inverse tangent (complex map of principal branch / 3D plot of principal branch / Riemann surface)
- Dual inverse tangent ([ dual map of principal branch] / [ 3D plot of principal branch])
- Dual two-argument inverse tangent ([ dual map of principal branch] / [ 3D plot of principal branch])
- Split-complex inverse tangent ([ split-complex map of principal branch] / [ 3D plot of principal branch])
- Split-complex two-argument inverse tangent ([ split-complex map of principal branch] / [ 3D plot of principal branch])
- Quaternionic inverse tangent (vectors of principal value / sliders of principal value / 3D rotations of principal value)
- Quaternionic two-argument inverse tangent ([ vectors] / [ sliders] / [ 3D rotations])
- Octonionic inverse tangent (vectors of principal value / sliders of principal value)
- Octonionic two-argument inverse tangent ([ vectors] / [ sliders])

#### 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])

#### Inverse cis

- Complex inverse cis (complex map of principal branch / 3D plot of principal branch / Riemann surface)
- is elementary
- is complex analytic
- is conformal
- is multivalued

#### Inverse cocis

- Complex inverse cocis (complex map of principal branch / 3D plot of principal branch / Riemann surface)
- is elementary
- is complex analytic
- is conformal
- is multivalued

#### 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])